Equivalence

INTRODUCTION

INTRODUCTION

First of all, before I even begin, please take a few moments to watch a hilarious video on "camera competition":

Hitler rants about D3x

and a quick lesson in phootography:

You stole my freakin' cameras

My apologies if linking to a Hitler parody offends anyone, or if the language is a bit rough in the second link, but sometimes funny is just funny.  :  )

It is my hope that this essay is useful and informative in explaining the differences between formats (sensor sizes), and what role both the sensor and glass play in terms of image quality (the section on IQ is a "must read" to keep the relevance of these differences in perspective, if you'll pardon the pun).  This is a very technical essay that explains technical aspects of photography, most notably noise, exposure, and DOF.  The target audience is for those that want to understand the physics of photography and how this applies to the engineering of modern digital cameras, and, more specifically, to how this relates to the different formats.  This essay is not targeted to people who want to know how to use their cameras to create "good" photographs.  As long as this essay is, that essay would be quite a bit longer.

Moving right along, let's cut to the chase and copy from the section on The Definition of Equivalence:

On the quick, larger sensor systems enjoy an IQ advantage over smaller sensor systems for images that are not fully equivalent to images from smaller sensor systems.  That is, when the larger sensor system is able to use a lower shutter speed or a more shallow DOF is preferable.  For fully equivalent images, the differences between systems is at a minimum.  This is not the whole story, of course, as the sensor efficiency, pixel count, and lens quality most certainly play a role.  Even more to the point is that the images must be displayed large enough, and/or processed heavily enough to where the IQ differences become noticeable (which depends heavily on the viewers QT -- "quality threshold").  The reason for the length of this essay is to cover all the angles in detail, which is why "a sentence or two" is not sufficient.

The concept of Equivalence is difficult for many to accept because it replaces the paradigm of Exposure, and its agent, f-ratio, with a new paradigm of Total Light, and its agent, aperture.  For many, this paradigm shift is heresy, just as when Copernicus declared that the Sun, and not the Earth, was at the center of our planetary system.  While the Copernican model appealed to many because of its simplicity, it was not until Kepler showed that the orbits were elliptical, not circular, and Newton demonstrated that the agent of this new geometry was a Universal Law of Gravitation, that the new heliocentric model made sense.  Similarly, when people understand that it is the light itself, and not the sensor, that is the primary source of noise in an image, the practical application of Equivalence becomes evident.

To that end, making the distinction between the following three terms, "exposure", "apparent exposure", and "total light", is critical.  The exposure represents the density of the light that makes up an image.  It is analogous to the depth of the water in a pool.  The apparent exposure is how bright or dark an image appears.  That is, an image that is "too dark" is said to be underexposed and an image that is "too bright" is said to be overexposed.  The image brightness can be adjusted in camera by adjusting the ISO to achieve the correct "apparent exposure".  Lastly, the total light is the total amount of light that was absorbed by the sensor to make up the image.  It corresponds to the total amount of water in a pool, rather than the depth of the water.  The reason that the total light that forms an image is so important is because it is the single most important factor in both the apparent noise and dynamic range of the image.

Central to understanding these concepts is differentiating between the terms "aperture" and "f-ratio".  While "aperture" is more accurately used to mean "entrance pupil diameter", it is often confusingly used to mean "f-ratio", most likely because "f-ratio", "aperture-ratio", and "relative aperture" are, in fact, synonyms.  The aperture diameter (entrance pupil diameter) is the diameter of the opening in a lens when viewed through the front element and is not, as some mistakenly believe, the diameter of the front element of the lens or the diameter of the physical aperture.  The f-ratio is the quotient of the focal length and the aperture diameter.  Conversely, we can compute the aperture diameter by taking the quotient of the focal length and the f-ratio.  For example, the f-ratio at 50mm with an aperture diameter of 25mm is 50mm / 25mm = f/2, and the aperture diameter at 50mm f/2 is 50mm / 2 = 25mm.  The concepts of, and connections between, total light, DOF, and noise, are much more easily understood in terms of aperture than they are in terms of f-ratio, especially when comparing different formats.

The rest of this essay discusses the Purpose of Equivalence, as well as the relevance/importance of the five parameters in the definition.  In addition, there are sections devoted to other matters of IQ in terms of equipment, such as the pixel count, lens quality, and sensor efficiency, none of which are principles of Equivalence, yet have a substantial impact on IQ.  However, what this essay does not discuss in detail are the operational differences between systems, such as focus accuracy/speed, size, weight, cost, build, etc., which often play a far more important role than the differences in IQ in determining which system is better for a particular photographer.

Nonetheless, whatever the differences in IQ may be, one needs to assess if the differences in the performance of the equipment make any difference in the context of one's skills as a photographer.  And, if so, if the IQ differences are substantial enough to overshadow the operational differences between systems.

Q&A

This section is a "quick and dirty" session where many misconceptions about the differences between systems with different formats are addressed.  All these points are addressed in more detail in other sections of the essay.

Before beginning the Q&A session, it's important to keep the capabilities of the equipment in context with the skills of the photographer, the types of photos they take, and the size at which they're displayed.  That is, even given that the equipment is capable does not give us any sense that the photographer can make full use of that potential, or that the differences will be meaningful for how the equipment is used.  Of course, just how much of a difference constitutes "meaningful" depends on the photographer or, perhaps more accurately, the target audience (which, in many cases, is the photographer!).  So, while one camera system may have higher IQ than another, this does not necessarily mean it's the best system for the job at hand for a particular photographer, nor does it necessarily mean that the differences in IQ between systems makes any meaningful difference for the end product to the target audience.  In the end, we must consider the system as a whole, both in terms of IQ and operation, and in conjunction with our needs, skills, and audience.  That all said, let's begin the Q&A:

Q:  What does "Equivalence" mean? 
A:  Equivalence simply refers to images that have the same perspective, framing, DOF, shutter speed, and display dimensions.

Q:  So what's the difference between "Equivalent" and "equal"? 
A:  While the five parameters listed above are obvious visual qualities to an image, they're not the whole story.  There's also detail, noise, bokeh, color, etc.

Q:  Don't Equivalent images have the same noise?
A:  For sensors of the same generation, it will usually be close, but it will never be exactly the same, since the sensors for different systems are never simply scaled versions of each other.  For example, some systems, like medium format digital, are using very old sensor tech, and fare rather poorly in terms of noise equivalence, since they are very inefficient with high read noise, whereas the efficiency of some compacts, like the Canon G11, have only recently been matched by Nikon's latest and greatest D3s.  See here for more discussion on this point.

Q:  So what makes the five parameters of Equivalence so special? 
A:  The parameters of equivalence are not affected by the technology of the equipment, so we can normalize these important visual qualities and provide a basis for an "apples to apples" comparison.

Q:  Isn't the real purpose of Equivalence to stack the deck against smaller sensor systems and promote FF?
A:  In fact, the exact opposite is true.  Larger sensor systems are at their worst (in terms of IQ) when comparing fully equivalent images.  The IQ advantages of FF come into play when they can use base ISO and still maintain the desired DOF and a shutter speed sufficiently high enough to avoid camera shake or motion blur (unless, of course, motion blur is desirable), or when a more shallow DOF is desirable.

Q:  What are the advantages of smaller sensor systems?
A:  Usually less size, weight, and cost for telephoto at deeper DOFs due to a greater pixel density and lenses with smaller maximum aperture diameters (not to be confused with f-ratio) than larger sensor systems.  Also, in some cases, depending on the lens (usually UWA), the extreme corners may be sharper with smaller formats for the same framing and DOF.  Furthermore, some features, such as in-camera IS, serve to sometimes give smaller sensor systems an IQ advantage, when the larger sensor system does not have in-camera IS or an IS lens at the desired AOV.  Another plus of smaller sensor systems is that they can often frame more tightly for a given AOV.  In addition, smaller sensor systems currently offer a wider spread of focus points in some cameras.  Lastly, compacts (including the Sigma DP series cameras and mFT) offer an even greater size advantage still since they lack a mirror box, although, currently, this may come at the expense of AF speed and/or accuracy.

Q:  What are the advantages of a larger sensor system? 
A:  Under optimal conditions, larger sensor systems usually offer superior IQ and give the option for a more shallow DOF.  Operationally, larger sensor systems have larger and brighter viewfinders.

Q:  Why do larger sensor systems usually offer more IQ? 
A:  In many cases, lenses perform better on larger sensor systems (see here for details) and, for sensors of the same generation, larger sensors usually have more pixels.  In addition, the available lenses for larger sensor systems often have larger aperture diameters (not to be confused with f-ratio) for a given AOV.  This gives larger sensor systems more DOF control (on the shallow side) as well as allowing more light to reach the sensor in low light situations, which results in less apparent noise.  Larger sensor systems are at their best compared to smaller sensor systems when they can remain at base ISO, stop the lens down to the desired DOF / sharpness, and not suffer ill effects due to motion blur and/or camera shake.  But for fully equivalent images, the IQ differential is at its least.  So, for those that would always (or usually) be taking photos at settings equivalent to smaller sensor systems, a larger sensor system may not be the best choice.

Q:  But don't larger sensor systems have softer corners and more vignetting than smaller sensor systems?
A:  It depends on the lens, but it's usually pretty close for the same framing and DOF.  Sometimes, the extreme corners for larger sensors may be softer, and exhibit more vignetting, even at the same DOF, (especially true with cheap UWAs), but the image will usually be sharper and more detailed elsewhere in the frame.

Q:  How can a softer lens on a larger sensor resolve more than a sharper lens on a smaller sensor?
A:  This point is directly addressed here.  On the quick, the following example explains the logic:  Consider the Zuiko 150 / 2 on 4/3 and the Canon 300 / 4L IS on 135, which are equivalent lenses on their respective formats -- that is, both have the same AOV and maximum aperture diameter.  The 150 / 2 tested at 49 lp/mm wide open, whereas the 300 / 4L IS tested at 36 lp/mm wide open.  Since the 4/3 sensor is 13mm tall, and the 135 sensor is 24mm tall, these figures translate to 49 lp/mm · 13mm/ih = 637 lp/ih for the 150/2 and 36 lp/mm · 24mm/ih = 864 lp/ih for the 300 / 4L IS.  In other words, even though the 150 / 2 is the sharper lens, the 300 / 4L IS out resolves it on the larger sensor.

Q:  But f/2 results in the same exposure regardless of the format, right?
A:  Yes, and this is discussed in more detail here.  While any given f-ratio results in the same exposure for a given scene and framing regardless of the focal length or the format, it is the total light that makes up an image, not the exposure, that is the primary determinant in image apparent noise and dynamic range.

Q:  Isn't the reason that larger sensor systems have less noise and more dynamic range because they have larger pixels?
A:  No.  The pixel size is irrelevant in terms of noise and DR (discussed in more detail here).  What is relevant, in terms of the apparent noise of an image is the total amount of light that makes up an image.  Larger sensor systems will enjoy a noise / DR advantage over smaller sensor systems when the available lenses have a larger aperture diameter for the larger sensor system (which will also result in a more shallow DOF), or when they can use a lower shutter speed to gather more light (base ISO in good light or using a tripod).

Q:  So won't larger sensor systems lose their noise advantage when they up the ISO to match the DOF and shutter speed of smaller sensor systems?
A:  In the cases when both the DOF and shutter speed need to be the same, larger sensor systems will lose the noise advantage they have over smaller sensor systems.  However, there are many common shooting situations where the larger sensor system can simply use a slower shutter speed.  For example, shooting in good light, using a tripod (or, to a lesser extent, monopod) when motion blur is not a factor, or when using flash (when the balance of ambient light and flash is not a factor).  In these circumstances, larger sensor systems suffer no IQ penalty in matching the DOF of smaller sensor systems.

Q:  Doesn't a higher ISO result in more noise?
A:  This is a common misconception.  Using a higher ISO results in either a faster shutter speed and/or a smaller aperture for a given apparent exposure.  The effect of either of these is to put less light on the sensor.  It is the lesser amount of light falling on the sensor that results in more apparent noise, not the higher ISO.  In fact, the higher ISO results in slightly less noise.  That is, if we took a pic of a scene at ISO 1600, and then took a pic of the same scene with the same f-ratio and shutter speed at ISO 100, and pushed the ISO 100 pic 4 stops in post to achieve the same apparent exposure, it would be more noisy than the ISO 1600 pic (discussed in more detail here).  So, the cause of the greater apparent noise is the lesser amount of light falling on the sensor, not the higher ISO.

Q:  Don't smaller sensors have more DOF than larger sensors?
A:  We have to be careful about spelling out the conditions of the DOF comparison.  Best to see the box here to cover the question for various scenarios.  However, on the quick, limiting the conditions of the comparison to the same perspective and framing, then smaller sensors will have a deeper DOF for the same f-ratio.  But, there's no reason that the larger system is compelled to use the same f-ratio -- it can simply stop down to maintain the same DOF.  If stopping down requires a concomitant increase in ISO to maintain the shutter speed (fully equivalent images), then the larger sensor system will have to give up its noise advantage to match the DOF of the smaller sensor system.  At the extreme deep end of the DOF spectrum, however, larger formats may have to employ inconvenient measures to match the DOF of smaller sensor systems.  For example, some lenses for 135 max out at f/16, which results in a larger aperture diameter than their equivalents for smaller sensor systems.  However, beyond this point we are well into the realm of diffraction limited photography.  But if the need for more DOF outweighs the negative impact of diffraction softening, the larger sensor system can use a shorter focal length and TC to compensate, although this will represent a definite operational inconvenience.  For example, rather than using 100mm f/16, one could use 50mm f/16 + 2x TC, which is equivalent to 100mm f/32.  The image degradation from the TC is negligible in comparison to the image degradation due to diffraction.  For people who shoot in "Auto" and "P" modes, however, they may find that larger sensor systems choose a more shallow DOF than smaller sensor systems.

Q:  Won't larger sensors suffer diffraction softening earlier than smaller sensors when stopping down for the same DOF?
A:  It depends on how you define "suffer".  So long as the larger sensor system has at least the same number of pixels as the smaller sensor system, it will resolve at least as much detail for the same perspective, framing, and DOF as the smaller sensor system.  However, for the system that has greater pixel count, regardless of sensor size, diffraction softening will begin to lessen the detail advantage afforded by the greater number of pixels at smaller DOFs.

Q:  Don't smaller sensor systems have more reach?
A:  Usually, but not always.  Effective reach is simply how many pixels the sensor has on the subject for a given perspective and focal length.  Since smaller sensor systems often have a much greater pixel density than larger sensor systems, they usually have a greater effective reach, but not in every instance.  For example, since the Canon 1DsIII (FF) and 20D (1.6x) both have the same size pixels, and thus the same pixel density, if we were to use the same focal length on both cameras, shoot the same scene from the same position, and crop the 1DsIII image to the same FOV as the 20D image, it would have the same number of pixels as the 20D image.  Thus, the effective reach is the same for both cameras.

Q:  Bottom line:  when do larger sensor systems deliver higher IQ?
A:  Generalizations are dangerous to make.  That said, larger sensor systems often have "higher overall IQ" when they can maintain the desired perspective, framing, DOF, and shutter speed (note:  "desired" does not necessarily mean "same") at the same ISO as the smaller sensor system.  How "often" and by how much this generalization is true depends on the relative difference between the size of the sensors and presumes, of course, that we are comparing sensors with the same, or nearly the same, efficiency, where the larger sensor system has at least as many pixels on the scene as the smaller sensor system.  Furthermore, it also presumes that we are comparing lenses of similar caliber on their respective systems.  For example, the Canon 24-105 / 4L IS on a 5D may well be beat by the Olympus 14-35 / 2 on an E30, but the Olympus lens costs nearly twice as much and has just over half the zoom range.  However, the Tamron 28-75 / 2.8 will fare as well overall, if not better, for the same AOV and DOF despite being 1/4 the price of the 14-35 / 2, likely due to the more comparable zoom range (discussed in more detail in the Lens vs Sensor section).

Q:  What's the deal with the "apparent" prefix?
A:  The idea behind the "apparent" prefix was to distinguish between different meanings of words and phrases that have ambiguous meanings.  For example, "exposure" means the density of the light that makes up an image (photons / area), as opposed to how bright or dark the image actually appears.  So, the term "apparent exposure" includes ISO (either in-camera or software push) to represent how brightness of the adjusted image appears.  In addition, noise can be used to represent both the density of the noise (NSR -- noise-to-signal ratio) in an image, which is what most people mean by "noise" (just as "apparent exposure" is what most people mean when they say "exposure"), as opposed to the total amount of noise in an image.  Thus, using the term "apparent noise" to represent the NSR, or how noisy an image appears, just as using the term "apparent exposure" to represent how bright and image appears, made sense in terms of useful terminology.

Q:  Last question -- Angelina Jolie or Jessica Alba?
A:  Well this is what Equivalence is all about.  At about the same age, Angelina Jolie, but that's not to take anything away from Jessica Alba.  But the current Angelina Jolie is a bit older design than the newer Jessica Alba.  So, while there are still some plusses to Angelina Jolie, if I were choosing today, I'd go with the younger model.  Of course, for the most part, it's not the equipment, but the operator.  Someone who knows what they're doing will produce much better "results" with Angelina Jolie than most could ever hope to accomplish with Jessica Alba.  Of course, those with the experience and skills know how to use, and make the best of, both.  : )

DEFINITIONS OF TERMS AND ABBREVIATIONS

Many of the misunderstandings come from people using different definitions for the same words. In particular, "f-ratio" is often confused with "aperture", and "exposure" is confused with "apparent exposure" and "total light". The importance of these distinctions is often overlooked or simply not understood, so a quick browse through this section would be helpful in understanding the rest of the essay.

IQ:  Image Quality
QT:  Quality Threshold
PP:  Post Processing
PPI:  Pixels per inch (not to be confused with DPI -- dots per inch -- which is a function of the printer)
NR:  Noise Reduction
AF:  Auto Focus

AOV:  Angle of View
FOV:  Field of View (framing)
UWA:  Ultra Wide Angle
SR:  Sensor Ratio (commonly referred to as "crop factor" and usually calculated as the ratio of the sensor diagonals for the same AOV)
Format:  Sensor Size (e.g. 1/1.8", 4/3, 1.5x, 1.6x, 35mm FF, etc.)
Aspect Ratio:  The ratio of the length to width of an image
135:  A sensor measuring 36mm x 24mm, sometimes simply referred to as "35mm FF" or just "FF" (Full Frame).
Output Size:  The number of pixels making up an image, or the dimensions of a print

Perspective:  The relative position of objects in the frame (a function only of subject-camera distance -- format and focal length independent)
FL:  Focal Length
EFL:  Effective Focal Length (the focal length that gives the same AOV in terms of 35mm FF)
Reach:  The native pixel count of an image for a given perspective, framing, and focal length.
TC:  Teleconverter (usually 1.4x or 2x)
DOF:  Depth of Field (the depth of the image from the focal plane that is considered to be in critical focus)
aperture:  As used in this essay, "aperture" is synonymous with "entrance pupil diameter" (the apparent diameter of the opening in a lens as seen by looking through the front element)
F-Ratio:  The ratio of the focal length and the aperture diameter (e.g. the f-ratio for a focal length of 50mm and an aperture diameter of 10mm is 50mm / 10mm = f/5)
Stop:  A difference of one stop represents a doubling, or halving, of a quantity

Exposure:  The density of light falling on the sensor (photons / mm²):  Exposure = Intensity x Time
Apparent Exposure:  The brightness of an image (what people usually think of as "exposure"):  Apparent Exposure = Exposure x ISO / 100
Total Light:  The total number of photons that falls on the sensor:  Total Light = Exposure x Light Collecting Area (of the sensor)
ev:  Exposure Value (in stops). A scene metered for f/1 and 1s has an ev of 0. Brighter scenes have higher ev's, darker scenes have lower ev's.
Noise:  The standard deviation of the recorded signal from the mean signal
Apparent Noise:  What people normally mean by "noise" -- the density of the noise in the image (NSR -- Noise-to-Signal Ratio), usually measured as a percent
Efficiency:  The percentage of light falling on the sensor that is recorded, along with the noise created by the sensor and supporting hardware
LMS:  Lowest Meaningful Signal -- usually taken to be the signal that results in 100% apparent noise (NSR = 1)
FWS / FWC:  Full Well Saturation / Full Well Capacity -- the maximum number of photons that the pixel can absorb before becoming oversaturated (blown)
DR:  Dynamic Range -- the difference (in stops) between the LMS and FWS
Tonal Gradations:  The number of different levels of brightness than can be measured within the dynamic range

Diffraction Softening:  Detail lost due to the diameter of the Airy disk exceeding the diagonal of a pixel (or two pixels for Bayer CFAs) due to the wave nature of light
Vignetting:  The radial light falloff from the center of an image
Distortion:  As used in this essay, the degree to which parallel lines stay parallel in the image
Bayer:  A color array where each pixel records one color (usually red, green, or blue)
Foveon:   A color array where each pixel records three colors

DEFINITION OF EQUIVALENCE

Equivalent images are images of a scene that share the following five parameters:

1) Same Perspective
2) Same Framing
3) Same DOF
4) Same Shutter Speed
5) Same Display Dimensions

Similarly, "equivalent lenses" are lenses that produce equivalent images on their respective formats, which means they will have the same AOV (angle-of-view) and the same aperture diameter.  For example, 50mm on 4/3 and 62.5mm on 1.6x are equivalent to 100mm on 135 since these focal lengths result in the same AOV on their respective formats.  Furthermore, and f/2 on 4/3 and f/2.5 on 1.6x are equivalent to f/4 on 135 since those f-ratios result in the same aperture diameter for the same AOV on their respective formats (50mm / 2 = 62.5mm / 2.5 = 100mm / 4 = 25mm).

