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 IQ (image quality).  However, what this essay does not discuss in detail are the operational differences between systems which often play a far more important role than IQ alone in determining which system is better for a particular photographer.

The essay is rather long and in-depth, and the length, unfortunately, is necessary to explain the concepts.  While a simple answer may do in many cases without all the "fluff", there are many times when the fluff is central to understanding the answer.  Like other questions, the myths of "common wisdom" often have more appeal because they appear to make sense at first glance.  For example, while we take it for granted that the Earth revolves about the Sun today, there was a time when this was seen as heresy.  Not only was the notion was met with much resistance since such an idea was in conflict with the dogma of The Church, it contradicted "common sense" -- if the Earth revolved about the Sun, then why do we not feel the Earth moving?  As if that were not enough, claiming that the Earth revolving about the Sun did not explain why the opposite, that the Sun revolved about the Earth, appeared to be true.  To understand the reality of what was happening, it was necessary to understand yet another concept, that a round Earth also revolves about its own axis, which gives the illusion that the Sun revolved about the Earth.  And when this is combined with yet more information, such as the size of both the Earth and its orbit, we understand why we do not feel any motion.  Adding even more information still, such as the tilt of the Earth's axis, we can even explain the seasons.

In other words, while all is not necessarily as simple as it seems at face value, when the underlying principles are understood, then all comes together.  But understanding the underlying principles takes time.  So while I have tried to present explanations in a manner that balances completeness with brevity, the balance I have chosen lies closer to completeness than brevity, even though I am fully aware that my treatment of the issues is far from complete.  But I want to provide at least enough explanation so that the principles presented do not come across as merely substituting one dogma with another.  In other words, I have done my best to describe the differences between formats as simply as I can while providing the necessary information to understand the concepts.  

Lastly, please keep in mind this key point when debating the IQ differences between systems:  do you take photos where the IQ differences between systems make any difference at all to your target audience for the types of photos you take and the skills that you have?
 

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.

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 are the advantages of a larger sensor system? 
A:  For the same generation of camera, larger sensors usually have more and larger pixels which give more detail and produce sharper images (except in the extreme corners in some instances), and have the option of a more shallow DOF for a given perspective and framing, if desired.  In addition, larger sensor systems have less noise for a given ISO and produce a sharper image for a given quality of lens and strength of AA filter.  Operationally, larger sensor systems often have a larger and brighter viewfinder.

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.  Also, in some cases, the extreme corners will be sharper with smaller formats as well.  In addition, for some situations, such as deep DOF high ISO shooting, smaller sensors may even have a noise advantage over larger sensors.  Furthermore, some features (such as in-camera IS), that are not always available in larger sensor systems, serve to sometimes give smaller sensor systems an IQ advantage.  Another plus of smaller sensor systems is that they can often frame more tightly for a given AOV.  Lastly, compacts (including the Sigma DP1) offer an even greater size advantage still since they lack a mirror box, although, currently, this comes at the expense of AF speed and accuracy.

Q:  Are larger sensor systems "better than" smaller sensor systems?
A:  For some, yes; for others, no.  Each person has a different balance point for IQ, operation, size, weight, and price for the type(s) of photography that they do with the skills they possess.

Q:  Do larger sensor systems produce a higher quality image than smaller sensor systems?
A:  I've found this too difficult a question to answer with a simple "yes" or "no".  Please read the section on IQ for why a blanket answer is simply not possible.  However, for cameras of the same generation, when the larger sensor system has at least the same number of pixels as the smaller sensor system, and can gather more light than the smaller sensor system while still maintaining the desired DOF, the answer is usually "yes" for most scenarios, although there may certainly be exceptions, such as the availability of IS at various focal lengths, as noted above.

Q:  Don't larger sensors always gather more light than smaller sensors?
A:  For the same perspective, framing, f-ratio, and shutter speed, yes, but not for the same perspective, framing, DOF, and shutter speed.  Larger sensor systems can often use a more shallow DOF than is available to the smaller sensor system to gather more light, and can sometimes use a lower shutter speed as well (when both systems are at base ISO).  However, when shutter speed is traded for more light rather than DOF, we must take care that it is still sufficiently high so that motion blur is not a factor, and, unless using a tripod, that camera shake is not a factor, either.  In addition, if the the ratio of flash to ambient light is not a factor, larger sensor systems can also gather more light by using flash.