Many object to the statement "f/2 on 4/3 is equivalent to f/4 on FF" or "f/2.5 on 1.6x is equivalent to f/4 on FF" since f/2 on 4/3 and f/2.5 on 1.6x both result in a different exposure than f/4 on 135 (for a given shutter speed).  This objection is thoroughly addressed in the section on exposure.  In addition, it is worthwhile to note that equivalent lenses often don't exist on different formats, which may well be a reason to choose one format over another.

Furthermore, it is also important to understand that Equivalent images are not "equal", but instead "corresponding, especially in effect or function" per one of the Webster's definitions of the term.  So, while equivalent images on different formats will usually have the most similar visual properties, they will not be identical, as other visual elements, such as noise, detail, flare, moiré, distortion, bokeh, etc., will not necessarily be the same, and sometimes, radically different.

In addition, the difference in aspect ratios between some formats (4:3 vs 3:2) means that to achieve the same framing for systems with different aspects ratios, either images from both systems need to be cropped to a common framing, or the photos from one system needs to be cropped to the same framing as the other, and is explained in more detail at the end of this section.  However, the difference between 4:3 and 3:2 is small enough to where this essay ignores this difference.

The most controversial visual property of equivalent images is that people incorrectly assume that Equivalence is based on equal noise.  Equivalence is based on the five principles listed above, which do not include noise, nor any other elements of IQ.  The primary elements in image noise, in order, are:

•  The Total Amount of Light that falls on the sensor (exposure · sensor area)
•  The percent of this light that is captured by the sensor (QE -- quantum efficiency)
•  The additional noise added by the sensor and supporting hardware (read noise)

Other factors, such as ISO and pixel count / size play a minor role in apparent noise compared to the above three factors.  Because equivalent images are made from the same total amount of light (since equivalent images, by definition, will have the same framing, aperture diameter, and shutter speed), and sensors of the same generation usually have similar QE / read noise, equivalent images from cameras of them same generation will usually have similar apparent noise for equivalent images.  People commonly believe that larger sensor systems have less apparent noise because they have better sensors, when, in fact, it is instead because they collect more total light for a given exposure.

Thus, breaking the properties of Equivalence down into the properties of the photo, lens, and sensor:

•  Photos with the same perspective, framing, display dimensions, and aperture diameter will have the same DOF
•  If we also include same shutter speed, then they will also have the same motion blur / camera shake, as well as be made from the same total amount of light
•  Differences in noise for equivalent images will primarily be a function of sensor efficiency and read noise, which are usually minor for sensors of the same generation

Note that it said above that "equivalent images on different formats will usually have the most similar visual properties" -- but not always.  For example, if one system has a significantly less efficient sensor than another system, if motion blur and/or camera shake are not an issue, then a longer exposure at a lower ISO on the system with the less efficient sensor may more closely match the shorter exposure on the system with more efficient sensor.  Or, if the extreme corners are of some importance in the composition, and one system has greater edge sharpness than the other, the system with the softer edges may need to stop down more to achieve sharper corners.  But, in most circumstances, Equivalent images will be, if not the most similar, very close.

A competent photographer will use their equipment to obtain the best image possible, which often means trading one IQ component for another (for example, when using a wider aperture to get less noise at the expense of less sharpness and greater vignetting).  However, it is important that we understand that these compromises represent choices that a photographer makes, and are not requirements imposed by the format.  Extending the example, it is disingenuous to say that a smaller format is superior to a larger format because it has more DOF, or is sharper, "wide open" than the larger format, when "wide open" is a choice, not a mandate, that results in a different (sometimes radically different) DOF and apparent noise, and the larger format can simply be stopped down for greater DOF and sharpness (although, if stopping down requires a concomitant increase in ISO to maintain a sufficient shutter speed, then the larger format may have to sacrifice some, or even all, of its noise advantage).

Thus, Equivalence is about the consequences of choices a photographer has in terms of IQ as a function of format.

As mentioned in the introduction, understanding the fundamental concepts of Equivalence requires making important distinctions between various terms which people often take to mean the same thing.  It is very much akin to making the distinction between "mass" and "weight", two terms which most people take to mean the same thing, when, in fact, they measure two different (but related) quantities.  While there are circumstances where making the distinction is unnecessary, there are other times when it is critical.

The first of these distinctions that needs to be made is between aperture and f-ratio.  While the term "aperture" is more accurately used to mean "entrance pupil diameter" it is often confusingly used to mean "f-ratio", most likely because "f-ratio", "aperture-ratio", and "relative aperture" are, in fact, synonyms.  The aperture diameter, as used in this essay, is the diameter of entrance pupil of the lens, which is the diameter of the opening in a lens when viewed through the front element, as opposed to the diameter of the front element, as some mistakenly believe, or the diameter of the physical aperture.  The f-ratio is defined as the quotient of the focal length and the aperture diameter.  Conversely, we can compute the entrance pupil diameter by taking the quotient of the focal length and the f-ratio.  For example, the f-ratio for a lens at 50mm with an aperture diameter of 25mm is 50mm / 25mm = f/2.  Likewise, the aperture diameter at 50mm f/2 is 50mm / 2 = 25mm.

The concepts of, and connections between, total light, DOF, and noise, are much more easily understood in terms of aperture rather than f-ratio, especially when comparing different formats.  While the same f-ratio will result in the same exposure across formats (see here), the aperture diameter, together with the shutter speed, determines the total amount of light that falls on the sensor, and, together with sensor efficiency and read noise, determines the apparent noise in the image.  In addition, for a given perspective, framing, display size, viewing distance, and visual acuity, the aperture diameter determines the DOF.

This naturally brings us to the distinction between total light, exposure, and apparent exposure.  The total light is simply the total amount of light that falls on the sensor, and is a function solely of the aperture and shutter speed for a given scene, perspective, and framing.  The importance of this measure cannot be understated as it, combined with the sensor efficiency, determines the apparent noise and dynamic range of an image.  The exposure, on the other hand, is the intensity of the light falling on the sensor, which is a function of the f-ratio rather than the aperture.  Since equivalent images are made from the same total amount of light, the light falling on the larger sensor will have a lower intensity (lower exposure) than a smaller sensor, just as a force applied over a larger area exerts less pressure than the same force applied over a smaller area.  To compensate for the lower exposure, the larger sensor system needs to use a higher ISO to amplify the image to get the appropriate level of brightness (apparent exposure).

The SR ("sensor ratio" -- more commonly referred to as the "crop factor") is the vehicle by which we compute equivalent settings for different formats.  For the same AOV, the SR is the ratio of the diagonal of the larger sensor to the diagonal of the smaller sensor.  If the aspect ratios (the ratio of the length and width of the sensor) are the same for the two systems, then the SR for the same AOV will be the same as the SR for the same framing as well.  However, if the aspect ratios are different, we will need to frame wider with one system, and then crop to the same framing as the other.  In this instance, we compute the SR as the ratio of the smaller dimensions of the sensors if cropping the more elongated image to the aspect ratio of the more square sensor, or the ratio of the longer dimensions of the sensors if we are cropping the more square image to the aspect ratio of the more elongated sensor.  It's often convenient to express the SR in stops and then round to the nearest 1/3 stop:  SR (in stops) = 2 log₂ SR (when rounding, it's helpful to recall that 1/3 ~ 0.33, and 2/3 ~ 0.67). Let's demonstrate this by working some examples between 135 (24mm x 36mm, 43.3mm diagonal) and 4/3 (13mm x 17.3mm, 21.6mm diagonal):

Same AOV:  SR = 43.3mm / 21.6mm = 2.00 (2 stops)
135 image cropped to same framing as 4/3 image:  SR = 24mm / 13.0mm = 1.85 (1.77 stops ~ 1 2/3 stops)
4/3 image cropped to same framing as 135 image:  SR = 36mm / 17.3mm = 2.08 (2.11 stops ~ 2 stops)

To calculate equivalent settings, we multiply the FL (focal length) used by the smaller sensor system by the SR to get the FL for the larger sensor system that will give the same AOV (or framing, depending on how we choose to compute the SR).  Similarly, we multiply the f-ratio of the smaller sensor system by the SR to get the f-ratio for the larger sensor system for the same DOF, and the ISO used by the smaller sensor system by the square of the SR to get the ISO for the larger sensor system that will give the same shutter speed for the same apparent exposure.  Alternatively, adding the SR (in stops) to both the f-ratio and ISO is usually preferable to the "messy" multiplication.  Let's work some examples to demonstrate:

25mm f/2 ISO 100 on 4/3 has the same AOV as 50mm (25mm x 2.00), the same aperture diameter as f/4 (f/2 x 2.00 = f/4), and the same shutter speed as ISO 400 (ISO 100 x 2.00² = ISO 400) on 135.  However, while we need to perform the multiplication to get the focal length, it is often easier to instead add the SR in stops for the f-ratio and ISO:  f/2 + 2 stops = f/4, ISO 100 + 2 stops = ISO 400.  Similarly, we could use an SR of 1.85 (1 2/3 stops) and get the settings required on 135 if we were going to crop the 135 image to the same framing as the 4/3 image, and use an SR of 2.08 (2 stops) if we were going to instead crop the 4/3 image to the same framing as the 135 image.

Since the most common aspect ratios, by far, for digital cameras are 3:2 and 4:3, we can see that the practical differences in the SR between the AOV and framing differ by less than 1/3 of a stop, so it is not a significant factor in terms of total light gathered, and thus apparent noise.  In addition, 1/9 of the pixels will be cropped away from the edges, which will have negligible impact on the PPI of a print, but may be important in terms of comparing corner sharpness.

The last step in making the images "equivalent" is to crop to the same framing, and either resample the images to the same display dimensions if comparing on a monitor (usually at least as large as the larger size image, but not necessarily so), or print at the same dimensions if comparing prints.

THE PURPOSE OF EQUIVALENCE

The motivation behind this essay was to dispel common myths about different formats which all sprang from one central fallacy:  to compare systems at the same f-ratio.  On the other hand, "equivalence" holds that there is not one parameter, but five, which are central to photography:

1) Perspective
2) Framing
3) DOF
4) Shutter Speed
5) Display Dimensions

An equivalent image is an image where all five of these parameters are the same.  As noted in the definition of equivalence in the section above, elements of IQ, such as detail, sharpness, apparent noise, vignetting, color, bokeh, etc., are not included in the definition since "equivalence" does not mean "equal", nor is it a mandate on how to use systems.  Furthermore, it is important to recognize that "equivalence" does not make any claims about which system is "better", as what constitutes "better" is entirely subjective, and involves operational considerations (size, weight, price, focus speed/accuracy, etc.) that are not addressed by equivalence (see the Q&A section for more on this).  Instead, equivalence is a set of parameters that serve as a starting point for comparing the IQ of systems.  However, it often makes more sense to compare the IQ between systems at non-equivalent settings to maximize the IQ of the system being used.

One of the most common situations where it makes sense to compare images that are not fully equivalent is when there is no need for the shutter speeds to be the same.  These types of images usually occur whenever a photographer is able to remain at base ISO and still maintain a "sufficient" shutter speed to avoid motion blur and/or camera shake, as well as an "appropriate" DOF for the scene.  Another scenario is the exact opposite -- to compare images at the same shutter speed and f-ratio (same exposure) rather than the same shutter speed and DOF.  This type of comparison is useful when apparent image noise and motion blur / camera shake play a more central role in the IQ of the image than do either DOF or sharpness (especially in the corners).  Another common scenario where fully equivalent images are not the "best" way to compare is when the photographer using a larger sensor system is focal length limited by the available lenses, and will compare systems at the same focal length and crop the image from the larger sensor system to the same framing as the smaller sensor system.  On the other hand, sometimes the entire point of a comparison is to compare images at the extreme limits of the equipment, such as at their minimum DOFs, maximum apparent magnifications, minimum focus distances, etc.  Lastly, we sometimes may compare images at settings that are not fully equivalent to make use of an operational advantage, such as intentionally dragging the shutter and making use of IS (either in-lens or in-camera) to create an artistic motion blur while maintaining a sharp background for pics where a tripod is impractical.

But it makes no sense to stage a comparison with settings that artificially handicaps one of the systems.  The most common scenarios for this are when systems are compared at the same f-ratio rather than the same DOF when comparing corner sharpness, or comparing images at the pixel level when one image has significantly more pixels than the other, or comparing apparent noise at different levels of detail.  Lastly, although not a postulate of equivalence,  comparing hardware on the basis of in-camera jpgs is a poor way to compare the potential of the equipment.  While it may be the most appropriate method of comparing for people who do not shoot RAW, it is hardly indicative of what the systems can achieve, in terms of IQ.

The point of photography is making photos.  As such, one doesn't choose the particular system to get images which are equivalent to another system.  A person chooses a particular system for the best balance of the factors that matter to the them, such as price, size, weight, IQ, DOF range, available lenses, and/or operation.  By understanding which settings on which system create equivalent images, the difference in their capabilities is more easily understood.  For example, a 50 / 1.4 on 35mm FF is equivalent to a 31 / 0.9 on 1.6x or a 25 / 0.7 on 2x, neither of which exist, and would be a reason for one person to choose a 35mm FF system if they needed such a lens.  On the other hand, a 4/3 system can get you a DSLR and a lens with an EFL of 28-84mm for less than the cost of the most inexpensive FF DSLR body alone.  Even more extreme, are compact digicams, such as the Canon G10, which deliver an EFL of 28-140mm (albeit having an effective minimum DOF from f/13 - f/21), and have, according to some, IQ as good as medium format for certain situations (please take a read of this article).

We can compare in many different ways.  The five parameters of Equivalence are simply guidelines to comparing systems in a fair and appropriate manner, and are not a mandate that systems must be compared in such a fashion.  Therefore, it is important to specify the purpose of the comparison, and then not artificially handicap one or the other system with the conditions of the comparison.  In addition, it is important to interpret the results of the comparison in the context of the circumstances where the conditions of the comparison are valid.  We choose one system over another on the basis of size, weight, operation, available lenses, IQ, DOF range, and, of course, price, so it is important to understand that IQ is only factor in determining which system is the best tool for the job.  How each person weighs the options will dictate their own unique choice for which system is best for their needs.

EQUIVALENCE AND PARTIAL EQUIVALENCE

As discussed in the section above on the Purpose of Equivalence, Equivalence is merely the baseline for a fair comparison between systems.  Often, it makes much more sense to compare systems on the bases of images that are not fully equivalent, in order to maximize the IQ of the systems being compared for specific shooting situations.  Specifically, this occurs when one system is able to use a lower shutter speed or more shallow DOF than another system for a particular image.

To that end, let's consider some comparisons between the Canon 5DII (35mm FF), Nikon D300 (1.5x), Canon 50D (1.6x), Olympus E30 (4/3).  The FM is 1.6 for the 5DII and D300, 1.6 for the 5DII and 50D, and 2 for the 5DII and E30 (for the same AOV only -- since the E30 has an aspect ratio of 4:3 as opposed to the 3:2 of the 5D, the framing will be slightly different).  We'll begin by comparing fully equivalent settings (rounded to the nearest 1/3 stop) on the four formats:

1)  5DII at 50mm, f/5.6, 1/200, ISO 400
2)  D300 at 33mm, f/3.5, 1/200, ISO 160
3)  50D at 31mm, f/3.5, 1/200, ISO 160
4)  E30 at 25mm, f/2.8, 1/200, ISO 100

If the same scene is shot from the same position, all four systems will have the same perspective (subject-camera distance) and AOV.  If the resulting images have the same display dimensions, they will also have the same DOF.  In addition, because the shutter speeds are also the same, they will also have the same apparent exposure.  The level of apparent noise will depend on the efficiency of the sensor, although, typically, for a given generation of camera and at the same level of detail, the apparent noise levels will generally be very close for equivalent settings.  However, the level of detail will depend on both the pixel count of the sensor and the sharpness of the lenses used.  In this scenario of fully equivalent images, the differences in IQ between the systems will be at a minimum.  As always, which system has the "IQ advantage" will be a subjective measure, but most likely will go to the system that is able to render the greatest amount of detail, which will often be the system that has the largest native pixel count and/or best performing lenses at the f-ratios used to capture the images.

The role of lens sharpness needs to be discussed in a bit more detail.  Note that for fully equivalent images, the lenses on the different systems do not use the same f-ratio.  Thus, while the available lenses for one system may be sharper than the available lenses for another system at the same f-ratio, we are not using the lenses at the same f-ratio for the same DOF, so this amounts to an improper comparison of the effect that the lens has on the captured image.  So, while one system may have superior lenses to another system, what matters, in terms of the captured image, is not how the lenses compare at the same focal length and f-ratio, but how the lenses compare at the settings that result in the same AOV and DOF.

Sometimes, we can get away with a slower shutter speed, rather than a higher ISO, and thus have lower apparent noise for the formats that are able to use lower ISOs.  The following comparisons are examples of partial equivalence where shutter speed is traded for ISO in the larger sensor systems, which will allow them to obtain a cleaner image with more detail (pixel count and lenses permitting) while still maintaining the same AOV, DOF, and exposure:

1)  5DII at 50mm, f/5.6, 1/50, ISO 100
2)  D300 at 33mm, f/3.5, 1/125, ISO 100
3)  50D at 31mm, f/3.5, 1/125, ISO 100
4)  E30 at 25mm, f/2.8, 1/200, ISO 100

Note the "danger" in comparing partially equivalent situations -- the lower shutter speed used to maintain the lower ISO will not always be feasible due to motion blur and/or camera shake, and is especially important to consider in lower light situations.  Regardless, being able to "safely" use a lower shutter speed with a larger sensor system is still a common scenario.  Of course, this can go in the opposite direction when one system has in-camera IS and/or in-lens IS that the other system does not, and motion blur is not a factor:

1)  5DII at 50mm, f/5.6, 1/50, ISO 1600
2)  D300 at 33mm, f/3.5, 1/50, ISO 640
3)  50D at 31mm, f/3.5, 1/50, ISO 640
4)  E30 at 25mm, f/2.8, 1/13, ISO 100

In this scenario, we are assuming a static scene and that the 5DII, D300, and 50D are not using IS lenses, whereas the E30 has sensor IS, and can thus use a much lower shutter speed and ISO to obtain a cleaner image at the desired DOF / sharpness.

Other times, we might rather use a more shallow DOF than a lower shutter speed to use a lower ISO and thus less apparent noise, either because we prefer a more shallow DOF, or we need a fast shutter but lower apparent noise is more important than the "side effects" (softer corners and more vignetting) of a more shallow DOF, but still the same AOV, shutter speed, and exposure:

1)  5DII at 50mm, f/2.8, 1/200, ISO 100
2)  D300 at 33mm, f/2.8, 1/200, ISO 100
3)  50D at 31mm, f/2.8, 1/200, ISO 100
4)  E30 at 25mm, f/2.8, 1/200, ISO 100

Some may have noticed that the D300 and 50D use the same f-ratio and ISO, but slightly different FLs.  The reason is that all numbers are rounded to the closest 1/3 stop, and the difference between sensor ratios of 1.6 and 1.5 produce is less than 1/3 of a stop.  The same type of minor correction for FL will happen if framing and cropping the 4:3 images to 3:2 or framing and cropping 3:2 to 4:3, but will be too small to see an effect on the f-ratio or the ISO.

Lastly, as noted a bit further up, it is important to note that systems with in-camera IS will often enjoy a distinct advantage over systems without in-camera IS, or IS lenses in the desired focal range, for handheld shooting of static scenes or handheld shooting where motion blur is a desired artistic effect.

THE FIVE POSTULATES OF EQUIVALENCE

PERSPECTIVE

Perspective is how objects appear in relation to other objects, and the effect it can have on the image is dramatically demonstrated with these images.  Perspective is a function only of the distance of the camera from the subject -- the only role the focal length plays is in determining which portion of the scene we are capturing, not how the scene is rendered.  Technically, it is a function of the distance from the subject to the lens aperture, but as long as we are not at macro, or near macro, distances, it is sufficient to think of the perspective simply as the subject-camera distance since this amounts to a difference of only a few inches.  Two photos taken from the same position will have the same perspective regardless of the focal length or sensor size regardless of the FL (focal length) of the lens used.

A good way to think of perspective is to consider two objects, one 10 ft from the camera, the other 30 ft from the camera.  If both objects are in the frame with the subject being the closer object, and we shoot at 50mm from 10 ft away, then the further object is three times as far away as the subject.  If, however, we step back another 10 ft and frame the subject in the same manner at 100mm, then, if the the further object is even still in the frame, then the subject will be 20 ft away and the other object 40 ft away -- only twice as far.  Conversely, if we get twice as close and frame at 25mm, now the subject is 5 ft away, and the other object is 25 feet away -- five times as far.

But the subject-camera distance change the perspective by changing the relative distances of subjects within the frame, it also changes, in a similar fashion, how widely separated they are in the frame.  In fact, when we use a longer perspective, we will often find that much of what was in the frame of a closer perspective is now outside the frame.  Inasmuch as the scene as a whole matters, rather than simply the actual subject, perspective can be one of the most striking elements of a photograph.

FRAMING

For a given perspective, the framing can be thought of as the whole of the captured scene, and is synonymous with the FOV (field of view), which is a combination of the horizontal and vertical AOV (angle of view).  Unless otherwise specified, the term "AOV" refers to the diagonal AOV.  The distinction between AOV and FOV need not be made when systems share the same aspect ratio, but the greater the difference in aspect ratios, the more important the distinction between the terms.

We can compute the horizontal, vertical, and diagonal AOVs (for rectilinear lenses at non-macro distances) with the following formula:

AOV = 2 · tan^-1 [ s · (d - FL) / (2 · d · FL) ]

where

d = distance to subject (mm = distance in meters · 1000 = distance in feet · 304.8)
s = sensor dimension (mm)
FL = focal length (mm)

Alternatively, we can express the AOV as a function of the magnification where the magnification (m) is approximated by:  m ~ FL / (d - FL):

AOV = 2 · tan^-1 { s / [ (2 · FL) · (1 + m) ] }

For infinity focus, both of these formulas can be reduced to:

AOV = 2 · tan^-1 [ s / [ (2 · FL) ]

Solving for focal length, we have:

FL = (s · d) / [ s + 2 · d · tan (AOV / 2) ] or, for infinity focus, FL = s / [ 2 · tan (AOV / 2) ]

This means that the effective focal length (EFL) of the lens for a subject at a distance d (mm) is given by:  EFL = (d · FL) / (d - FL)

For example, the diagonal, horizontal, and vertical AOV for infinity focus (m=0) on 35mm FF at 50mm is:

Diagonal AOV for 50mm on 35mm FF = 2 · tan^-1 [43.3mm / (2 · 50mm)] ~ 47°
Horizontal AOV for 50mm on 35mm FF = 2 · tan^-1 [36mm / (2 · 50mm)] ~ 40°
Vertical AOV for 50mm on 35mm FF = 2 · tan^-1 [24mm / (2 · 50mm)] ~ 27°

Next, let's repeat for a subject at 3 ft (914.4mm):

Diagonal AOV for 50mm on 35mm FF = 2 · tan^-1 [ 43.3mm · (914.4mm - 50mm) / (2 · 914.4mm · 50mm) ] ~ 45°
Horizontal AOV for 50mm on 35mm FF = 2 · tan^-1 [ 36mm · (914.4mm - 33mm) / (2 · 914.4mm · 33mm) ] ~ 38°
Vertical AOV for 50mm on 35mm FF = 2 · tan^-1 [ 24mm · (914.4mm - 31mm) / (2 · 914.4mm · 31mm) ] ~ 26°

Let's now compute the focal length for 35mm FF, 1.5x, 1.6x, and 4/3 for a diagonal AOV of 47° at infinity:

FL for FF = 43.3mm / [ 2 · tan (47° / 2) ] ~ 50mm
FL for 1.5x = 28.4mm / [ 2 · tan (47° / 2) ] ~ 33mm
FL for 1.6x = 26.7mm / [ 2 · tan (47° / 2) ] ~ 31mm
FL for 4/3 = 21.6mm / [ 2 · tan (47° / 2) ] ~ 25mm

Note that these focal lengths are all proportional to the sensor ratio:

50mm / 1.5 ~ 33mm
50mm / 1.6 ~ 31mm
50mm / 2 ~ 25mm

Now we'll repeat for a horizontal AOV of 40° at infinity:

FL for FF = 36mm / [ 2 · tan (40° / 2) ] ~ 50mm
FL for 1.5x = 23.7mm / [ 2 · tan (40° / 2) ] ~ 33mm
FL for 1.6x = 22.2mm / [ 2 · tan (40° / 2) ] ~ 31mm
FL for 4/3 = 17.3mm / [ 2 · tan (40° / 2) ] ~ 24mm

Once again, we see these are proportional to the sensor ratio:

50mm / 1.5 ~ 33mm
50mm / 1.6 ~ 31mm
50mm / 2.08 ~ 24mm

And for a vertical AOV of 27° at infinity:

FL for FF = 24mm / [ 2 · tan (27° / 2) ] ~ 50mm
FL for 1.5x = 15.7mm / [ 2 · tan (27° / 2) ] ~ 33mm
FL for 1.6x = 14.8mm / [ 2 · tan (27° / 2) ] ~ 31mm
FL for 4/3 = 13mm / [ 2 · tan (27° / 2) ] ~ 27mm

And, again, these focal lengths are proportional to the sensor ratio:

50mm / 1.5 ~ 33mm
50mm / 1.6 ~ 31mm
50mm / 1.85 ~ 27mm

As noted further above, the EFL for a lens will change as a function of focal distance.  The following table demonstrates the effect of focal distance on the EFL of a 50mm lens:
 

A useful relationship between focal length, sensor size, distance to subject, and the height or width of the focal plane in the photo is:

FL = (s · d) / (s + h)

where all variables below are given in mm (1m = 1000mm, 1 ft = 304.8mm)

FL = focal length
s    = sensor dimension (sensor height for landscape orientation, sensor length for portrait orientation -- given in the table just a bit further down)
d   = distance to subject
h   = height of frame

For most situations, where d > 10·FL, a very good approximation of the relationship is given by:

FL / s = d / h

What might we use this for?  Let's say we have a landscape oriented photo of a model who is standing approximately 5' 8" (1727mm) taken on 35mm FF with an 85mm lens and would like to know what the subject distance was.  The calculation is as follows:

85mm / 24mm = d / 1727mm → d = 6116mm = 20 ft.

 

Listed below are tables of common SRs (sensor ratios) in relation to 35mm FF for images using the same AOV.  When given in stops, the SR is rounded to the nearest 1/3 stop.  The reason that 35mm FF (24mm x 36mm) is chosen as a standard is due to its popularity in the days of film and the fact that there are more lenses made for this particular format which many of the smaller sensor DSLRs also use, but we can use any format as a reference.  Due to different aspect ratios, when cropping to the dimensions of the more square sensor, we use the ratio of the shorter dimensions of the sensor to compute the SR, and when cropping to the dimensions of the more elongated sensor, we use the ratio of the longer sensor dimensions.  In the case of 3:2 being cropped to 4:3, or vice-versa, this will result in less than a 1/3 stop difference.

One side effect of cropping 3:2 images to 4:3 is that it greatly mitigates any softness that might show in the extreme corners.  However, we must also realize that this comes at the expense of removing 1/9 of the pixels from the image.  But as 3:2 systems generally have more pixels than 4:3 systems of the same generation, this can be done without any detail penalty when comparing systems.  Realistically, however, the extreme corners make up so little of the image, and are so close between systems anyway at the same DOF that it is only a consideration for the most hardcore of "pixel-peepers".  Please see this image as an example of what would be called a "huge" difference in the corners of different systems at the same DOF.  I simply see it as a non-issue, especially considering that the differences elsewhere in the frame matter more by far, but others see it as a serious disadvantage.  In any event, framing slightly wider and cropping to 4:3 will basically eliminate even that extreme case.

Compacts:
 

Sensor Size	Dimensions (mm)	Diagonal (mm)	Area (mm²)	FM	FM (stops)
1/2.7”	4.035 x 5.371	6.72	21.7	6.44x	5 1/3
1/2.5”	4.290 x 5.760	7.18	24.7	6.02x	5 1/3
1/2.33"	4.60 x 6.13	7.66	28.2	5.65x	5
1/1.8”	5.319 x 7.716	8.93	41.0	4.84x	4 1/2
1/1.7”	5.7 x 7.6	9.5	43.3	4.55x	4 1/3
2/3”	6.6 x 8.8	11.0	58.1	3.93x	4

DSLRs:

IMAGE QUALITY

Depending on the image, various elements of IQ will have varying levels of importance.  For example, apparent noise will usually play little role in ISO 100 images, edge sharpness will play basically no role in shallow DOF images, sharpness will play little role in images where motion blur is used for artistic effect, etc., etc.  So, while we can discuss the differences in IQ between systems, we cannot say which elements of IQ are more important than others.  Thus, while one system may have significantly more appeal on the basis of IQ to a vast majority, that does not mean that it will have higher IQ in the eyes of all.  Hence, when comparing systems, as mentioned further above, it is best not to compare on the basis of IQ, but on the basis of specific elements of IQ.

So, is it simply a waste of time to compare IQ between systems?  Some believe so, but I disagree.  Some elements of IQ that most people value are predictable and quantifiable on the basis of the sensor and available lenses.  This essay discusses the relationship between the glass and the sensor in how they determine some aspects of IQ, in particular, detail, sharpness, contrast, vignetting, and apparent noise.  However, it is also important to note the aspects of IQ that this essay does not discuss, such as bokeh, color, and distortion.

All these qualifiers and disclaimers said, a critical consideration to IQ is the individual's QT (quality threshold), that is, the point at which additional IQ makes no difference to the viewer at a given output size.  For example, System A may satisfy one person's QT at 8x12, but fail to do so at 12x18.  Or, one system may fail to satisfy a viewer's QT at any output size due to factors that are independent of the image dimensions (bokeh, for example).

Regardless, it's still not possible to reach universal agreement that one image, or system, has higher IQ than another.  The reason for this is that images from two different systems are never identical, and whatever differences there are between them may appeal to different people differently, as people value different aspects of IQ differently.  For example, let's say one image is sharper everywhere than another, except in the extreme corners.  Which image has the higher IQ?  Different people will have different answers depending both on the type of photography they do or enjoy, and on the degree to which the differences in sharpness vary in the images.  Another difficulty is when one system shows higher IQ in one circumstance, but lower IQ in another.  Likewise, a sensor with a weaker AA filter will render a sharper image, but be more subject to moiré, so in some instances it will have higher IQ and in other instances lower IQ, depending on the scene.  In other words, there's still a great deal of subjectivity even within this very narrow set of parameters for IQ.

Apparent noise is perhaps the most hotly contested of the IQ parameters.  As mentioned earlier, it is not simply the total amount of apparent noise, but the quality of the noise -- the distribution of the noise in the various color channels, the balance of color vs luminosity noise, and the grain of the apparent noise (which is a function of the native pixel count of the sensor).  But while noise can even have a pleasing effect in some images, I've never heard of anyone saying the same for pattern noise and banding.  Thus, a noisy image without pattern noise or banding will likely look significantly better than a cleaner image with pattern noise or banding, depending on the pattern, degree of banding, and how large the difference in total apparent noise is.  Furthermore, since different cameras will apply NR (noise reduction) to various degrees (some even to RAW files), it is important to recognize that while one image may be more noisy than another, it may also yield more detail, which may well matter more than the apparent noise.  If not, then we should apply NR and/or downsampling the more detailed image to match the level of detail of the less detailed image before comparing apparent noise.

Another critical factor that needs to be mentioned is in-camera processing and PP (post-processing).  For example, comparing images from different systems based on in-camera jpgs tests the in-camera jpg engine (firmware) as much, if not more than, as it does the camera hardware.  For people who loathe PP, comparing systems on the basis of in-camera jpgs, of course, makes the most sense.  But such a comparison will have less to do with the IQ potential of a system and more to do with operational convenience.  However, for people looking to get the most out of their hardware, the "appropriate" format is RAW.  To this end, it is important that we choose a RAW conversion that portrays each system at its best.  Unfortunately, we are right back to the subjective with what looks "best".

In addition, the IQ differential, while present, may not always be noticeable.  Let me explain that odd statement, since it would seem obvious that if you can't see a difference in IQ, then there is no difference in IQ.  Well, yes and no.  True, if for a particular image you cannot see a difference, then there is no meaningful difference in IQ.  But depending on how much processing is applied to the image, we may find that one image withstands that processing much better than the other.  In addition, as mentioned earlier, the IQ differential may not show at one print size, but become apparent at another.  Thus, the "hidden" IQ of an image may become apparent only under strong PP or larger prints.  It's for that exact reason that so many shoot RAW instead of jpg.  In many cases, the IQ differential between jpg and RAW conversions are completely insignificant, whereas in some cases, the differences are substantial.  So just as RAW has higher IQ than jpg, one system may have higher IQ than another, but that higher IQ does not always manifest itself.  Hence, while for one person the IQ difference is non-existent, for another, the IQ difference is significant. 

Furthermore, it is fair to say that the elements of IQ that can be corrected with PP matter less than the elements of IQ that are resistant to PP.  For example, vignetting and distortion are easy corrections in post (and, in fact, can be automatically "corrected" in some RAW converters, along with even PF), whereas detail and DR are not.  Other attributes are intermediary -- apparent noise can be lowered, but this comes at the expense of detail.  Sharpness can be enhanced, but this comes at the expense of artifacts.  Still other effects are primary:  bokeh, flare, and moire are often beyond the abilities of PP (unless one wishes to painstakingly hand-edit every portion of the image), but these attributes occur in only certain types of photos, and thus may not be important considerations to some people.  Nonetheless, despite the fact that there is no way around the subjective elements of IQ and the narrow definition used in this essay, generalizations about the IQ of different systems can be made.

Lastly, I would like to more thoroughly address the issue of output size, which is a critical consideration in determining what level of IQ, especially in terms of sharpness and detail, really matters.  For many, if not most, the web is their primary venue for displaying images.  Thus, even a 1.3 MP image is good for a 1280x1024 presentation.  However, for those that print their images, 300 PPI (pixels per inch -- not to be confused with DPI, dots per inch, which is a function of the printer) is considered about as good as a person will see even with their nose right up against the image.  It's certainly well beyond the QT of the "average" viewer, and 150 PPI is considered "quite decent".  Of course, it depends on your needs and standards -- "decent" to one person is "garbage" to another.

And, since I bring up printing, it's no small point that the printer and paper used for the final image is a critical component of the final image.  However, this topic of this essay is comparing camera systems (camera and available lenses), and it is presumed that we are taking care to process the images as best we can and use the same quality printer and paper for both systems.

To that end, let's consider the PPIs for common print sizes (in inches).  The table gives the PPIs for 10, 20, 30, and 40 MP images for the with a native 3:2 aspect ratio / 4:3 aspect ratio cropped to the given print dimensions:

It's worth noting that since Bayer arrays record only one color per pixel, the PPIs in the above table may be more accurately represented by pixel counts twice as large as given.  That is, to truly achieve 323 PPI for an 8x12 print, we may need 20 MP, not 10 MP.  However, that is a debate outside the scope of this essay.

Anyway, as we can see, the aspect ratio really doesn't affect the image PPI by any significant amount.  More importantly, the table above serves to demonstrate how important it is to keep our IQ "needs" in context with the size we print and the expected viewing distance.  Basically, what we see is that if 10 MP is "good enough" for up to 13x19 inch prints, then 20 MP of equal quality pixels will be "good enough" for up to 18x24 inch prints, but would offer no meaningful advantage for 13x19 inch prints.  But we must pay special attention to the qualifier -- "equal quality pixels".  We wouldn't expect 10 MP from a compact to deliver the same quality as 10 MP from a DSLR, for example, nor can we simply upsample a 10 MP image to 20 MP and expect a marked improvement (in fact, the utility of upsampling for the purposes of increasing print quality is of debatable value).  Note also how doubling the pixel count doesn't make that much of a difference in the print size, either, and that for print sizes of 8x12 or smaller, 10 MP is already past the point where we can distinguish any differences for equal quality pixels.

That said, the reality is that for deep DOF pics at base ISO and smaller print sizes (8x12 inches and smaller, and even larger, depending on the scene and QT of the viewer), few will be able to distinguish, or care, about the differences in IQ between most formats.  An interesting article on that point is Michael Reichmann's "You've Got to be Kidding -- No, I'm Not".

In any event, there are many elements to IQ that matter even at smaller print sizes, such as bokeh and DR.  Thus, even though one system may be able to output a sharper and more detailed image at larger dimensions, these qualities may not be as important as the other qualities of IQ, depending on the image.  Of course, the artistic considerations almost always outweigh the technical considerations of an image, which brings us back to the distinction between a quality image and image quality.

So what IQ advantages does a larger sensor have?  Typically, the larger sensor system will deliver "higher overall IQ" over smaller sensor systems of the same generation in the following ways:

1) Larger sensor systems have less apparent noise and more dynamic range at any given ISO
2) Larger sensor systems usually allow for the option of a more shallow DOF
3) Larger sensor systems often have more pixels which means more detail and a finer grain of noise

This, of course, invites the question as to when smaller sensor systems will have "higher IQ".  This can happen when:

1) The lenses designed for the smaller sensor system can resolve more pixels than the lenses for the larger sensor
2) The lenses designed for the smaller sensor system is a superior optic in terms of bokeh, flare, distortion, etc.
3) The smaller sensor system has an operational advantage such as more accurate AF or in-camera IS

If we think about all these situations, it's easy to see how the balance of these advantages and disadvantages play into the type of photography a person does.  The more narrow the scope of photography, the easier it is for one system to be superior to another for the particular application.  The more broad the scope, the more difficult it is for a single system to be able to be a clear winner overall.

Typically, for cameras of the same generation, the larger sensor is usually at least as efficient as the smaller sensor, since it is featured in the company's "top-end" cameras.  However, much of the glass being used on larger sensors is of an older design and outperformed by the newer generation glass.  On the other hand, from images I've seen (and linked in the evidence section of this essay), for equivalent images, even much of the older 35mm FF glass delivers sharper images everywhere except the extreme corners with some UWAs, although, as noted, there is more to IQ than just sharpness.  As for focus accuracy, unfortunately, I don't know of any "reliable" reports that compare the accuracy and speed of different systems.  But in-camera IS is a very powerful plus for smaller sensor cameras, since it is not yet available on 35mm FF cameras to date, with the exception of the Sony A900.  While many argue that in-lens IS is superior to in-camera IS (but neither are as good as using a chicken -- click here for a demonstration), it is definitely not superior if the lenses you use do not have it.

Regardless of what IQ differences there may be between systems, we have to decide when, if ever, these differences in IQ have any meaning.  For example, a Suzuki GSXR-1000 may significantly outperform a Yamaha R-6 on a track, presuming the driver is skilled enough to make use of the extra performance.  But if all you use the bikes for is traveling back and forth to work or school, the difference in performance between the bikes is meaningless -- it is more a matter of comfort, gas mileage, and other aspects of the bike that matter more by far.

Thus, it is my opinion that for the sizes that most people print (or display on the web), the differences in IQ between modern systems are insignificant for the vast majority, just as the performance differences in bullet bikes is insignificant for most riders.  Instead, the the primary consideration for most people when choosing a system is not the merely the IQ of the images it produces, but the the types of images the system can produce and the operation of the system.

MYTHS AND COMMON MISUNDERSTANDINGS

The motivation behind this essay on "equivalence" was prompted by the many myths about the differences between formats.  In particular, the following myths and misunderstandings are common:

1) f/2 = f/2 = f/2

This is perhaps the single most misunderstood concept when comparing formats.  Saying "f/2 = f/2 = f/2" is like saying "50mm = 50mm = 50mm" -- true, but meaningless out of context.  We choose a particular focal length to obtain a particular AOV (angle-of-view).  Naturally, we would not use the same focal length on different formats to get the same AOV.  Likewise, we choose a particular f-ratio to get a certain DOF, a desired level of sharpness, or to minimize noise or maximize shutter speed by shooting wide open.  All of these effects are functions of the aperture diameter, which is the quotient of the focal length and the f-ratio.  For example, the aperture diameter for 50mm f/2 is 50mm / 2 = 25mm..

The central role that the size of the aperture plays in photography, as opposed to the f-ratio, cannot be understated.  While a given f-ratio does result in the same exposure (light density) for a given scene and shutter speed, it is the total amount of light, as opposed to the density of the light, that is of central importance to the final image.  For a given scene, perspective, and shutter speed, the aperture diameter, as opposed to the f-ratio, determines the total amount of light that falls on the sensor.  The reason the total amount of light that makes up an image is so important, as opposed to the density of the light, is because the total amount of light is the primary factor in the apparent noise in an image.  Furthermore, for a given perspective, framing, display size, and viewing distance, the aperture diameter will determine the DOF of an image.  In addition, it is important to note that all systems suffer the effects of diffraction softening at the same DOF.

For example, let's consider a lens with a 25mm maximum aperture diameter.  If the focal length is 50mm, the minimum f-ratio is, by definition, 50mm / 25mm = f/2.  Likewise, if the focal length is 63mm, the minimum f-ratio is 63mm / 25mm = f/2.5, and if the focal length is 100mm, the minimum f-ratio is 100mm / 25mm = f/4.  Now, since 25mm on 4/3, 63mm on 1.6x, and 100mm on 135 all have the same AOV, we say that 25mm on 4/3 is equivalent to 63mm on 1.6x which is equivalent to 100mm on 135.  Likewise, since f/2 on 4/3, f/2.5 on 1.6x, and f/4 on 135 all have the same aperture diameters for the same AOV, f/2 on 4/3 is equivalent to f/2.5 on 1.6x which is equivalent to f/4 on 135.

This does not mean, of course, that 50mm f/2 on 4/3 results in the same shutter speed as 63mm f/2.5 on 1.6x or 100mm f/4 on 135.  Clearly, the larger sensor systems will require higher ISOs to maintain the same shutter speed or underexpose and push in post.  A common misconception is that higher ISOs result in more noise, which comes from confusing cause and effect.  A higher ISO results in a faster shutter speed, smaller aperture diameter, or both, all of which result in less light falling on the sensor.  It is the lesser amount of light, and not the higher ISO per se, that results in greater apparent noise.  In fact, higher ISOs are usually somewhat less noisy than using a lower ISO and pushing the image in post (discussed in more detail in the Noise section).