Q:  Doesn't the same f-ratio gather the same light (f/2 = f/2 = f/2) no matter what the sensor size is?
A:  Not for the same perspective and framing.  The same f-ratio will yield the same intensity of light, and thus the same exposure, regardless of the sensor size, but it will yield neither the same DOF nor the same the total amount of light, and thus not the same amount of noise.  This is discussed in detail here.

Q:  Do larger sensors have less noise?
A:  Not necessarily.  Assuming the same efficiency of sensor, a larger sensor will have less noise only in the situations when it can gather more light, as explained above.

Q:  Don't larger pixels have less noise?
A:  Yes.  But, for a given sensor size, smaller pixels allow more detail since there are more pixels.  If we compare noise at the same level of detail, either by using NR and/or resampling the more detailed image to the same number of pixels as the smaller image, the total noise will be the same for the same efficiency of sensor.  However, the quality of the noise is at least as important as the quantity of the noise, and more pixels means a finer grain, which often may give the image with more pixels a more pleasing appearance, as long as the total quantity of noise is not significantly different.

Q:  Do smaller sensors have more DOF than larger sensors?
A:  For the same framing and f-ratio, yes.  However, larger sensors can match the DOF by stopping down.

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, if the larger sensor system has more pixels than the smaller sensor system, then maximum image detail will be realized at more shallow DOFs if enough of the image is within the DOF and the lens is sharp enough to resolve the pixels at the wider apertures, but will still resolve at least the same amount of detail as the smaller sensor system at the same DOF.  This is discussed in more detail here.

Q:  Won't larger sensors have more noise than smaller sensors when they have to use a higher ISO to maintain the same shutter speed as smaller sensors when stopping down for the same DOF?
A:  For the same level of detail and generation of modern digital cameras, the differences are all but insignificant until we reach ISOs past 6400 on 35mm FF.  Once pushed past that point, however, smaller sensor systems do appear to have an advantage for the same DOF and shutter speed.

Q:  Do larger sensors require sharper glass?
A:  Just the opposite.  Larger sensors are more tolerant of glass than smaller sensors.  Thus, sharper glass on a smaller sensor does not necessarily equate to superior results.  This is discussed in more detail in Myth #4.

Q:  But don't larger sensor systems have softer corners than smaller sensor systems?
A:  It depends on the lens, but generally not for the same FOV and DOF.  Sometimes, the extreme corners for larger sensors may be slightly softer, even at the same DOF, (especially true with cheap UWAs), but will be sharper elsewhere in the image.

Q:  Do larger sensor systems have more vignetting?
A:  Only at the same FOV and f-ratio, not for the same FOV and DOF.

Q:  What's the advantage of a larger sensor system if it has to stop down to match the DOF/sharpness/vignetting of a smaller sensor system?
A:  First of all, you're not always trying to match the DOF of the smaller sensor system -- the more shallow DOF ability of larger sensor systems is a major plus to some.  But if shallow DOF is not your thing, then for the times that you stop the larger sensor down to match the DOF, and can use a lower shutter speed rather than up the ISO to match the shutter speed, then it will deliver a sharper and more detailed pic (so long as it has at least the same number of pixels as the smaller sensor), with no more vignetting, and even having less noise to boot.  Much more on this here.

Q:  Don't smaller sensor systems have more reach?
A:  Effective reach is a function of pixel density, not the sensor size as many think.  Since smaller sensor systems often have a greater pixel density, they usually have a greater effective reach for the same focal length, but not because their sensors are smaller.  For example, since the Canon 1DsIII and 20D both have the same size pixels, using the same lens on both cameras, the 1DsIII image can be cropped to the same FOV as the 20D image and have the same number of pixels as the 20D image, thus the effective reach is the same for both cameras.

Q:  Aren't larger sensor systems larger, heavier, and more expensive than smaller sensor systems?
A:  For the same effective reach, if light collecting ability and DOF are not taken into consideration, this is often true.  However, for equal light collecting ability and DOF, the larger sensor system is usually favored in terms of price and weight.