One of the most misunderstood concepts in cross-format comparisons is that many feel if only the sensors for smaller sensor systems could be made "as good as" the sensors of larger sensor systems, then smaller sensor systems could match the noise performance of the larger sensor systems.  However, while sensor efficiency most certainly plays a role in image noise as well, the differences in sensor efficiencies for sensors of the same generation, regardless of format, are usually quite minor in comparison to the differences in the ability to collect more light due the differences in size of either the sensor (at base ISO), or the aperture (when not at base ISO).

At base ISO, larger sensor systems can use the same aperture diameter as a smaller sensor system to achieve the same DOF for a given perspective and framing, but use a longer shutter speed to compensate for the higher f-ratio that is required for the same aperture diameter (being mindful of camera shake and/or motion blur).  Since the aperture diameter is the same, but the shutter speed is longer, more more light will fall on the larger sensor than the smaller sensor.

Alternatively, if we are above base ISO, then the shutter speed is necessarily an important factor (else we'd not be using a higher ISO).  In this scenario, larger sensor systems can often achieve less noise than smaller sensor systems by using larger aperture diameters (with a concomitant more shallow DOF, softer corners, and more vignetting) to collect more light.  However, for the same AOV, DOF, and shutter speed, all systems will collect the same total amount of light.  The differences in apparent noise from equivalent images will come from differences in sensor efficiencies, which vary from camera to camera even of the same format, and especially for different generations of bodies.  However, for the same exposure, unless the formats are very close in size, it is very unlikely that the differences in sensor efficiencies will outweigh the differences in the light collecting capability due to sheer size.

In other words, equivalent images will not necessarily have the same amount of noise in the same way that even two cameras from the same format will not necessarily have the same noise for the same exposure.  Equivalence is not based on equal noise (see Misunderstanding #7), but for sensors of the same generation, noise equivalence is usually fairly close.  In general, for a given scene, perspective, shutter speed, and display dimension of the final image, 50mm f/2 on 4/3, 63mm f/2.5 on 1.6x, and 100mm f/4 on 135 will look substantially more similar than 50mm f/2 on 4/3, 63mm f/2 on 1.6x, and 100mm f/2 on 135.  So, while a given f-ratio will result in the same exposure across formats (but nothing else), it is more useful to say that f/2 on 4/3 is equivalent to f/2.5 on 1.6x is equivalent to f/4 on 135, since these images will usually share the most similar visual properties.
 

2) Larger sensor systems are bulky and heavy

While larger sensor systems usually are more bulky and heavy than smaller sensor systems, this is not necessarily the case.  In fact, sometimes even the exact opposite is true.  The reason is not as much due to the larger sensor as it is due to the fact that the lenses designed for larger sensor systems usually have larger maximum aperture diameters than lenses designed for smaller sensors.  But when equivalent lenses do exist in both systems, such as the 35-100 / 2 on 4/3 vs the 70-200 / 4L IS on 35mm FF, the lenses for the larger sensor systems are usually lighter (but often longer for the telephoto lenses) and less expensive.  There are exceptions, of course, such as the Canon 300 / 2.8L IS on 1.6x vs the Canon 500 / 4L IS on FF.  But if reach is the primary consideration, and light gathering ability secondary, then smaller sensor systems will usually have a size/weight/price advantage, the most extreme example of this being the 12x zooms of compact digicams.  Thus, smaller sensor systems are usually significantly smaller, lighter, and less expensive when compared only for the same AOV, but not when compared for the same AOV and aperture diameter.
 

3) Larger sensor systems have a DOF that is "too shallow"; smaller sensor systems have more DOF

Larger sensor systems usually have the option of a more shallow DOF than smaller sensor systems with existing lenses, since the lenses have larger maximum aperture diameters for a given AOV.  But one can still stop down for a deeper DOF, well into the realm where the effects of diffraction softening are the primary factor in terms of IQ.  So, while for the same perspective, framing, and f-ratio, larger sensors do have a more shallow DOF, one needs only stop down to get a deeper DOF, keeping in mind that, for a given pixel count, the effects of diffraction softening are the same across formats at the same DOF, not the same f-ratio, and that while more pixels do result in the ability to resolve more detail, it is a negligible difference at "diffraction limited" f-ratios.  Even in a limited light situation where the shutter speed needs to be maintained and the larger sensor system will have to up its ISO accordingly, the apparent noise will be the same for the same level of detail if the sensors of the two systems have the same efficiency.

However, for the extreme end of the deeper DOFs, the lenses for smaller sensor systems will often, but not always, be able to go deeper.  However, at such DOFs, the effects of diffraction softening will be severe (although not necessarily apparent for images resized for web display).  For example, the 50/2 macro on 4/3 attains it's minimum aperture at f/22 (equivalent to 100mm f/45 on 35mm FF), the 60 / 2.8 macro on 1.6x attains it's minimum aperture at f/32 (equivalent to 96mm f/51 on 35mm FF), and the Canon 100 / 2.8 macro on FF attains its minimum aperture at f/32 (but the Sigma 105 / 2.8 macro stops down to f/45).  However, if the FF shooter needs deeper DOFs, they can simply use the same focal length as 4/3 or 1.6x, in conjunction with a 2x or 1.4x TC, respectively.  For example, the 50 / 2.8 macro used on 35mm FF attains its minimum aperture at f/32, and is equivalent to a 100 / 5.6 macro with a maximum f-ratio of f/64 if used with a 2x TC.  It is important to note that at such small apertures, that image degradation from the TC is insignificant in comparison to detail loss from diffraction softening.

Of course, what's good for the goose is good for the gander.  That is, all systems can use TCs to increase their DOFs.  However, at f-ratios at and beyond f/22 (in terms of 35mm FF equivalents), the effects of diffraction softening are so strong that they overwhelm any other IQ differences between systems, keeping in mind that all systems suffer the effects of diffraction softening equally for the same DOF at a given level of detail.  Thus, at the extreme deep end of the DOF spectrum, there is virtually no difference between systems in terms of IQ.
 

4) Larger sensors require sharper glass

In fact, the exact opposite is true.  First of all, as discussed in Myth #1, it is important to compare systems at the same DOF when discussing sharpness, since if we don't, the system with the more shallow DOF will have less of the scene within the DOF, and thus appear less sharp.  So, given that we are comparing systems at the same DOF, consider the following scenario:  we have two targets, each with the same number of squares on them covering the entire area.  Since both targets have the same number of squares, the squares on the larger target will be larger than the squares on the smaller target.  Thus, when trying to hit the squares on the smaller target, we need to be more accurate than when aiming at the squares on the larger target.  For example, if the larger target has twice the length and width of the smaller target, then we need to be twice as accurate to hit the smaller squares on the smaller target.  This is why the MTFs for 4/3 lenses use 20/60 as compared to 10/30 for FF.  In the same way, a lens on a larger sensor does not need to be as sharp as a lens on smaller sensor to resolve the same amount of detail.  Lenses that are able to resolve the same detail on sensors with the same number of pixels on their respective formats have the same relative sharpness.

For example, consider the Zuiko 150 / 2 on 4/3 and the Canon 300 / 4L IS on 135, which are equivalent lenses on their respective formats -- that is, both have the same AOV and maximum aperture diameter.  The 150 / 2 tested at 49 lp/mm wide open, whereas the 300 / 4L IS tested at 36 lp/mm wide open.  Since the 4/3 sensor is 13mm tall, and the 135 sensor is 24mm tall, these figures translate to 49 lp/mm · 13mm/ih = 637 lp/ih for the 150/2 and 36 lp/mm · 24mm/ih = 864 lp/ih for the 300 / 4L IS.  In other words, even though the 150 / 2 is the sharper lens, the 300 / 4L IS out resolves it on the larger sensor.

However, since lenses for FF can be used on both crop and FF, the manufacturer MTFs overstate the lens performance on cropped sensors since they are reported at 10/30 instead of 15/45 (1.5x) or 16/48 (1.6x).  Another issue, of course, is that MTF charts usually show only wide open and f/8 performance, which means we are unable to use these charts to compare at the same, or even similar, DOF.  Thus, we need to rely on other tests to make system comparisons.  However, all the web "lens tests" are actually system tests.  That is, they evaluate the performance of the lens on a particular camera.  The problem with this form of testing is that it confounds both the sensor resolution and the effect of the AA filter with the lens performance.  So, while system "lens tests" are more useful for comparing the actual systems tested, they need to be continually updated to reflect current sensor resolutions and AA filter strengths.

Thus, the pixel count of the sensor, along with the AA filter, are both critical to system resolution.  Many subscribe to the myth that system resolution is largely limited by the lens in the case of modern sensors, but the reality is that current pixel densities are far from making systems "lens limited".  More pixels will always resolve more detail, but not necessarily as much more detail as the increase in pixels suggests.  For example, a 20 MP sensor will resolve less than double the detail (41% more linear resolution) of a 10 MP sensor unless the lens resolution is much greater than the sensor resolution.  As the pixel count continues to increase, the return becomes disproportionately smaller as the sensor resolution approaches the lens resolution.  However, as lenses get updated with newer and sharper versions, the limiting pixel densities will concomitantly increase.  In any event, for lenses with the same relative sharpness, the system with the greater pixel count will resolve more detail.  Thus, the level of detail in an image depends both the on how many pixels the sensor has and how well the glass is able to resolve those pixels.  This is also why FF glass will almost always perform better on FF sensors than on cropped sensors, unless the glass is significantly higher than the sensor resolution.  For the scenario when lens resolution is well beyond sensor resolution, the system performance will be primarily a function of the pixel count.

One issue that the lenses for FF sometimes suffer is that the sharpness is not as even across the frame as it is for smaller sensor systems.  This is usually not an issue at larger aperture diameters, except for far off-center composition, since the corners are rarely within the DOF at large aperture diameters.  However, some lenses far poorly in the corners even at relatively deep DOFs, such as the Nikon 70-200 / 2.8 VR, which is discussed in Myth 5 below.  In these cases, while the FF system may resolve better in the central region of the image, they may resolve worse in the corners.  Thus, while the overall sharpness is often the same or better with FF, the issue of evenness of frame needs to be considered, and taken on a lens-by-lens basis.

Thus, while smaller sensor systems usually have sharper glass, that does not necessarily give them sharper end results -- they need that extra sharpness just to "break even".  In practice, however, for the same AOV and DOF, the comparable glass for smaller sensors does not appear to hit that break-even point until the edges of the image, where, in some cases (usually UWA), they outperform FF glass in the extreme corners.  But since the larger sensor systems almost always have more pixels and resolve more detail in the central area of the image, if the extreme corners are not satisfactory, you have the option to frame wider and crop.  What is meant by "comparable glass"?  This is tricky, but generally lenses at, or near, the same price-point.  For example, we wouldn't call the Olympus 14-35 / 2 on 4/3 ($1840) or the Nikon 17-55 / 2.8 ($1130) on 1.5x "comparable" to the Canon 24-85 / 3.5-4.5 on FF ($310), since the prices are so different, even though they have nearly the same AOV and range of aperture diameters.  But we could call them comparable to the Canon 24-70 / 2.8L ($1255) even though the Olympus lens still costs significantly more and both have smaller aperture diameters, since we are now comparing the best against the best from each manufacturer that have comparable AOV ranges on their respective systems.  Alternatively, we could also call the Canon 24-105 / 4L IS ($1060) "comparable" since it is also "top glass" for Canon FF, has the same aperture diameter, and a zoom range that includes the AOVs of the aforementioned competitors' lenses.

When comparing systems, then, we must carefully articulate the reasons for choosing the lenses used in the comparison, since those reasons may be "invalid" depending on the use of the lens as it is rare to find two lenses from two different systems that enjoy the same range of AOV, aperture diameters, and price.
 

5) Larger sensor systems have softer edges and more vignetting than smaller sensor systems

Once again, as discussed in Myth #1, this belief is a result of people comparing systems at the same f-ratio rather than the same aperture diameter.  At the same f-ratio, the larger sensor system will have a larger aperture diameter, and thus a more shallow DOF, which will result in the areas of the scene outside the DOF being OOF (out-of-focus), as well as greater vignetting.  A more fair comparison for edge sharpness is to compare at the same DOF, or, often even more appropriate, at the lenses' sharpest settings, since it is rare that edge sharpness plays a role in high ISO photography.  However, it is disingenuous to compare edge sharpness and vignetting by artificially handicapping the larger sensor system with the same f-ratio as the smaller sensor system.

Of course, as we know, glass does not have the same sharpness across the image.  For example, the issue of telecentricity for UWAs causes a sharp drop in the MTF for many UWAs in 35mm FF lenses (the Nikon 14-24 / 2.8 being a remarkable exception).  Thus, the image may be "sharp enough" in the center, but too soft in the corners.  This is what happens when comparing, for example, 4/3 lenses with 35mm FF lenses.  The 4/3 lenses are sharper than the 35mm lenses (on average), but they need to be sharper to resolve the smaller pixels of their sensors.  And while 35mm FF glass is easily "sharp enough" for the center of the image, it is sometimes not "sharp enough" for the extreme corners (for some UWAs) even at the same DOF, despite the larger pixels.  Thus, near the edges, the sharper glass on a smaller sensor may outperform the less sharp glass on a larger sensor, but the amount of the corners where this reversal in sharpness occurs is dependent on which lenses are being compared, and must be taken on a lens by lens basis.  In fact, sometimes the larger sensor system will have the sharper corners (the evidence section of this essay gives examples).

An interesting case is the earlier version of the Nikon 70-200 / 2.8 VR which, according to a test conducted by DPR, performs significantly better in the corners on 1.5x than it does on FF even for the same perspective, FOV, and DOF (although, again, the rest of the image is sharper on FF).  However, the reviewer noted that this is almost certainly since the lens was optimized for 1.5x, since Nikon had no FF DSLRs, or even plans for one, at the time of the introduction of the lens, and is quite different from how Canon's 70-200 / 2.8L IS performs.  Another good example is the Canon 24 / 1.4L II on 1.6x vs the Canon 35 / 1.4L on FF.  The two have nearly identical performance at the same DOF in all areas that www.slrgear.com tests.  Of course, the 24 / 1.4L II is a newer lens and costs significantly more than the 35 / 1.4L, but such a comparison is fair to make since they are both top level lenses with the same FOV for their respective systems.  The question, then, is how would the 24 / 1.4L II on a 50D compare to a 35 / 1.4L on a 5D II?  The answer is that the 5DII image would likely deliver more detail, since it has more (and larger) pixels, as well as deliver an optional more shallow DOF, if desired.

However, back to the UWA situation, there is another angle to this story.  DSLRs with a 3:2 aspect ratio must shoot wider and then crop to match the FOV of a 4:3 aspect ratio, and this cropping all but eliminates the soft corners, if they even exist.  For example, for a Canon 5D to match the perspective, framing, and DOF of an Olympus E3 shooting at 7mm f/4, it would have to shoot at 12.5mm f/7.1 and crop to a 4:3 aspect ratio.  This would leave 10 MP on the 5D image, which would still match the pixel count and framing of the E3 image, while eliminating the extreme corners.  Likewise, the Canon 5DII (FF) has more pixels than the Canon 50D (1.6x), which also gives it more cropping latitude.  However, a 50D has more pixels than a 5D, so the 5D would have no such luxury, except if the lens were unable to sufficiently resolve the smaller pixels of the 50D.  In this case, a cropped image from the 5D, despite having less pixels, would likely retain the same, or even more detail, in the instances that we would need to frame wider and crop the corners out.

Stopping the larger system's lens down to normalize the DOF has the additional benefit of increasing the sharpness of the lens (especially in the corners) and reducing vignetting.  Many 4/3 proponents like to cite their glass as being "sharp wide open" with no significant vignetting.  However, "wide open" for 4/3 is "stopped down" for 35mm FF.  For example, let's compare the Leica 25 / 1.4 on 4/3 with the Canon 50 / 1.4 on 35mm FF.  With both lenses at f/1.4, the 4/3 lens will surely have the superior image in terms of sharpness and vignetting, but the 4/3 image with have a DOF that is be two stops deeper.  Stopping the Canon lens down to the same DOF (f/2.8) will produce a sharper image (even in the corners) with the same or even less vignetting.  If the 35mm FF system must also raise the ISO two stops to match the shutter speed, all this means is that the 35mm FF system loses its advantage in apparent noise, but it is not at a disadvantage for apparent noise (for sensors with the same efficiency and images with the same level of detail).

Of course, this is not to say that the corners and vignetting are exactly the same between systems at the same DOF -- it most certainly varies from lens to lens.  However, it makes little sense to compare corners and vignetting at different DOFs, and it is because people compare at the same f-ratio rather than the same DOF, that the myth of 35mm FF having softer edges and greater vignetting exists.
 

6) Assuming "equivalent" means "equal"

It is important to distinguish between "equivalent" and "equal" -- "equal" is a much stronger condition than "equivalent".  As stated in the Definition of Equivalence, equivalent images are images that share the following five attributes:

1) Same Perspective
2) Same Framing
3) Same DOF
4) Same Shutter Speed
5) Same Display Dimensions

As a corollary, equivalent lenses are lenses that produce equivalent images on different systems (same AOV and aperture diameter).  If the images were "equal", then they would also render the same amount of apparent noise, detail, the same color, the same bokeh, etc., etc., etc.  These elements of IQ are what make "equal" a much stronger condition than "equivalent", and are a function of the sensor efficiency, pixel count, lens design, AA filter, CFA, etc., etc., etc.  However, for different systems, equivalent images will be the images that look most similar in appearance.

The most talked about aspect of equivalent images is that the apparent noise will be the same.  This notion is predicated on the premise that since equivalent images are made from the same total amount of light, then the photon noise will be the same.  However, differences in sensor efficiencies will affect not only how efficiently the sensor captures the light that falls on it (and thus affect the photon noise), but also how efficiently that noise is processed (read noise).  In practice, differences in sensor efficiencies for sensors of a given generation are much less significant than the total amount of light that falls on the sensor, so equivalent images will have roughly the same total apparent noise.

There is one way to get images that are "equal", however -- if the larger sensor system had the same pixel density as the smaller sensor system, and we could use the same lens on both systems, then we could get equal images from both systems using the same settings (focal length, f-ratio, shutter speed, and ISO) and cropping the larger sensor image to the framing of the smaller sensor image.  While using the same lens and settings on the larger sensor system sometimes makes sense for macro and long telephoto, it is a rather odd way to use a camera in any other application.
 

7) Assuming Equivalence is based on equal noise

The most controversial visual property of equivalent images is that people incorrectly assume that Equivalence is based on equal noise.  Equivalence is based on the five principles listed above, which do not include noise, nor any other elements of IQ.  The primary elements in image noise, in order, are:

•  The Total Amount of Light that falls on the sensor (exposure · sensor area)
•  The percent of this light that is captured by the sensor (QE -- quantum efficiency)
•  The additional noise added by the sensor and supporting hardware (read noise)

Other factors, such as ISO and pixel count / size play a minor role in apparent noise compared to the above three factors.  Since equivalent images are made from the same total amount of light, and sensors of the same generation usually have similar QE / read noise, equivalent images from cameras of them same generation will usually have similar apparent noise for equivalent images.  People commonly believe that larger sensor systems have less apparent noise because they have better sensors, when, in fact, it is instead because they collect more total light for a given exposure.

Thus, breaking the properties of Equivalence down into the properties of the photo, lens, and sensor:

•  Photos with the same perspective, framing, display dimensions, and aperture diameter will have the same DOF
•  If we also include same shutter speed, then they will also have the same motion blur / camera shake, as well as be made from the same total amount of light
•  Differences in noise for equivalent images will primarily be a function of sensor efficiency and read noise, which are usually minor for sensors of the same generation

 

8) Larger sensor systems have less noise because they have larger pixels / higher ISOs result in more noise

The reason so many feel that smaller pixels result in more apparent noise is that smaller sensor systems usually have smaller pixels than larger sensor systems.  However, it is the size of the sensor and the diameter of the aperture that determines the total amount of light that makes up an image, and that, in combination with the sensor efficiency, determines the apparent noise, not the pixel size.  For fully equivalent images, where both the DOF and shutter speeds are the same, however, all systems will collect the same amount of light, and the system with the more efficient sensor will have the least apparent noise.

Furthermore, the belief that higher ISOs result in more noise is a common misinterpretation as to what is actually taking place.  Yes, higher ISO images usually result in more apparent noise, but this is because using a higher ISO results in either a faster shutter speed and/or a smaller aperture for a given apparent exposure.  The effect of either of these is to put less light on the sensor.  It is the lesser amount of light falling on the sensor that results in more apparent noise, not the higher ISO per se.  In fact, the higher ISO results in slightly less noise for a given exposure (that is, for a given f-ratio and shutter speed, the higher ISO setting will result in less apparent noise).  For example, if we took a pic of a scene at ISO 1600, and then took a pic of the same scene with the same f-ratio and shutter speed at ISO 100, and pushed the ISO 100 pic 4 stops in post to achieve the same apparent exposure, it would be more noisy than the ISO 1600 pic (discussed in more detail here).  The same would be true if we took a pic at ISO 1600 and pulled it down four stops to match the apparent exposure of the ISO 100 pic, but at the expense of four stops of highlight detail.  So, the cause of the greater apparent noise is the lesser amount of light falling on the sensor, not the higher ISO.

So while it is not news to anyone that a higher exposure results in less apparent noise, it is news that it is not the higher ISO setting of lower exposures that causes more noise.  Instead, it is the lesser amount of light falling on the sensor.  To minimize the apparent noise in an image, we want to maximize the exposure within the constraints of how much of the scene we are willing to oversaturate, the DOF / sharpness we wish to achieve, and the shutter speed necessary to offset motion blur and/or camera shake.

9) Comparing images at 100% rather than the same display dimensions

It is common for people to compare images at 100% -- that is, to compare images at the pixel level.  However, such a comparison would only make sense if each image was made from the same number of pixels.  For example, it makes no sense to compare a 4x6 print with an 8x12 print, just as it makes no sense to compare, for example, a 2000 x 3000 pixel image with a 4000 x 6000 image.  To properly compare image, we need to compare at the same display dimensions, that is, at the same print size and/or the same number of pixels.  Comparing otherwise is like comparing lap times of different vehicles where one has a longer course than the other.

Some argue that the process of resampling the image with the smaller pixel count to the dimensions of the image with the larger pixel count is unfair to the smaller image since the upsampling introduces a new variable into the comparison.  However, this variable is always introduced regardless.  That is, we either resize an image for web display, or the printer will automatically interpolate the image for printing, regardless of whether we upsample or not.  The most fair method for comparing at 100% is to carefully resample both images to a common dimension, so that neither system is favored.  Usually, we would choose a dimension at least as large as the larger image, since downsampling the larger image reduces detail.
 

10) Larger sensor systems gather more light and have less noise than smaller sensor systems

For the same AOV, lenses for larger sensor systems often have larger aperture diameters which gather more light than smaller sensor systems, and thus deliver less noisy images even if the sensor for the larger sensor system is less efficient (to a degree).  However, choosing a larger aperture diameter also results in a more shallow DOF, more vignetting, and softer corners.  For fully equivalent images, however, all systems gather the same total amount of light.  Thus, any differences in the apparent noise and dynamic range will be due to differences in the sensor efficiencies, and, contrary to popular belief, larger sensors are not necessarily more efficient than smaller sensors.  On the other hand, in situations where motion blur is not an issue, or even desirable, systems that have in-camera IS or IS lenses can gather more light by using a slower shutter speed and achieve an advantage in apparent noise over other systems lacking IS when a tripod is not used.

EXPOSURE, APPARENT EXPOSURE,  AND TOTAL LIGHT

As mentioned in the introduction of this essay, the concept of Equivalence is controversial because it replaces the paradigm of exposure, and its agent, f-ratio, with a new paradigm of total light, and its agent, aperture.  The first step in explaining this paradigm shift is to define exposure, apparent exposure, and total light.

The exposure tells us the density of the light that falls on the sensor, where density means the number of photons per unit area, and does not include ISO.  For a given scene, the exposure is determined solely by the f-ratio and shutter speed.  For example, two pics of the same scene, one at f/2.8 1/200 ISO 100 and another at f/2.8 1/200 ISO 400 will both have the same exposure, because the same number of photons per unit area will fall on the sensor regardless of what the ISO is set at (the ISO simply applies a gain to the actual exposure).

The apparent exposure is brightness of the final image after a gain is applied to the exposure (usually by adjusting the ISO), and is often what people mean when they say "exposure".   Using the same example as above, pics of the same scene at f/2.8 1/200 ISO 100 and f/2.8 1/200 ISO 400, while both will have the same exposure, their apparent exposures will be two stops apart.

Lastly, the total light is the total number of photons that fall on the sensor during the exposure.  In terms of IQ, this is the relevant measure, because the apparent noise and DR (dynamic range), which are both discussed in detail in the next section, are largely a function of total light.  For the same perspective and framing, the total light depends only on the aperture diameter and shutter speed (as opposed to the f-ratio and shutter speed for exposure).  Fully equivalent images on different formats will not have the same exposure, but they will share the same apparent exposure and be created with the same total amount of light.

Mathematically, we can express these three quantities rather simply:

Exposure                 = Intensity · Time
Apparent Exposure = Intensity · Time · Gain
Total Light              = Intensity · Time · Sensor Area

where intensity is defined in photons per unit time per unit area that fall on the sensor, and the gain is the amplification applied to the signal (for example, at ISO 1600, the gain is 1600 / 100 = 16x for a camera with a base ISO of 100).  Note that we can represent both apparent exposure and total light as functions of exposure:

Apparent Exposure = Exposure · Gain
Total Light              = Exposure · Sensor Area

The role of exposure in digital photography is nothing more than noise and/or highlight control.  A higher exposure will use more light to create the image, resulting in less apparent noise.  However, the more light we use to create the image, the more we will run the risk of oversaturation (blowing highlights).  The only way to increase the exposure is by using a slower shutter speed (thus increasing the chance/effects of motion blur and/or camera shake), using a wider aperture (thus decreasing the DOF and reducing the image detail/sharpness, particularly in the corners), or increasing the amount of light on the scene (e.g. flash photography).

Thus, it does not necessarily make sense to compare systems with the same exposure.  For example, if we compare systems at the same f-ratio and shutter speed (same exposure), the larger sensor system will have a more shallow DOF, which may, or may not, be desirable.  If we instead compare systems at the same exposure and DOF, the larger sensor system will have to use a larger f-ratio and a concomitantly lower shutter speed, which will increase the risk of motion blur and/or camera shake.

However, if we instead compare systems with the same total amount of light, and thus different exposures, then the DOF and effects of motion blur / camera shake will be the same.  Any differences in apparent noise will be due to differences in the sensor efficiencies, and not because the larger sensor system will require a higher ISO for the same apparent exposure.  In other words, for equivalent images, the visual properties will be rather similar, but for non-equivalent images, the visual properties may be radically different.  Sometimes this difference will favor the larger sensor system, sometimes it will not -- it depends on the scene.  However, if non-equivalent settings put the larger sensor system at a disadvantage, the photographer can instead always choose equivalent settings instead.  Indeed, the photographer is served best by choosing the optimal settings for their system, keeping in mind that what constitutes "optimal" is not only subjective, but highly dependent on the scene.  However, while equivalent settings are not necessarily optimal for the larger sensor system, these settings remove the subjectivity from the comparison, and are applicable to all scenes.

It is instructive to understand why the same f-ratio results in the same exposure for the same scene and framing, regardless of the focal length or format.  We have three things going on:

• The amount of light from the scene

• The amount of that light reaching the lens

• The amount of that light passing through the aperture

The amount of light from the scene depends on how wide we frame -- the wider we frame, the more light we will capture, since we are gathering light from a larger scene.  If we assume a uniformly lit scene, framing twice as wide, for example, will result in four times as much area, and thus four times as much light. The amount of that light reaching the lens depends on how far we are from the scene -- the further away we are, the less of that light that reaches the lens.  For example, if we are twice as far away, the density of the light reaching us is 1/4 as much (the "inverse square law" for light), so only 1/4 as much light will fall on the lens in any given time interval.  Finally the amount of that light that passes through the aperture depends on the size of the aperture.  If we double the diameter of the aperture, the area will quadruple, so four times as much light can pass through in any given time interval.  As it turns out, for the same f-ratio and shutter speed, the density of the light falling on the sensor (exposure) will be the same regardless of framing, perspective, and format (for a uniformly lit scene).

Let's work a few examples. Say a photographer takes a "properly exposed" pic of a subject 10 ft away at 40mm f/4 1/100 (aperture diameter = 40mm / 4 = 10mm). If they step back to 20 ft away and use 80mm f/4 1/100, the aperture diameter has doubled (80mm / 4 = 20mm) and the aperture area has quadrupled (area is proportional to the square of the diameter).  However, the amount of light from the scene reaching the lens is 1/4 as much since they're twice as far away. Since the aperture area is four times as much, it exactly compensates, and the same amount of light will pass through the aperture onto the sensor.

Alternatively, let's say they don't step back, but instead remained at 10ft and shot at 80mm f/4 1/100.  At 80mm, the framing will be twice as tight, and thus record only 1/4 the area of the scene as they would at 40mm. Thus, despite the fact that 1/4 as much light is reaching the lens, since the aperture area is four times as great, it exactly compensates once again.  An excellent video on this can be seen here.

The above two examples demonstrate how the same f-ratio and shutter speed results in the same exposure for a given scene regardless of focal length and format.  For a given format and scene, the same exposure will also result in the same total light.  However, for different formats, the same exposure does not result in the same total amount of light falling on the sensor.  The last example, then, is to compare equivalent images on different formats.

Let's say one photographer using 4/3 is using 40mm f/2 1/100 and another photographer with 135 is shooting the same scene from the same distance at 80mm f/4 1/100.  In both cases, the framing is the same (ignoring the minor differences in aspect ratio between the systems, which amounts to a mere 4% difference), the aperture diameters are the same (40mm / 2 = 80mm / 4 = 20mm), and the distances from the scene are the same.  Thus, the same amount of light will fall on the sensors.  Alternatively, the 135 photographer might instead use 40mm f/4 1/100 and shoot the scene from half the distance to maintain the same framing (but a significantly different perspective).  In this case, the aperture diameter is half as much as for the 4/3 photographer (40mm / 4 = 10mm as opposed to 40mm / 2 = 20mm), but since they are twice as close, four times as much light from the scene is reaching the lens.  Thus, once again, the same amount of light will fall on the sensor.  However, in both cases, the 135 sensor has four times the area as the 4/3 sensor, so the density of the light (exposure) is 1/4 as much (2 stops less).

On the other hand, if the photographer using 135 shot the same scene from the same position at 80mm f/2 1/100, the aperture diameter would now be 80mm / 2 = 40mm as opposed to 20mm.  Since the aperture diameter is twice the size, the aperture area is four times as large, and four times as much light will fall on the sensor.  But, since the sensor has four times the area, the density of the light on the sensor would be the same as the 4/3 sensor, so the exposures would be the same.  Likewise, if they moved in twice as close and used 40mm f/2 1/100, while the aperture diameters would be the same, four times as much light is reaching the lens, so, once again, four times as much light will fall on the sensor.

In other words, f/2 = f/2 in terms of exposure, regardless of format, but in terms of total light (and DOF), f/2 (on 4/3) is equivalent to f/4 on 135.  The reason, once again, that total light is so important, is because it is the total light that makes up the image (along with sensor efficiency) that determines the apparent noise, not the exposure.  Of course, for a given format, the distinction between exposure and total light is not necessary to make.  But when comparing between formats, the distinction is crucial.

Hence, the same f-ratio will result in the same exposure for the same scene and framing regardless of format.  However, two different formats cannot simultaneously have the same exposure and same total amount of light for the same perspective and framing, since the same amount of light is being distributed on different areas.  There is one exception:  if we use the same perspective and focal length on both formats, and then crop the larger sensor image to the same framing as the smaller sensor image, then we will have the same exposure and total light (as well as the same DOF) if we use the same f-ratio.  However, in practice, the only time I know of when an image from a larger format uses the same perspective and focal length, and is subsequently cropped to the same framing as the smaller format, is when the larger format is focal length limited, or for greater apparent magnification (macro).

In other words, the exposure matters only inasmuch as it is a component of the apparent exposure and total light -- it is not an important measure in and of itself.  That is, when we look at an image, we can see how bright or dark it appears (apparent exposure) and we can see the apparent noise and DR in the image (total light).  But we cannot see the exposure itself, so it is not important except as a means to an end.  This is a radical statement that many have difficulty coming to terms with, but it is a key point to understanding Equivalence, so let's take some time to discuss this critical point in more detail.

So why bother to meter the scene are all?  That is, why even bother setting an ISO if it's the total light that determines the apparent noise in the image?  For an ideal camera where the sensor counts each and every photon that lies on it and has no saturation limit, we would not have any need for an ISO setting.  We would simply set the aperture to achieve the DOF / sharpness we want and the shutter speed to whatever is necessary to avoid motion blur and/or camera shake.  We could then select whatever apparent exposure we wanted after the capture.  However, such sensors, of course, do not exist.  At the bright end of the spectrum, it's easy to see why we can't always simply set the f-ratio and shutter speed  -- the sensor could oversaturate (blow out) many portions of the scene, if not the entire scene, where we wish to record detail if the shutter speed is too low.  As a side note, it's important to understand that a "properly" metered scene does not mean that portions of it are not oversaturated, since the camera has limited DR (dynamic range) due to the finite saturation point.

Consider a landscape with the sun in it, for example.  The sun will most certainly be blown.  Thus, we choose a metering mode (or meter manually) so that we minimize the amount of the scene that is blown while maximizing the total light recorded to keep apparent noise down.  In other words, there is no objective "correct" exposure.  If there were, then cameras would only have one metering mode.  There is simply the best balance between oversaturation (blown highlights) and total light (low apparent noise).  For scenes with narrow DR, we want them exposed as far to the right (of the histogram) as possible to maximize total light and thus minimize apparent noise.  But for scenes with wide DR, we need to make a choice of how much of the scene we are willing to blow out to keep apparent noise down.

On the dark end, however, it's not so obvious why we need to set the ISO higher rather than merely record the image at base ISO and push the capture to the desired apparent exposure (either in-camera for jpg or via software in a RAW conversion).  Well, we could do that by using M mode, where the photographer sets both the f-ratio and shutter speed.  However, while the primary source of apparent noise in the midtones and highlights (photon noise) is usually determined by the total light, the sensor efficiency also plays a role in the read noise which plays a more dominant role in the shadows.  Ironically, higher ISOs actually have less apparent read noise than lower ISOs (see this demonstration).  If they didn't, then the camera would only have a single ISO and then push or pull for the correct apparent exposure.  Thus, we still want to know how the scene meters so that we can set the apparent exposure with the appropriate ISO to minimize the apparent noise.

The camera determines the exposure by directly measuring the brightness of the scene at the sensor via a dedicated photovoltaic sensor.  Since the camera meters (and focuses) wide open, the camera can now determine the relative difference between the brightness of the scene at the sensor with the lens wide open and the brightness of the scene at the sensor with the aperture at a particular value.  In combination with shutter speed and ISO, the camera can now determine the apparent exposure of the captured image.  If the aperture for the capture is predetermined (Av mode), the camera chooses the proper shutter speed for the desired apparent exposure.  If the shutter speed is predetermined (Tv mode), the camera chooses the proper aperture for the desired apparent exposure.  If neither aperture nor shutter speed are predetermined (P and Auto modes), the camera will adjust both according to a programmed balance between the two.  This is why some feel that FF has a DOF that is too shallow -- in P and Auto modes, FF cameras choose an aperture that is wider than what smaller sensor cameras choose in these modes.  For example, we will find that if a camera chooses f/5 1/200 on a 1.6x DSLR in P or Auto mode, it will choose the same on FF, rather than f/8 1/80 instead, which would result in the same exposure and DOF.  Of course, the complaint would then be that FF chooses a shutter speed that is too slow compared to crop, rather than a DOF that is too narrow.  The solution to this problem is to incorporate "Auto ISO", where the FF camera chooses the same apparent exposure in P and Auto Modes as the smaller sensor camera, rather than the same exposure.  For example, if the 1.6x camera were to choose f/5 1/200 ISO 100 in P or Auto mode, the FF camera would choose f/8 1/200 ISO 250, which would result in the same DOF (given the same perspective and framing), the same apparent exposure (same image brightness), and the same total light (same apparent image noise given equally efficient sensors).  Of course, now the FF camera loses its advantage in apparent noise over the 1.6x camera.  Regardless, these problems only occur in P and Auto modes.  In any event, exposure, per se, is irrelevant, since, in terms of the photograph, what we care about are the brightness, DOF, and apparent noise.

The bottom line is that exposure is important to the internal workings of the camera since, in terms of how the camera operates, since sensors have a saturation limit, and analog amplification (higher ISOs) results in less apparent read noise than using a lower ISO and pushing the exposure in post (discussed in the next section).  But, in terms of comparing systems, it is easier to conceptualize that for a given perspective and framing, the total light depends only on the aperture diameter and shutter speed.  For fully equivalent images, the aperture diameters and shutter speeds will be the same, and hence the total amount of light falling on the sensors will also be the same.  Any differences in noise or dynamic range between systems will be due to the differences in efficiencies of the sensors.  Thus, a larger sensor system gives the option of getting less apparent noise by trading shutter speed, DOF, or a combination of both, but that this advantage can be offset, and even reversed, in some instances when the smaller system has IS lenses or sensor IS that the larger sensor system does not.  However, these advantages are often overstated since there are situations when these trades cannot be made, and the larger sensor will not have an advantage in apparent noise over smaller sensors, or IS will be of limited use due to motion blur of moving subjects.

In the end, images are created with light, and it is the total amount of that light, and not the exposure, that is the important measure in terms of IQ.  Thus, it is important to understand that equivalent images are based on the same total light and not the same exposure.  At first read, this sounds ludicrous, but when we discover that we normally mean "apparent exposure" when we say "exposure", just as people usually mean "f-ratio" when they say "aperture", all falls into place.  These distinctions are key, and in combination with same output size, the basis of pretty much all the confusion in understanding equivalence.  We now bring the essence of a photograph back to the fundamental point that it is the total amount of light used to create the image, not the intensity of that light, that is central.

NOISE

Noise.  That's where the controversy over Equivalence begins.  People think that the "equivalence argument" is based on the presumption that Equivalent images have the same noise.  Even different cameras from the same format will have not have the exactly the same noise, either in quantity or quality, so, clearly, neither will Equivalent images from different formats.  However, for sensors of the same generation, Equivalent images will usually be fairly similar in terms of noise, and certainly more similar than images from different formats with the same exposure (same f-ratio and shutter speed).  Furthermore, it is of utmost importance to distinguish between noise at the pixel level, and noise at the image level.  Since pixels are the building blocks of digital images, this section begins with the discussion of noise at the pixel level, followed by a discussion of how these pixels, in aggregate, relate to the image as a whole (beginning here).

When people refer to noise in an image, what they mean is what this essay calls "apparent noise", which is the density of the noise in an image (NSR -- noise-to-signal ratio) and is often represented as a percent.  Often, we hear the term "SNR" (signal-to-noise ratio), which is the reciprocal of the NSR (SNR = 1 / NSR).  However, since a "noisy image" corresponds to a high NSR, and a clean image corresponds to a low NSR, whereas it is exactly opposite for SNR, it is less confusing to think in terms of NSR than SNR.

In terms of photography, there are two principle sources of noise:  photon noise and read noise.  Except for the deep shadows in an image, the photon noise is the primary source of noise in an image.  It is an inherent characteristic of incoherent light (the kind of light in almost all situations -- see the diagrams at the bottom of this page), and unavoidable -- one of those "Laws of Physics" things, as opposed to "an engineering challenge".  Light has the properties of both a wave and particle (called photons), and the noise is measured in terms of its particle characteristics.  The photons are collected and focused by the lens onto the sensor, where they are converted into electrons, and the signal is processed and recorded.  The only role the sensor plays in the photon noise is what proportion of the photons falling on the sensor are converted into electrons, since the electrons are the source of the electrical current that is processed by the hardware.  The read noise, discussed in more detail further down, is how much noise is added when collecting and processing the signal produced by the photons.

Let's begin with an analogy to traffic to understand photon noise.  Imagine we drew a line across a busy freeway, and counted cars crossing the line.  The number of cars crossing the line in any given time interval represents the total light falling on the sensor.  If we are talking about short intervals of time, like seconds, or, at most, minutes, then we can safely assume that there is a constant average flow of traffic.  But we also know that it is very unlikely that any two equal time intervals will contain exactly the same number of cars.  This variation from the "true average" is what we call "noise".  The larger the time interval, the larger the noise will be, but the less significant it will be in terms of the total number of cars counted.

For example, let's say that, on average, 10 cars pass by our line every second, and let's say we take three one-second samples that come up with 8, 11, and 13 cars.  These three "photographs" would have a noise of 2, 1, and 3 cars, respectively, which correspond to an apparent noise of 20%, 10%, and 30%.  Now, let's say we extend the time interval to ten seconds.  The expected number of cars would now be 100 cars (10 cars / second · 10 seconds = 100 cars).  Let's say that, once again, we take three "photographs" and count 93, 98, and 112 cars.  The noise is now 7, 2, and 12 cars -- much more than before.  But the apparent noise is 7%, 2%, and 12% -- much less than before.

So, the question is, then, how do we know what the "true average" number of cars is?  Well, we don't.  But what we do know is that the arrival of photons for incoherent light is described by a Poisson Distribution, that the standard deviation for phenomena that is described with a Poisson Distribution is equal to the square root of the mean (average), and that the standard deviation is the photon noise (often called the Shot Noise, which is a more general term).  Thus, since the magnitude of the noise is equal to the square root of the number of recorded photons, the noise increases with more light.  But since the NSR is the ratio of the noise and the recorded signal, the apparent noise decreases with more light.  For example, whenever we double the amount of light, the photon noise increases by 41%, but the apparent photon noise decreases by 41%.

A good way to visualize the role the lens and sensor play in photon noise is to think of rain falling on a flat surface through an opening.  The size of the opening corresponds to the aperture area, and length of time the rain falls corresponds to the shutter speed.  If a lot of rain is falling (lots of light), then the surface will quickly be covered in water having a very smooth appearance (low apparent noise).  However, let's imagine a much lighter rain (low light).  At first, we will see splotches of water here and there in a random and irregular pattern (noisy image).  As more water falls (the total amount of light increases), either by letting more time pass (longer shutter speed) and/or by making the opening larger (larger aperture area), the pattern becomes smoother (less noisy).

Let's now pack a large number of cups on the surface to collect the water.  The cups are analogous to the pixels on the sensor.  Larger cups will collect more water than smaller cups, but smaller cups will give us a better idea of the pattern of the rain.  If we compare an array of cups covering the same area, the water in the larger cups will be more uniformly filled (less noisy) than the smaller cups.  This is why sensors with larger pixels appear less noisy than sensors with smaller pixels.  However, the amount of water in the smaller cups gives us a much better idea of the pattern of rain that has fallen (resolve greater detail).  If the resulting "image" formed by the water in the cups is too noisy to our liking with the smaller cups, we could replace the smaller cups with larger cups and pour the water collected in the smaller cups into the larger cups, achieving the same smoothness we had if we had used the larger cups from the beginning (binning).  Alternatively, we can always siphon water from one cup and add it to an adjacent cup to smooth the appearance to look like the pattern of water formed with the larger cups which would still retain much of the detail of the original pattern, but with more smoothness (noise reduction).

There are, of course, important considerations.  How closely are the cups packed together?  How deep is each cup?  How thick is the glass in each cup?  These concerns are all analogous to the efficiency of the pixel, and has much to do with whether or not a sensor with more pixels can accurately achieve the lower noise of a sensor with fewer pixels via binning.  For the same technology, it appears as though this is very much the case.  Thus, smaller pixels offer more options of detail vs noise than do larger pixels.  For images composed with a large amount of light, smaller pixels basically resolve proportionately more detail with the same apparent noise as larger pixels.  It is only for images, or portions thereof, that are created with little light that sensors with smaller pixels much choose, via post processing, to have greater detail and more apparent noise, or less detail with less apparent noise.

Hence, it is not merely the total amount of light that falls on the sensor, but how many of those photons make it through the color filters into the pixel, how many of those are converted into electrons (QE -- quantum efficiency), and how many electrons a pixel can release (FWS -- full well saturation, or FWC -- full well capacity).  After that, we need to figure in the noise associated with the sensor and the supporting hardware (read noise).

For example, the Canon 5D has a QE of 23%, the Canon 5DII has a QE of 32%, and the Nikon D3 has a QE of 40%.  What this means is that the Nikon D3 will record twice as much (1 stop) more total light for a given exposure than the 5D, and 25% more light (1/3 of a stop) than the 5DII.  Thus, if there were no read noise for all three (which, of course, is not the case), the D3 would have 1 stop better noise performance than the 5D and 1/3 stop better noise performance than the 5DII.  It is important to note that as the QE improves with successive generations of cameras, there little more improvement to be made.  The maximum possible QE is, of course, 100%, which is but 1 1/3 stops more than the Nikon D3.

The read noise (R), is the sum of the pixel noise (P) and electronic noise (E).  The total noise (N) is the sum of the photon noise (p) and the read noise (R).  Since noise is a standard deviation, it does not sum linearly -- that is, R ≠ P + E.  For independent random phenomena, like noise, it is instead the variances (the squares of the standard deviations) that sum linearly:  R² = P² + E².  Thus, the read noise is the square root of the sum of the squares:  R = sqrt (P² + E²).  We can represent the total noise (N) as the sum of the photon noise and read noise, N = sqrt (p² + R²), or as the sum of the three components discussed (photon noise, pixel noise, and electronic noise):   N = sqrt (p² + P² + E²).

For example, the Nikon D3X pixels have a FWS of 68,000 electrons.  For the sake of discussion, let's say that the output voltage of the pixel is 1V at full saturation.  Then one electron would correspond to 1/68000 V ~ 14.7 µV.  Thus, a pixel noise of 66 µV would correspond to 66 µV / 14.7 µV ~ 4.5 electrons.  Finally, other signal processing, such as the conversion of the analog signal into a digital signal, add in additional electronic noise.  Again, for example, if the electronic noise were 37 µV, then it would correspond to 37 µV /  14.7 µV ~ 2.5 electrons.  The total read noise can be computed as sqrt (4.5² + 2.5²) ~ 5.1 electrons (as opposed to the 7 electrons we would get for a simple linear sum).  If we now consider a signal of 250 photons, and assume QE = 40%, then  250 · 0.4 = 100 electrons are released by the pixel.  Thus, the photon noise would be p = sqrt (100) = 10 electrons.  This would result in total noise for the pixel of N = sqrt (10² + 4.5² + 2.5²) ~ 11.2 electrons (again, considerably less than the 17 electrons of a linear sum), and an NSR of 11.2 / 100 ~ 11.2%.

Let's now consider the effect of ISO on read noise.  If the camera is at any setting other than base ISO, a gain is applied to that signal.  For example, if the base ISO is 100, then at ISO 1600, a 16x gain is applied to both the signal and the pixel noise.  This means that both the signal and the pixel noise are effectively 16 times larger.  In other words, while the number of electrons released by photon capture and the number of electrons randomly released by the pixel have not changed, the voltage associated with their release has been increased by a factor of 16, so it's as if the signal, photon noise, and pixel noise were 16 times larger. After the gain is applied (if at other than base ISO), additional electronic signal processing is performed.

Let's continue the example with the Nikon D3x.  If the ISO is set to 1600 (16x gain), the signal is equivalent to 16 · 100 electrons = 1600 electrons, the photon noise is equivalent to 16 · sqrt 100 = 160 electrons, and the pixel noise is equivalent to 4.5 electrons x 16 = 72 electrons.  We now add in the electronic noise of 2.5 electrons, which remains unchanged since no gain was applied to the electronic noise, for a total effective noise for the pixel of sqrt (160² + 72² + 2.5²) ~ 175 electrons, and thus an NSR of 175 / 1600 ~ 11%, which is lower than the apparent noise at ISO 100!  Of course, this doesn't mean that we should go around shooting higher ISOs as a matter of course, since most of the pixels will oversaturate.  The best way to minimize noise in an image is to get as much signal as possible, and that means to use as long a shutter speed or as large and aperture as we can for a properly exposed image.  But what it does mean is that we should not shoot ISO 100 and push the image in post to reduce apparent noise, because, in fact, the exact opposite will happen, as demonstrated here.

It's worth working an example with the 5DII to show the effect of sensors with different efficiencies, even for the same format.  The 5DII has a pixel noise of 2.5 electrons and an electronic noise of 20 electrons.  This gives us a read noise of sqrt (2.5² + 20²) ~ 20.2 electrons.  For an ISO 100 image and a signal of 100 photons, a pixel on the 5DII would have a noise of N = sqrt (10² + 2.5² + 20²) ~ 22.5 electrons as opposed to the 11.2 electrons for the D3X.  This will result give an NSR of 22.5 / 100 ~ 22.5% which is double the apparent noise of the D3X.  Thus, at base ISO, the D3X will have cleaner shadows than the 5DII.  But, at ISO 1600, the pixel noise for the 5DII is now 2.5 electrons x 16 = 40 electrons, and, as with the D3X, the signal of 100 photons translates to 1600 electrons and a photon noise of 160 electrons the 16x gain (again, for QE = 1).  This makes for a total noise of N = sqrt (160² + 40² + 20²) = 166 electrons -- less than the D3X -- making for an NSR of 166 / 1600 ~ 10.4% -- so slightly less apparent noise.  Thus, depending on the balance of the pixel noise and the electronic noise, for equivalent images, we may find the noise / DR performance of the systems to flip-flop depending on the ISO.  But the fact that larger sensor systems need to use a higher ISO for equivalent images (to maintain the same DOF and shutter speed) does not put them at a disadvantage.  In fact, quite the opposite as higher ISOs result in less apparent noise for a given signal.

It's more than worthwhile to note that the photon noise is the dominant source of noise in the midtones and highlights, whereas the read noise only becomes significant in the shadows.  In the examples above, we considered a signal of 100 electrons which is 4.3 stops up from the noise floor (bottom of the midtones) at base ISO.  Thus, even if there were no read noise whatsoever, the apparent noise in the image would still amount to 10%, and today's sensors are delivering NSRs of about 11% at the bottom of the midtones.  On the other hand, were we to consider a signal of 25 electrons, just two stops up from the noise floor (deep shadow), the effect of the read noise would have been more significant.  In this case, the photon noise, sqrt 25 = 5 electrons, is almost the same as the read noise.  Thus, the total noise for the pixel would be sqrt (5² + 5.1²) ~ 7.1 electrons.  While this is less than the 11.2 electrons for a signal of 100 photons, the apparent noise will be greater since there is less signal.  The NSR for 100 photons is 11.2 / 100 = 11.2% whereas the NSR for 25 photons is 7.1 / 25 = 28.4% -- 2.5 times the apparent noise.

The photon noise will dominate the image in the midtones and highlights, so the read noise plays little role except in the shadows.  Many systems deal with shadow noise simply by clipping them earlier, which, of course, removes the detail as well.  However, much more important than the shadow noise itself is banding, which is most noticeable and distracting in the shadows due to the fact that banding is a regular pattern as opposed to the random nature of noise.  For example, a sensor with low read noise and banding may well produce a significantly worse image in the shadows than a sensor with more read noise without the banding.  That said, the issue of banding is separate from the issue of noise, and not discussed in this essay as banding has nothing to do with sensor size.

There are, of course, other sources of noise, such as thermal noise, which plays a central role in long exposures, PRNU (Pixel Response Non-Uniformity) noise, which plays an important role in the highlights of the image, as well as other sources of noise.  So, noise, is, of course, even more complicated than this essay makes it appear, and for some specific forms of photography (such as astrophotography) we may find that the noise is very different for equivalent images in some situations, much in the same way that corner sharpness is very different for equivalent images in some situations.

However, there can be a more common disparity in noise due to the CFA (color filter array).  Color noise arises from the fact that the color filters are not perfect.  For example, the green filter may only admit 60% of the green light that falls on it, but also admit 10% of other colors that fall on it.  If we use a weaker filter to increase the transmissivity, we will reduce the luminance noise (more total light will pass through the filter and onto the sensor), but also increase the transmission error by concomitantly allowing a greater percentage of other colors to also pass through.  Thus, different manufacturers may strike different balances between luminance noise vs color noise in their choice of color filters in the same way they strike a different balance between pixel noise and electronic noise, as in the D3X and 5DII.

In short, the quantity of noise in an image is determined by the following three factors:

1) the total amount of light that falls on the sensor
2) how efficiently the sensor captures this light
3) how efficiently this signal is processed

For a given scene, perspective, and framing, the total amount of light is determined solely by the aperture diameter (where aperture diameter = focal length / f-ratio) and the shutter speed.  The role the sensor size plays in total light is that a larger sensor can absorb more light before oversaturating.  By using a larger aperture, we also force a more shallow DOF, because the aperture plays an integral role for both total light and DOF.  So, if a more shallow DOF is not desirable (and/or the concomitant effects that can occur, such as softer corners and more vignetting), then the only way to decrease noise is to decrease the shutter speed, at the risk of overexposing more of the image (this technique is known as ETTR -- expose to the right), and works best for images with a narrow DR (dynamic range).  The efficiency by which this light is captured is a function of many variables, primarily the CFA, the efficiency of the microlens covering, the percentage of the light transmitted by the color filters, and the QE (quantum efficiency) of the sensor.  The efficiency of the signal amplification is a function of not only the sensor, but the supporting hardware, such as the ADUs (Analog-to-Digital Units).

In terms of the total signal, the amount of light lost by traveling through the glass of the lens is insignificant, so there is no improvement to be made there.  The microlens covering over the sensor, which directs the light that falls on the sensor into the pixels is also near 100% efficiency for modern cameras, so, again, there is no improvement to be made there, either.  However, for Bayer sensors, each pixel records only one color (usually 25% red, 25% blue, and 50% green), which results in around another stop of light lost.  Modern DSLRs have a QE of anywhere from 30% - 40%, so there is about another 1.5 stops more gain that can be made there.  Thus, modern DSLRs with Bayer CFA's are around 2.5 stops away from the maximum possible improvement in photon noise, with one more stop for a scheme that does not require color filters.

While alternative schemes from Bayer may one day be able to give us this extra 1-2 stops improvement in photon noise and possibly render double the detail for a given pixel count as well, so far these technologies have run into significant problems of their own, such as the Foveon sensor where each pixel records three colors as opposed to one.  While ostensibly a very good idea, and one that may eventually take hold, in its current implementation it is actually more noisy than a Bayer CFA.  Another scheme is to use prisms or dichromatic mirrors to direct the different colors to three sensors ("3CCD" video cameras, for example), but these alternatives currently have significant technical problems of their own to overcome.

Dynamic Range (DR) and noise are two sides of the same coin.  The DR is the number of stops from the lowest "meaningful" signal (LMS) to the full well capacity (FWC):  DR = log₂ (FWS / LMS), where the LMS is the signal that results in a 100% NSR:  LMS = [1 + sqrt (1 + 4·R²) ] / 2 where R represents the read noise for a pixel.  For example, the Nikon DX3 has a read noise of 5.1 electrons at base ISO giving a LMS of [1 + sqrt (1 + 4·5.1²) ] / 2 ~ 5.6 electrons, which results in a DR of log₂ (68000 / 5.1) ~ 13.7 stops.  Likewise, for the Canon 5DII at base ISO, which has a read noise of 22.5 electrons and the same pixel saturation point as the D3X (despite the fact that the 5DII has larger pixels than the D3X), the LMS is [1 + sqrt (1 + 4·22.5²) ] / 2 ~ 23 electrons, giving a DR of log₂ (68000 / 23) ~ 11.5 stops.  Both of these computations correspond almost perfectly with DXO Mark's DR tests.  DR can be increased in one of two ways:  decreasing the read noise or increasing the full well capacity.  For example, if the D3X and 5DII had zero read noise, the DR would be log₂ (68000 / 1) ~ 16 stops.  If read noise and FWC scale with pixel area, the DR will be relatively unaffected by pixel size (if using a 100% NSR for the LMS).  In other words, pixel size has no effect on DR for equally efficient sensors with the same read noise, and there is no evidence of any correlation between pixel size, sensor efficiency, and read noise for existing sensors.

However, it is important to discuss an often overlooked facet of IQ -- tonal gradation, which is how smoothly the image levels change from pixel to pixel.  So, while two sensors may have the same DR, the sensor with more pixels will record a smoother tonal range.  Even if one sensor has more DR than another, as long as the DR for both is sufficient for the display media, the greater smoothness in tonal gradation may be a more important factor than DR, depending on the image.  This is the same situation we come across when comparing noise at the pixel level between two images.  Even though one image may have greater noise at the pixel level, the finer "grain" of the noise due to the smaller pixels will likely deliver the more pleasing image.  In other words, we cannot separate apparent noise and DR from the number of pixels that make up an image.  Contrary to popular belief, more pixels result in higher IQ, not lower.  How different, is, of course, subjective, and highly dependent on not only the scene, but the display dimensions of the final image.

So far, the discussion of noise has been focused on noise at the pixel level.  Most believe that because a larger pixel gathers more light for a given exposure, that larger pixels result in less apparent noise.  However, for a given sensor size, the smaller the pixel, the more pixels you have.  So, while a larger pixel will have less noise than a smaller pixel since it gathers more light for the same exposure, the image as a whole will be made from the same total amount of light regardless of the number of pixels.

This is not to say that the number of pixels has no effect on the apparent noise in an image.  In terms of the IQ of the final image, we are best served by viewing noise as a vector measure (multiple components) rather than a scalar measure (single valued).  Noise has both an amplitude and frequency.  The amplitude of the noise is the standard deviation from the mean signal, and the frequency of the noise is the number of samples taken.  For a given sensor size, smaller pixels will result in an NSR with a higher amplitude, but also a higher frequency.  The problem with noise comparisons on sensors with different pixel counts is that people often compare noise at different frequencies by comparing 100% crops rather than equal areas of the image displayed at the same dimensions.

Consider, for example, a 10 MP sensor and a 40 MP sensor of the same size and efficiency.  If we display both images at the same pixel density, say 300 PPI, the 40 MP image would have twice the dimensions of the 10 MP image and also appear more noisy.  But what if we instead displayed both images at the same dimensions?  They would still have the same amount of apparent noise for any given spatial frequency, but the 40 MP image would resolve spatial frequencies that would just be a blur with the 10 MP image (see this example).  If the noise at the higher spatial frequencies is objectionable, then we need only downsample and/or apply NR to the 40 MP image to match the noise at all spatial frequencies at the expense of the captured detail.  In fact, NR will often be able to preserve more detail than downsampling, and result in either a cleaner image of a given level of detail, or a more detailed image for a given level of noise.

In other words, the more detailed image will have more noise at higher spatial frequencies because it can resolve it.  A good way to think of this is that the noise from the scene already exists, and the closer you look, the more clearly you see it:

A sensor with more pixels will more clearly resolve the noise that already exists, whereas a sensor with fewer pixels will simply blur the existing noise.

If, on the other hand, the sensor with more pixels is less efficient than the sensor with fewer pixels, then we have to decide which we prefer -- more detail with more noise or less detail with less noise.  The choice, of course, has everything to do with how different the efficiencies are.  However, modern digital cameras show no correlation between pixel count and sensor efficiency, so there's no reason to opt for smaller pixel counts, at least in terms of noise consideration (there are, of course, other arguments against higher pixel counts for those that don't need the extra detail, such as memory usage, frame rate, processing time, etc.).

Let's work an example with two hypothetical sensors with the same size and efficiency (and, for the sake of simplicity, assume QE = 100% for each), but the pixels for Sensor B are half the size (1/4 the area) as the pixels of Sensor A.  Without any loss in generality, let us simplify the comparison to consider 1 pixel from Sensor A and 4 pixels from Sensor B, since they will each cover the same area of the sensor (and resulting image).  Let's say the FWC for the pixels in sensor A is 80000 electrons.  Then the FWC for the pixels of Sensor B will be 1/4 as much, since they have 1/4 the area -- 20000 electrons.  Likewise, if the read noise for Sensor A is 8 electrons, then the read noise for Sensor B will be 2 electrons.  The LMS for a pixel on Sensor A will be  [1 + sqrt (1 + 4·8²) ] / 2 ~ 8.5 electrons, and the LMS for a pixel on Sensor B will be [1 + sqrt (1 + 4·2²) ] / 2 ~ 2.6 electrons.  Thus, the DR for Sensor A will be log₂ (80000 / 8.5) ~ 13.2 stops and the DR for Sensor B will be log₂ (20000 / 2.6) ~ 12.9 stops -- virtually identical.

If we assume, then, that a pixel on Sensor A receives a signal of 64 photons, then, also assuming relatively uniform illumination, each pixel of Sensor B will receive a signal of 16 photons (1/4 as much).  This will result in a photon noise of sqrt 64 = 8 electrons for Sensor A and sqrt 16 = 4 electrons for Sensor B.   Thus, the total noise for a pixel of Sensor A will be sqrt (8² + 8²) ~ 11.3 electrons, and the total noise for a pixel of Sensor B will be sqrt (4² + 2²) ~ 4.5 electrons.  This will result in an apparent noise of 11.3 / 64 ~ 18% for each pixel on Sensor A and 4.5 / 16 ~ 28% for each pixel on Sensor B.  Thus, the price for the increased detail on Sensor B is more apparent noise.  In other words, if we displayed the image from Sensor B with double the dimensions as the image from Sensor A, and viewed from the same distance, it would appear to have up to double the linear detail (depending on the performance of the lens and the effects of diffraction softening), and 56% more apparent noise.

If, however, we instead downsized the image from Sensor B to the same output dimensions as Sensor A, then the effect would be the same as if we binned the pixels together.  Thus, the signal for four binned pixels would the same as one pixel from Sensor A, and the noise for a "binned pixel" on Sensor A would be sqrt (4.5² + 4.5² + 4.5² + 4.5²) ~ 9 electrons.  This gives us an apparent noise of 9 / 64 ~ 14% which is actually lower than the apparent noise of a pixel from Sensor A!  However, even more effective would be to instead apply NR (noise reduction) to the more detailed image, as this process is more efficient than binning in trading detail for a cleaner image.  Regardless, we need to keep in mind that more pixels do not result in more noise, but rather show the noise more clearly along with the extra detail.

In more simple terms, it meaningless to discuss apparent noise without considering the detail of the image.  Just as it makes since to compare the sharpness and detail of images at the same output size, it only makes sense to compare the noise in images at the same level of detail.  In other words, it makes no sense to say that one image has less noise than another, when it also has less detail, since NR (noise reduction) can be applied to the more detailed image to get a cleaner image at the expense of detail.  Thus, for a fair comparison of apparent noise between images, we would first apply NR to the more detailed image until it matches the level of detail of the less detailed image.  Interestingly, this means that the sharpness of the lens has an effect on apparent image noise since a sharp image is able to withstand more NR than is a soft image.  Even more important, in some instances, is focus accuracy, since even a small focus error can often lead to a significant loss of sharpness (please see this article).  In any event, it is often more an issue of how the noise is processed, rather than the quantity of noise, that is the primary issue.  For example, some manufacturers may select a higher black point that gives cleaner shadows, but also destroys all detail.  So, we must take care to once again consider apparent noise vs detail.

All that said, while we often speak of the apparent noise of Camera A vs Camera B, what many overlook is that it is not merely the quantity of apparent noise, but the quality of the noise, that is important.  It is not only possible, but likely, that one image may be have more apparent noise than another, yet have a much more pleasing look due to the quality of the noise.  For example, color noise is usually much more distracting than luminescence noise.  In addition, higher frequency apparent noise (finer grain) with the accompanying greater detail is usually considered much more appealing than lower frequency apparent noise (coarse grain) with less detail.  In other words, a more noisy image with a finer "grain" may well look better than a less noisy image with a clumpier "grain", depending on how close the overall quantities of noise are and the differences in detail rendered (an excellent demonstration of this is given here and here).  To this end, having more pixels, even at the expense of more apparent noise, can lead to a more appealing overall image, but this is most certainly subjective.  Of course, if the more detailed image has the same, or even less, apparent noise than the less detailed image after NR is applied to match the level of detail, then the system with the more detailed, yet more noisy, image will have a substantial IQ advantage by being able to better balance noise and detail in post.  Regardless, it is important to consider the types of images where apparent noise is even an issue.  This, of course, depends greatly on both the QT (quality threshold) of the viewer which is strongly influenced by print size and the viewer's "noise floor" -- that is, the point at which less apparent noise has no noticeable impact on the IQ of the image.  For example, while an ISO 100 image from 35mm FF has less apparent noise than an ISO 100 image from 4/3, the advantage in apparent noise of 35mm FF may be unnoticeable to the viewer at ISO 100.  Of course, the "noise floor" is likely a function of the print size and viewing distance as well.  For example, the apparent noise in an image may not be distracting in a 5x7 print, but become an issue in a 12x18 print.  Furthermore, the impact of the apparent noise is greatly dependent on the scene.  The apparent noise may go overlooked in areas with lots of detail, but stick out in areas with low detail, such as sky noise.

In addition to the mere quantity of apparent noise, we have to consider the balance of apparent noise in the different color channels, which is a function of the CFA (color filter array) that is used on the sensor.  One image may be less noisy than another overall, but exhibit significantly more apparent noise in one of the color channels which will give it a less appealing overall look.  Furthermore, both photon and read noise are completely random which makes for a significantly more pleasing appearance than banding which has a regular pattern.  In other words, while apparent noise is most certainly an important consideration in the IQ of an image, the quantity of the apparent noise most likely is less important than the quality of the apparent noise.

Lastly, we have to take into account that different RAW converters and JPG engines will deal with noise differently, so the noise/detail present in the final image is not necessarily an accurate representation of the actual hardware.  NR (noise reduction) can be applied even with a setting of "0", blacks can be clipped early to hide shadow noise (albeit at the cost of erasing detail), etc.  In fact, even the same RAW converter might use different settings for different cameras since the programmers decided that such-and-such a look was "better".  So, for sure, it is the final image that matters.  But the final image is not necessarily representative of the capability of the hardware.  In fact, many compare images on the basis of in-camera jpgs, which, of course, is a very poor way in which to compare the hardware performance.  But it is the best way to compare if you shoot jpg!

Thus, the advantage in apparent noise of larger sensor systems is limited to situations when they can use a lower shutter speed than the smaller sensor systems, such as good light, tripod use where motion blur is not a factor, flash photography when the balance of the light from the flash and the ambient light is not an issue, or when a more shallow DOF is used by trading f-ratio for ISO.  And, once again, all these factors only matter if we are talking about sensors that have the same, or nearly the same, efficiency.  Regardless, it is likely that it is the quality of the apparent noise, more so than quantity of the apparent noise, that is the primary factor in distinguishing between the IQ of Equivalent images in terms of apparent noise.  Just as with any element of IQ, apparent noise is very subjective, and different people will reach different conclusions about which image is more pleasing, even if the numbers clearly point to one image or the other as having more overall apparent noise.

LENS VS SENSOR

A digital image is made with a lens and a sensor, but which matters more?  The simple answer is that neither is more important than the other.  The sensor size and efficiency in combination with the aperture diameter of the lens determine how much light makes up the image for a given shutter speed, which is the primary source of apparent noise in an image.  The detail captured depends on both the sharpness of the lens and the number of pixels on the sensor (it is a common myth that modern sensors out resolve the lenses), and the size of the sensor.

Many are unaware that the size of the sensor contributes greatly to the sharpness of the image.  The reason the size matters is because an image captured on a larger sensor is magnified less for a given display size than an image from a smaller sensor.  The sharpness of the lens is measured in lp/mm (linear pairs per mm) or lw/mm (line widths per mm) where lw/mm is simply double the value of lp/mm.  For example, 50 lp/mm = 100 lw/mm.  The sharpness of the system is measured in lp/ih (linear pairs per image height) or lw/ih (linear width per image height).  To convert lp/mm to lp/ih, we simply multiply the value by the sensor height.  For example, let's consider a lens that resolves 50 lp/mm and convert to lp/ih:

4/3  :  50 lp/mm · 13.0mm / ih =   650 lp/ih
1.6x:  50 lp/mm · 14.8mm / ih =   740 lp/ih
1.5x:  50 lp/mm · 15.7mm / ih =   785 lp/ih
135 :  50 lp/mm · 24.0mm / ih = 1200 lp/ih

Thus, for a given sharpness of lens, a larger sensor will resolve more detail.  In reality, all lens test I know of are system tests, not lens tests.  They simply express the results in lp/mm or lp/ih.  That is, the native measurement it lp/ih, and they simply divide by the sensor height to get lp/mm.  The problem with this method, of course, is that the AA filter contributes a great deal to the sharpness of the system, as do the number of pixels on the sensor (see the DPR test of the 50/2 macro on the L10 vs E3 for an excellent example of the effect of the AA filter).  So, while DPR measures the results as lp/ih, another website may record the same measurements, divide by the sensor height, and record lp/mm.  Thus, we really don't know how sharp the actual lens is from lens tests.

However, the lenses specifically designed for smaller formats are often sharper than the lenses designed for larger formats.  Whether or not this is enough to overcome the difference in sensor size depends on the specific lens-sensor combination, and, of course, whether or not we are comparing at the same AOV and DOF.  For example, consider the Zuiko 150 / 2 on 4/3 and the Canon 300 / 4L IS on 135, which are equivalent lenses on their respective formats -- that is, both have the same AOV and maximum aperture diameter.  The 150 / 2 tested at 49 lp/mm wide open, whereas the 300 / 4L IS tested at 36 lp/mm wide open.  Since the 4/3 sensor is 13mm tall, and the 135 sensor is 24mm tall, these figures translate to 49 lp/mm · 13mm/ih = 637 lp/ih for the 150/2 and 36 lp/mm · 24mm/ih = 864 lp/ih for the 300 / 4L IS.  In other words, even though the 150 / 2 is the sharper lens, the 300 / 4L IS out resolves it on the larger sensor.

Other factors to consider are flatness-of-field (how the image sharpness varies from center to corner) and also the contrast.  For example, DPR compares with a contrast level of 50% (MTF-50), and manufacturer MTF charts typically use computer simulations to compare contrast at 10 lp/mm and 30 lp/mm (Olympus uses 20 lp/mm and 60 lp/mm).

In addition, it's worthwhile to note that the system with the sharper lens has the option of trading the extra detail for less noise by using stronger NR (noise reduction).  In other words, lens sharpness indirectly affects the noise performance of the camera.  However, there are other characteristics of IQ that are entirely dependent on the lens, such as bokeh, flare, and distortion, which often matter more than sharpness alone.  And then there are other IQ attributes, such as color, vignetting, and PF (purple fringing), which are properties of both the sensor and lens.  However, as the latter two elements of IQ are relatively simple fixes in PP (and can even be automatically corrected for either in-camera or with many RAW converters), they are of significantly lesser concern.

Existing lenses for larger sensor systems almost always allow for a larger maximum aperture diameter, and thus the option for a more shallow DOF, if desired, and less apparent noise.  However, the downside is that lenses for larger sensor systems (usually wide-angle) sometimes have a sudden drop-off in their MTF curves at the edges of the image circle, which makes for softer corners, although this is greatly mitigated, and sometimes even reversed, when compared at the same DOF and same output size.  Nonetheless, as the Nikon 14-24 / 2.8 has shown, wide open sharpness corner-to-corner (presuming, of course, that the entire scene is within the DOF) is certainly attainable for 35mm FF UWA.  On the other hand, other modern lenses, such as the Canon 14 / 2.8L II, do not hit full stride in the corners until f/5.6.  However, this may be a design compromise necessary to deliver desired bokeh characteristics and low distortion, or it could be a cost-cutting solution where designers felt that sharp corners are unnecessary at f-ratios less than f/5.6 on 35mm FF (as the corners are often outside the DOF regardless), or a compromise required to keep the lens small and compact.  Whatever the case may be, the Nikon lens has shown that sharp corners on 35mm FF UWA is not endemic to the format, but a choice in the design of the lens.  What compromises need to be made for these choices, is something only the lens designers know.

Nonetheless, we can summarize the advantages of lenses for larger sensor systems as follows:

1)  Most lenses offer larger aperture diameters, which allows more light, and thus less apparent noise and more DR, as well as the option of a more shallow DOF.
2)  Lenses for larger sensors often resolve more detail than lenses for smaller sensor systems, for lenses at a comparable price-point or grade.
3)  For a given maximum aperture diameter, lenses for larger sensor systems are usually lighter and less expensive.
4)  Larger sensors often have more pixels, allowing for more detail, depending on the lens.

and the advantages of lenses for smaller sensor systems:

1)  Often smaller for the same reach
2)  Usually lighter since they have smaller maximum aperture diameters for similar AOVs.
3)  Sometimes sharper in the corners (usually UWA).
4)  Usually closer MFDs (minimum focusing distances) for the same AOV.

The bottom line is that comparing lenses or sensors independently leads to inaccurate perceptions about the images that different systems are able to produce.  In terms of the IQ of the final image, it is the specific lens-sensor system that needs to be evaluated, not one or the other.

MEGAPIXELS:  QUALITY VS QUANTITY

In earlier times in digital photography, most of the talk about IQ revolved about megapixels, and it was commonly believed that more were better.  When Foveon came out with their sensor, where each "sensel" records three colors as opposed to the one color per sensel of the Bayer sensors, the debate between quality and quantity began.  While the debate between Bayer and Foveon is not discussed in this essay, most seem to agree that, in most circumstances, a Foveon sensor resolves as well as a Bayer sensor that has twice the pixel count (it is much more complicated than this, but, as I said, that discussion is left out of this essay).

In recent times, however, as the megapixels of Bayer sensors have continued to rise, there are many who believe that there is a "sweet spot" for the number of megapixels, and exceeding that number actually reduces IQ.  This belief arises primarily from the notion that more pixels increase apparent noise and decrease DR (dynamic range) since, for a given sensor size, more pixels means smaller pixels.  Unfortunately, the propensity of many to compare images at 100% rather than at the same display dimensions makes for incorrect conclusions.

It should be obvious that more pixels will resolve more detail.  But what is not obvious is how much more detail more pixels will resolve.  Under ideal circumstances, the increase in resolution will be proportional to the increase in pixels.  For example, a 50% increase in pixels will ideally result in 50% more captured detail (or a 22% increase in linear resolution, since sqrt 1.5 ~ 1.22).  However, whether or not this magnitude of increase can be obtained depends on the sharpness of the lens, the speed of objects in the frame (motion blur), and the steadiness with which the camera is held (camera shake).  For example, if a 10 MP sensor needs 1/100s to avoid motion blur and camera shake, then a 15 MP sensor would need 1/125s to resolve the pixels equally as well (lens permitting, of course).

On the other hand, sensors with smaller pixels do not need as strong an AA filter as sensors with larger pixels.  Thus, smaller pixels will resolve more detail not only by the virtue of their being more pixels, but because they require less blur to ameliorate aliasing.  In fact, having a sensor with pixels smaller than the lens can resolve can be seen as a good thing, as no AA filter would be necessary -- the blur of the lens would act as the AA filter.

However, of more concern to many is not so much detail, as it is apparent noise.  For many, even 10 MP is easily enough detail for the size they display their images, but the high ISOs are not as clean as they would like them to be.  So, the question is simple:  do smaller pixels result in more apparent noise?  This depends on how the efficiency of the sensor scales with pixel size.  To date, there is no evidence that pixel size has any correlation with efficiency.  In practice, however, since no two sensors of the same quality and technology made at the same time by the same team with different pixel counts, we cannot say for certain.  However, unless the sensor with smaller pixels is less efficient, it's safe to say that more pixels result in higher IQ (see this demonstration).  However, whether or not that higher IQ will be realized is another matter, and depends greatly on how large the image is displayed.

One simple example demonstrating pixel quality vs pixel quantity would be to shoot the same scene from the same position with both a Canon 5D (FF) and Canon 40D (1.6x) using the same lens and the same settings (focal length, f-ratio, shutter speed, and ISO).  Then crop the 5D image to the same framing as the 40D image, and print (or display) both at the same size.  The crop from the 5D image will look identical to the 40D image except it will have less detail, as the crop will have 5 MP as opposed to the 10 MP of the 40D image.  At 100%, the 40D image will appear more noisy, but that is simply because you can see the noise more clearly, as it is resolving more detail.  In other words, the noise already exists, as opposed to being created by the smaller pixels, since the dominant source of noise in an image is the photon noise, but the smaller pixels resolve it more clearly.  However, we can apply NR to the 40D image to sacrifice the detail and match the 5D crop in terms of apparent noise and detail, if we choose.  But we cannot sacrifice the apparent noise and DR of the 5D crop to match the detail of the 40D image.  Furthermore, the final output size of the image cannot be overlooked as playing a significant role in how many pixels are "enough".  For example, as only 8.64 MP are necessary for 300 PPI on an 8x12 print, the differences in IQ between the 40D image and 5D crop may be entirely insignificant at that print size.  The advantage of more pixels would only make a difference in larger prints, and even then, only when detail matters more than apparent noise and DR.

The relationship between pixel count and detail is further complicated by the ability of a lens to resolve a pixel, which depends not only on the pixel size, but on the aperture diameter of the lens and how far the pixel is from the center of the image.  For example, consider the Canon 50 / 1.4 on a Canon 5D (click the FF tab).  At f/1.4, it is somewhat soft in the center and has horrific corners (of course, at f/1.4 the DOF is so shallow that the corners are all but meaningless since few, if any, scenes at f/1.4 would have the corners are within the DOF, unless, of course, the focal point is in the corner).  By f/2, the center is looking very good, but the corners are still atrocious (although, as above, the DOF at f/2 is still so shallow that this is a non-issue).  By f/2.8, the lens is performing admirably across the image circle, and by f/5.6, where the corners could well matter, the lens is all but perfect (in terms of sharpness across the frame).

But the 5D has 12.7 MP.  What might we expect on the 5DII, which has 21 MP?  While www.slrgear.com has not run the test on the 5DII (or 1DsIII, which also has 21 MP), DPR has tested the Canon 50 / 1.4 on the 1DsIII (also 21 MP).  The results show that while relatively soft wide open, by f/5.6 the lens can still resolve the smaller pixels all the way to the extreme corners.  In other words, the extra pixels of the 5DII may not help much, if at all, at f/1.4, but may show considerable increase in detail by f/5.6.  Thus, depending on the lens and the aperture diameter used, the benefits of a greater pixel density can greatly vary.  Further complicating the issue is that diffraction softening will begin to take effect earlier on a 5DII compared to a 5D since it has smaller pixels, which will mitigate the advantage of the greater pixel density at the opposite end of the DOF range as well.  Lastly, we need to consider camera shake and motion blur.  To take advantage of smaller pixels, the shutter speed must be proportionally higher for moving objects, which may require an increase in ISO (thus increasing apparent noise).  For stationary subjects, while we may not need to up the ISO for a higher shutter speed for reasons of motion blur, we may need to do so to account for camera shake in many circumstances.

So, depending on the lens and aperture diameter used, and where the image needs to be sharp (center vs corners), and the ISO needed to maintain the necessary shutter speed, the IQ advantage of more pixels may vary greatly. Nonetheless, regardless of the lens performance, more pixels will not produce an inferior image at the image level, so long as the sensor is at least as efficient.  For example, downsampling the 21 MP 5DII image to the 12.7 MP of the 5D, or the upsampling the 5D image to the dimensions of the 5DII image, any lens will would perform better overall on the 5DII due to the larger pixel count.  However, the increase in detail will never be the full 65% improvement (29% improvement in linear detail) that the differences in pixel counts suggests, and will be highly dependent on the both the lens and the aperture diameter used, as well as the circumstances of the shot.

This brings us to the consideration of the sensor size.  For a given pixel count, a larger sensor will have larger pixels and thus be less demanding on the lens.  Smaller sensors, on the other hand, require sharper lenses to resolve the smaller pixels.  The question becomes, then, if the lenses for the smaller sensor system are sharper by a great enough margin to overcome this disadvantage.  In fact, some might argue that they might be more than sharp enough and even be able to resolve more pixels than the larger sensor system.  Well, of course, this depends on the individual lenses being compared.  However, from numerous comparisons between lenses on various formats at www.slrgear.com as well as the lens tests on DPR, I have found that at the same AOV and DOF (not the same focal length and f-ratio), that lenses for larger sensor systems usually outperform lenses on smaller sensor systems overall for the same tier of lenses, the exceptions usually being the extreme corners of some UWA lenses (but not always).  Lastly, we need to discuss what effect the smaller pixels due to larger pixel counts have on apparent noise.  There may be more apparent noise, depending on how the read noise scales with the size of the pixel.  Either way, however, smaller pixels will render more detail.  So, even if there is more apparent noise with more pixels, by applying NR (noise reduction) to the more detailed image to match the detail of the image with the smaller native pixel count, we can regain the noise performance by sacrificing the additional detail.  In fact, we almost always end up with more detail for the same levels of apparent noise.

One might ask what the utility of the additional pixels is if NR must be applied to match the apparent noise level and DR of an image made from fewer, but larger, pixels.  The answer is simple:  depending on the image, detail often matters more than apparent noise -- with the more detailed image we have the option of more detail with less apparent noise, or the same detail with the same apparent noise (and, as mentioned earlier, usually more detail for the same apparent noise).  The system with the smaller native pixel count never has the option for more detail.  In terms of DR, it depends on how the pixel noise scales with pixel size.  For equally efficient sensors, the pixel noise scales with the area of the pixel, so DR remains unchanged.  However, even if the sensor with smaller pixels were less efficient, we need to consider the balance between DR and the fineness of the tonal gradations.  For example, consider two sensors:  Sensor A has 40 MP and Sensor B has 10 MP and assume that the smaller pixels of Sensor A are less efficient than the larger pixels of Sensor B.  Thus, Sensor A will have more apparent noise than Sensor B, since it will have more pixels, as well as less DR, but it will also render more detail and have finer tonal gradations.  However, the detail and finer tonal gradations can be traded for lower apparent noise and greater DR with the use of NR.

For example, if a pixel of Sensor A saturates at 20000 photons, then a pixel from Sensor B will saturate at 80000 pixels, since it has four times the area.  Let's assume that 60000 photons land on a pixel of Sensor B.  In that same area, then, 60000 photons would land on 4 different pixels of Sensor A.  Let's say that 10000 photons land on one pixel, 18000 land on another, 19000 on another, and thus 13000 on the last.  So, while the pixel from Sensor B is well below its saturation limit of 80000 photons, two of the pixels from Sensor A are blown (oversaturated).  However, even with two blown pixels, Sensor A will deliver a smoother tonal gradation, even though it will have less DR.  But with proper downsampling or by using NR, Sensor A will be able to match the apparent noise and DR of Sensor B, at the expense of the extra detail.

So, are more pixels still better?  In terms of IQ, yes.  But the IQ advantages of more pixels is not as extreme as the difference in pixel counts seems to suggest, due to the fact that lenses are not infinitely sharp and that, unless a higher shutter speed can be used without raising the ISO, the effects of motion blur and/or camera shake may degrade the additional detail afforded by more pixels.  But unless the system with the higher pixel count has a less efficient sensor, it will never have lower IQ, and even with a less efficient sensor, may still render higher IQ in many instances.  The question, then, is at what point the additional IQ of more megapixels passes the point of diminishing returns and becomes more of a burden than it's worth, especially given that more megapixels requires more memory, more time to process to realize the potential, and likely a lower frame rate.  The answer to that question, of course, depends on the size the image is displayed, the quality of the lenses being used, and the QT (quality threshold) of the viewer.  Given that 8.6 MP results in 300 PPI for an 8x12 inch print, for many, we are well past the point of diminishing returns already.

EQUIVALENT LENSES

The definition of an equivalent lens is a lens that produces an equivalent image that another lens produces on another format.  In other words, equivalent lenses will have the same aperture diameter (as opposed to f-ratio) for the same AOV.  For example, the 135 / 2L on 35mm FF is equivalent to an 85 / 1.2L on 1.6x and a 70-200 / 4L (IS) on 35mm FF is equivalent to a 35-100 / 2 on 4/3.  Many people very much dislike this terminology and consider it "misleading" and even "dishonest".  Typically, they feel that it is sufficient to think in terms of AOV and exposure, and ignore the importance of DOF, as well as the significance of difference between exposure and total light.  To this end, we often hear people saying "f/2 is f/2 is f/2" regardless of format.  However, that statement is every bit as misleading as saying "50mm is 50mm is 50mm" regardless of format.  Just as 50mm yields different AOVs on different formats, f/2 will result in a different aperture diameter for a given AOV and thus a different DOF as well as admitting a different total amount of light onto the sensor which will result in different quantities of apparent noise.

Recalling one of the top ten misunderstandings of equivalence, that "equivalence" does not mean "equal", we need to realize that equivalent lenses are not identical, of course, and there can be important operational differences between them.  For example, while the Canon 135 / 2L on FF is equivalent to the 85 / 1.2L II on 1.6x, the 135 / 2L is much less expensive, larger (but lighter), and focuses much faster.  Another example is the Canon 24-105 / 4L IS on 35mm FF.  Its closest equivalent on 1.6x is the 17-55 / 2.8 IS (equivalent to a 28-88 / 4.5 IS on 35mm FF).  But while the 35mm FF lens has the advantage of more range at both the wide and long end, it suffers the disadvantage of not being able to use the high precision f/2.8 AF sensor as the 17-55 / 2.8 IS on 1.6x can.  Nonetheless, while there most certainly may be operational differences in many instances, equivalent lenses produce equivalent images on their respective formats.  Of course, we must remember that "equivalent" does not mean "equal", and there may be important IQ considerations to consider between equivalent lenses that are not subject to equivalence, such as bokeh, flare, PF, etc.

An important consideration when choosing systems is to compare available lenses in equivalent terms of the same format.  Often, lenses may have the same AOV, but not the same maximum aperture diameter.  For example, the Canon 50 / 1.4 has the same AOV as the 30 / 1.4 on 1.6x and the 25 / 1.4 on 4/3, but it's max aperture diameter (50mm / 1.4 = 36mm) is larger than either.  To match the aperture diameter of the 50 / 1.4 on FF, we would need a 30 / 0.9 on 1.6x and a 25 / 0.7 on 4/3, neither of which exist.  On the other hand, smaller formats will often have smaller and lighter lenses when such larger aperture diameters are not needed.  For example, the Canon 400 / 5.6L on 1.6x is equivalent to a 640 / 9L on 35mm FF, which does not exist.  And even if it did, the AF system would not function at that f-ratio.  In fact, 35mm FF does not even currently have a 600 / 5.6L as an option.  Thus, FF shooters are "forced" to use a 600 / 4L IS, which is huge and expensive, or crop the images for shorter glass yielding less pixels on the subject, and thus less detail.

So while lenses for 35mm FF typically have the advantage for more shallow DOFs (if desired) and more light gathering ability, FF often lacks smaller and lighter lenses with smaller aperture diameters to achieve the same reach as smaller formats.  In addition to this drawback, FF lenses sometimes have the same minimum focusing distance for the same FL, not for the same effective reach.  For example, the minimum focusing distance of the 135 / 2L is 0.9m whether on 1.6x or 35mm FF, but 135mm on 1.6x has an EFL (effective focal length) of 216mm on 35mm FF, and the 35mm FF equivalent of the 135 / 2L on 1.6x is the 200 / 2.8L, which has a minimum AF distance of 1.2m.  Hence, the smaller sensor system can almost always frame more closely.

And then there are some lenses which have no equivalents in either AOV or DOF between systems.  For example, the Tokina 10-17 / 3.5-4.5 FE on 1.6x corresponds to a 16-27 / 5.6-7.1 FE on 35mm FF.  There is no lens even remotely like that available, and, even if there were, it would not AF on anything less than a Canon 1-series body.  Also, the 70-300 / 4-5.6 on 4/3 corresponds to a 140-600/ 8-11 on 35mm FF.  If such a lens were made for 35mm FF, and it would AF, I'm sure there are many 35mm FF shooters who would love such a lens.

Lastly, there is the notion that some lenses on one system are "superior" to the lenses available on another system.  So, while this section has discussed "equivalent lenses" in terms of AOV and DOF, it has not addressed "equivalent lenses" in terms of IQ.  First of all, it's important to understand what role the sensor size plays in terms of image sharpness.  A sharper lens on a smaller sensor does not necessarily deliver a sharper image (see Myth #4 for more explanation on this point).  Of course, this must be taken on a lens by lens basis, and the individual properties of sharpness, bokeh, distortion, flare resistance, etc., may well be superior for the lens of one system for some elements of IQ, and inferior for other elements of IQ.  In addition, we must also take care to compare the lenses as they perform on their respective systems, rather than on their own merits.  For example, it makes no sense in terms of the capability of the system to say "Lens A is sharper than Lens B" if it does not produce a sharper image for equivalent settings (same AOV, DOF, and output size) on the format the lens is used on, since the sensor size plays a significant role in how well the lens performs on a given system.  Likewise, if we are comparing bokeh or distortion, we have to be careful to compare at the same AOV and DOF, lest me make inaccurate assessments about the lens performance on the system it will be used on.

Of course some might argue that the lens lasts longer than the sensor technology, and consider evaluating the lens alone in terms of an "investment" in the system.  However, how a lens performs on a system has everything to do with the size of the sensor and not the technology level of the sensor.  The technology level of the sensor determines noise performance (and, to some extent, vignetting and chromatic aberration), not sharpness, bokeh, flare, distortion, etc. 

In the end, we must consider the available lenses for a system when choosing which system best suits our needs, but we must understand how these lenses perform on the sensor that will be recording the images.  To this end, understanding that equivalent lenses produce equivalent images (same AOV and DOF), along with the operational differences between the lenses, is an important consideration when choosing a system.

IQ VS OPERATION

There are, of course, many other operational advantages to smaller sensor cameras, not the least of which are size, weight, and cost.  For example, compacts fit in your pocket and even have video, which give them quite an advantage over DSLRs in that regard.  It should not merely be noted, but stressed, that for the sizes that most people print, and the DOFs that most people prefer, IQ is likely the least of their concerns with modern cameras, and operation is the overwhelming difference by which to choose.

Among the most critical of the operational differences to consider is the camera's AF system -- IQ means nothing if the pic is OOF (out-of-focus) or a focus lock cannot be achieved.  In fact, it's amazing how much attention the megapixel counts and how little the AF system receives, since even a tiny focus error can greatly reduce the detail of an image.  A great write-up on the importance of accurate AF is given here at www.slrgear.com.  To a lesser degree, but sometimes just as important, if not more so, is shutter lag -- the time lag between when the shutter is depressed and when the capture is taken.  For some types of photography, the moment can be lost in that split-second (although, from personal experience, more often than not it is the hesitation from the operator that is usually the dominant factor for missed shots of narrow opportunity).   In addition, it must be said that the availability of a feature, such as in-camera IS, will most certainly, under many circumstances, go a long way to creating a higher quality image.  Yet in-camera IS is available only on some 1.5x and 4/3 DSLRs as well as compact digicams.  FF DSLRs currently rely on in-lens IS which is not available for all, or even most, lenses (Sony's upcoming FF DSLR may change all that).  Of course, one can argue that the larger sensor DSLR has better noise performance, but this can only be achieved if there is enough light to sacrifice shutter speed, or if one sacrifices DOF to use the same f-ratio.  Furthermore, certain lenses that are specifically designed for the cropped DSLRs may have various IQ advantages (less PF, flare, distortion, etc.).  In addition, if the smaller sensor camera has a more efficient sensor, it will have less apparent noise for the same DOF.

And, of course, none of these points matter if you don't have the camera with you in the first place.  In other words, if the system is too heavy to carry up the mountain, or the weather is wet and the camera is not weather sealed, or the camera is too large to be used discreetly, then it really doesn't matter how good the IQ is if the camera is not going to be used.

But we must keep in mind that images created from systems with smaller sensors do not have more DOF, do not have sharper corners (except, in some instances, usually wide angle, in the very extreme corners for the same DOF), do not vignette less, and do not suffer diffraction softening better, than their larger sensor counterparts.  These myths are wholly attributable to people comparing systems at the same f-ratio rather than the same DOF.

In general, the advantage of larger sensor systems are basically shallow DOF and higher IQ for times when shutter speed may be "safely" lowered to maintain the desired DOF.  On the other hand, smaller sensor DSLR systems often have size, weight, operational, and cost advantages that outweigh the DOF and IQ differences between the systems.

It is always paramount to compare the systems in terms of IQ, operation, and available lenses/accessories.  Each individual must the seek the best balance of these considerations and choose the system that best meets their needs.

HYPOTHETICAL COMPARISON

The purpose of this section is to demonstrate the principles of equivalence without muddying the water with differences between various systems in terms of operation, available lenses, and unequal pixel counts.  Of course, the operation, available lenses, and pixel count are not only important considerations, but many times the primary consideration, in choosing a system.  However, by eliminating these variables in this section, it is my intent to more clearly illustrate the relationship between the sensor and the lenses in terms of IQ.

We begin with a hypothetical Olympus FF DSLR system, which I will call the "F3", that is simply a scaled version of the current Olympus E3.  It will use a 34.6mm x 26mm sensor (twice the length and width of the E3 sensor and same 43.3mm diagonal of 3:2 FF sensors) that has the same pixel count, design, and efficiency of the current sensor in the E3.

Since the F3 sensor has twice the dimensions as the E3 sensor and has the same pixel count, the F3 pixels will have have twice the dimensions as the E3 pixels.  Thus, the glass for the F3 only needs to be half as sharp as the glass for the E3 to resolve the pixels.  In addition, the F3 pixels would have four times the area as the E3 pixels, which means the F3 will have the same apparent noise as the E3 when using an ISO four times higher, as we are assuming the same efficiency of sensor.  In addition, since we are using twice the FL on the F3 as the E3 to get the same perspective and FOV, we will also be using four times the ISO and half the f-ratio and to get the same total amount of light, and consequently DOF, for the same shutter speed.  Thus, the IQ of the systems will be virtually identical, since they gather the same total amount of light, have the same efficiency of sensor, and the glass for each system is equally sharp relative to the pixel size.

Let's now consider the following equivalent systems:

E3 (ISO 100-3200)
12-60 / 2.8-4
50-200 / 2.8-3.5
50 / 2 1:2 macro

and

F3 (ISO 400-12800)
24-120 / 5.6-8
100-400 / 5.6-7
100 / 4 1:1 macro

As we can see, the lenses for the F3 system seem almost painfully slow but they will nonetheless produce identical images to the E3 system, since we are assuming that the sensors have the same efficiency and the F-system glass is exactly half as sharp as the E-system glass.  The lenses all have the same effective reach, the same aperture diameters (which means they will admit the same total amount of light and produce the same DOF range), and will produce equally sharp images with the same apparent noise for the same DOF and shutter speed.

The advantages of the lenses for the F-series system is that they would be often be lighter (the two stops smaller f-ratio affects the weight more than the larger image circle and longer FL) and less expensive (it is much less expensive to make a lens that has double the image circle and is half as sharp).  The disadvantage of the F-series lenses is that, despite often weighing less, they would be physically longer, which, while probably not much of an issue below 100mm, may be a substantial minus above 200mm.  In addition, it's possible that at least some of the lenses, if not most, for the E-system would likely be able to frame closer for the same AOV, which can often be a big plus.

Another operational advantage of the E-system is that it may be able to have a more "generous" spread of AF points.  All FF systems have their AF points clustered near the center, and I do not know if this is simply how manufacturers choose to group them, or if the tighter grouping is a necessary consequence of the larger sensor.  On the other hand, since the same size AF sensors would cover a smaller relative area of the scene, the F-system would likely have more accurate AF.  This brings us to the viewfinder size and brightness.  The lenses for the E-system in this equivalent example are twice as bright, but transmit the same total amount of light.  Thus, for the same size viewfinder, both the E and F-systems would have the same brightness despite the differences in the lens brightness.  But if the F-system has a viewfinder that is proportionally larger (four times the area) than the E-system, then it's viewfinder will necessarily be dimmer.

OK.  Let's consider the following two high-end systems:

E3 (ISO 100-3200)
7-14 / 4
14-35 / 2
35-100 / 2
50 / 2 macro

and

F3 (ISO 400-12800)
14-28 / 8
28-70 / 4
70-200 / 4
100 / 4 macro

Once again, the images created with the two systems would be virtually indistinguishable even given that the F-system lenses are half as sharp.  However, the differences in price and weight will be even more extreme than in the previous example, since the E-system lenses are getting much closer to the maximum possible f-ratio of f / 0.5, with the design and cost of such lenses becoming increasingly difficult and expensive.

Let's now discuss about the future expandability of the systems. Olympus could make faster F-series lenses that many people would enjoy, and still at moderate size, weight, and cost. For example:

F3 (ISO 400-12800)
14-28 / 5.6
28-70 / 2.8
70-200 / 2.8
100 / 2.8 1:1 macro

whereas to do that for the E-system would be ridiculously heavy and expensive, if even possible:

E3 (ISO 100-3200)
7-14 / 2.8
14-35 / 1.4
35-100 / 1.4
50 / 1.4 macro

This would give the F3 an optional more shallow DOF when desired, and the option to trade DOF for a lower ISO to get a cleaner image.  On the other hand, for slower lenses that cover a lot of range:

E3 (ISO 100-3200)
14-54 / 2.8-3.5
40-150 / 4-5.6

and

F3 (ISO 400-12800)
28-108 / 5.6-7
80-300 / 8-11

the size/weight advantage will likely go to the E-system, which will most certainly cost less as well.

So we see that the advantages afforded by a larger sensor system have everything to do with the economics and availability of the available glass as well as the number of lenses, their speed, and focal length.  The equivalent glass for the larger sensor system will often be physically longer (at least for the longer FLs, since it needs double the focal length for the same perspective and AOV), but be significantly less costly and often lighter for larger aperture diameter glass.  The break-even point likely falls around f/5.6 on FF (f/2.8 on 4/3), with cost and weight favoring the F-system for faster glass, and favoring the E-system for slower glass.  In addition, we need to consider the likely advantages of closer MFDs for a given reach and a greater spread of AF points.

In the end, it's important to reiterate that this is a hypothetical situation to describe the relationship between the glass and the sensor in terms of the IQ of the image.  As of now, the choice between systems is not so straightforward, as there are no FF DSLRs that are simply scaled up versions of smaller sensor DSLRs, nor are there lenses specifically designed for each sensor that have perfect equivalents in the other formats.  Thus, operation, available glass, and IQ must all be taken together in choosing which system is best for the individual.

EVIDENCE

Here are links to threads on this subject that contain pics (and one graph) that provide empirical evidence to many of the claims made in this essay.  The last two links contain fullsize UWA images from 35mm FF, but do not compare against smaller formats.

www.slrgear.com lens tests

Interpreting the Blur Charts at www.slrgear.com

DxO Mark Sensor Tests

Pixel Density vs Apparent Noise

Lenses on FF vs Crop

5DII vs 50D

D700 vs D300

Canon G1-G10 Detail / Apparent Noise Comparison

Everything you wanted to know about noise, and then some

20D (1.6x) vs 5D (FF) apparent noise equivalency

S3 IS (6x) vs 5D (FF) apparent noise equivalency

30D @ 85mm vs 5D @ 135mm vignetting / edge sharpness / apparent noise equivalency

30D @ 21mm vs 5D @ 35mm vignetting / edge sharpness / apparent noise equivalency

Canon 30D + 35 / 1.4L vs Canon 5D + 50 / 1.4

20D + 17-55 / 2.8 IS vs 5D + 24-105 / 4L IS vignetting / distortion equivalency

20D + 10-22 / 3.5-4.5 vs 5D + 17-40 / 4L (pay special note that they are compared at the same f-ratio, instead of same DOF)

Olympus E400 + 7-14 / 4 vs Canon 5D + 16-35 / 2.8L II

Olympus E410 + 14-42 / 3.5-5.6  vs Canon 5D + 28 / 2.8

Olympus E330 + 11-22 / 2.8-3.5 vs Canon 5D + 24-105L IS (please compare f/8 on the E330 to f/16 on the 5D for the same DOF)

Canon 5D + 50 / 1.4 vs Olympus E420 + 25 / 2.8 Bokeh Comparison

Canon 5D + 50 / 1.4 vs Olympus E420 + 25 / 2.8 DOF Comparison

Sigma DP1 vs Olympus E420 vs Canon 5D (I)

Sigma DP1 vs Olympus E420 vs Canon 5D (II)

Panasonic G1 vs Nikon D700

5DII vs K20D

Apparent Noise and Pushed ISOs

Diffraction Demonstration

Diffraction Demonstration with Sigma 50 / 2.8 macro

Bokeh Demonstration (1.6x vs 35mm FF)

Compact vs 35mm FF for Deep DOF High ISO

Detail vs Apparent Noise (I)

Detail vs Apparent Noise (II)

Detail vs Apparent Noise 1DIII, 5D2, D700, A900

Canon 16-35 / 2.8L on a 5D

Canon 24 / 1.4L on a 5D

RELATED ARTICLES

The links below are articles related to the discussion in this essay.  However, the inclusion of these articles does not necessarily mean that I agree or endorse the entirety of their content (although, for the most part, I do).  They are simply additional resources on the same subject.

Paul van Walree's outstanding articles on optics

A similar article on different formats

F-ratio and Aperture

Fantastic Article on DOF

Lots of great tutorials (DOF, noise, etc.)

More great tutorials (Bob Atkins)

DOF Tutorial

Sensor Size, Pixels, Noise -- the whole nine yards (similar to this essay, but much more in-depth and technical)

Daniel Browning's Six Post Treatment of Pixel Size vs Noise

Comprehensive Treatment of Noise

Noise and DR (Clarkvision)

Noise and Quantum Efficiency (Clarkvision)

Noise, DR, and Bit Range

Noise and Dynamic Range

Dynamic Range

Sensor Size

Lens / Sensor Limits

Practical Use of DOF and Diffraction

DOF and Diffraction

More on Sensor Size and Diffraction (plus Diffraction Calculator)

More on DOF (plus DOF Calculator)

Effects of the AA Filter

DOF (extremely technical)

DOF (Wikipedia)

DOF Calculator

Equivalent Lenses

Equivalent Lens Calculator

Image Quality

Bokeh Tutorial

The Importance of Accurate Focus

Canon G10 vs Medium Format

CONCLUSION

Photography is all about the image.  But before we talk about IQ, we must first get the image.  In other words, IQ plays no role in an image that is out of focus.  IQ plays no role in an image that is missed due to slow focus or shutter lag.  IQ plays no role in an image that was not captured because the equipment was too bulky to be carried to the mountain top or too conspicuous to be used.

However, just as the amazing television series, Planet Earth, is significantly more impressive on a 52 inch HDTV than it is on a 36 inch conventional TV set, there are plenty of TV shows and movies where the type of TV set they were viewed on would make no significant difference to the viewer.  Thus, while the IQ of an image, as well as the optional ability to achieve a more shallow DOF, can greatly enhance the impact that an image has, this impact depends greatly on the image as well as the display dimensions of the image.  In other words, sometimes IQ is paramount, and sometimes it is not.  Just as we do not all watch the same TV shows, or even have the same opinions about the value of the shows that we do watch, different photographers will not take the same types of photographs or give the various elements of IQ the same value as another.  Each photographer must balance the operation of a system against its IQ potential not only in concert with the display size of the image, but also with both their skills in photography and post-processing, to decide what system best gets the job done for the type of photography that they do.

The debate between different sensor formats is very much like the debate between primes and zooms.  While top-quality primes may have higher IQ, allow for a more shallow DOF, and be better suited for low-light photography, they do not zoom.  That singular advantage of a zoom trumps all the advantages of a prime for many photographers, and so it is when comparing formats.   It is not only a matter of whether one system is "better" than the other in terms of IQ, but at what display dimensions this difference becomes significant.  For many, and likely most, it is more often a matter of available lenses, differences in DOF capabilities, and operational convenience, than it is a matter of IQ alone.

The bottom line is that we use a camera to create images.  It is important to understand the advantages of any particular system as a whole, both in terms of IQ and operation.  The purpose of equivalence is to help evaluate the IQ end of that consideration, and, in conjunction with our individual "quality threshold", make an informed choice as to which system, or systems, best meet our personal needs for the photography that we do.

ACKNOWLEDGMENTS

This essay was possible only because of the immense help and education I have received from others.  At the risk of slighting many of those who have helped me by failing to mention them by name (especially those whose images are linked in the Evidence section of this essay), I would like to particularly thank Lee Jay for giving me the bulk of my initial education as well as providing many of the examples, Amin Sabet for challenging my biases and also contributing many of the examples, and Steen for arguing with me so vigorously over the writing of this work and helping me shape the language and tone, as well as pointing out numerous errors.  In addition, much thanks to Bob Newman who taught most of what I know about noise.  Without the help from many others, this essay would not have been possible, and I am indebted to them for their help and contributions.

Also, a special shout-out to "bad doggie", who was one of the most vocal opponents of Equivalence, and with whom I was constantly fighting.  The arguments didn't really help me form this essay, but I just loved that guy.  He was, by far, the most fun of the "bad guys".  I also liked a few of the pics he posted.  More a photographer and less a gear-head.  So, "bad doggie", if you're reading, take it easy!  :  )

-- joseph james