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
FM:  focal multiplier (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
35mm FF:  A sensor measuring 24mm x 36mm, sometimes simply referred to as "FF" (Full Frame), or the 135 Format.
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 FL in terms of 35mm FF that will give the same AOV)
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:  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 (e.g. the f-ratio for a focal length of 50mm and an aperture of 10mm is 50mm / 10mm = f/5)
Stop:  A difference of one stop represents a doubling, or halving, of a quantity

Exposure:  The product of the intensity of the light and the time the shutter is open:  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:  Confusingly used to mean the total amount of noise, the NSR (Noise-to-Signal Ratio), and the SNR (Signal-to-Noise Ratio)
Efficiency:  The percentage of light falling on the sensor that is recorded, the noise created by the sensor, and how cleanly the signal is amplified
DR:  Dynamic Range:  the difference (in stops) between the brightest and darkest area that can be captured
Tonal Gradations:  The number of different levels of brightness than can be measured within the dynamic range

Diffraction Softening:  Lost detail due to the diameter of the Airy disk exceeding the diagonal of a pixel 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

Equivalent images are images from two different cameras that look as similar as they possibly can.  It is critical to note that "equivalent" does not mean "equal" -- I cannot stress this point enough.  That said, the definition of "equivalent images" is as follows:

A necessary consequence of equivalent settings is that equivalent images will automatically have the same apparent exposure, so it is unnecessary to make this requirement an additional postulate.  Of paramount importance is that the conditions of "same FOV", "same DOF", and "same shutter speed" necessarily mean that different formats will not use the same focal length, f-ratio, or ISO for equivalent images.

It is important to note that "same exposure" is omitted from the definition.  Since "equivalent images are images from two different cameras that look as similar as they possibly can", it makes sense that equivalent images are made with the same total amount of light, and it is the total amount of light that matters in terms of IQ, not exposure.  This is a major point of confusion since many people erroneously believe that the exposure represents the total amount of light.  It does not.  To understand why, we need to realize that exposure is intensity x time, whereas the total amount of light is the exposure x the light collecting area of the sensor.  Since equivalent images are made with the same total amount of light, and that light is spread out on sensors of different sizes, we will necessarily use different intensities (f-ratios) on different formats for equivalent images.  By increasing the ISO the same number of stops as the f-ratio to maintain the same shutter speed and DOF, the total amount of light and apparent exposures will be the same for equivalent images, but the exposures will be different.  The impatient reader may wish to jump ahead to the section on exposure if there are questions on this crucial point.

Also note that the definition of equivalence does not discuss elements of IQ.  This is because "equivalence" is not about "same IQ", it is about the conditions for a fair comparison of IQ.  In particular is the omission of "same noise" as a postulate of equivalence.  First of all, the only factors that contribute to the total amount of noise are:

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 amplified

If the sensors have the same efficiency, then the total noise for the same level of detail will be the same for equivalent images because the images are made with the same total amount of light (this is discussed in more detail, along with an important exception, in the Noise section of the essay).

Note the condition of "same level of detail" for making a claim of equal noise.  If the pixel counts of the sensors are different, then, for the image with the higher pixel count, there will be more detail, assuming the glass can sufficiently resolve the pixels.  This additional detail may, or may not, come at the expense of additional noise, depending on the efficiency of the sensor, which is discussed in detail here.  Regardless, when comparing noise levels, a fair comparison requires us to do so at the same level of detail.  This requirement cannot be met unless both sensors have the same number of pixels and the lenses for the respective systems are capable of resolving those pixels.  Naturally, the system with more pixels will have a detail advantage, unless its glass is incapable of resolving those pixels.  So, while we sacrifice detail from the more detailed image to compare noise by downsampling and/or using NR, the requirement of "same detail" itself is not a condition of equivalence.

To create equivalent images on systems with different formats, we first compute the FM ("focal multiplier", often called the "crop factor"), which  is the ratio of the diagonals of the larger sensor to the smaller sensor of the systems we are comparing (or, alternatively, the ratio of the lengths or widths of the sensors to get the same FOV, instead of same AOV, due to differing aspect ratios, such as 4:3 and 3:2).  For example, the FM for 35mm FF and 4/3 is 43.3mm / 21.6mm = 2.00 for the same AOV, 36mm / 17.3mm = 2.08 for the 4/3 image cropped to 3:2, or 24mm / 13.0mm = 1.85 for the 35mm FF image cropped to 4:3.

We then multiply the FL and f-ratio of the larger sensor system by the FM to get the same AOV (or FOV, depending on how we compute the FM) and DOF, and multiply the ISO by the square of the FM to get the same shutter speed.  However, it's usually easier to convert the FM into stops, and then add the FM (in stops) to both the f-ratio and ISO of the larger sensor system (FM in stops = 2 log₂ FM).  Lastly, we resample the images to the same output size (usually at least as large as the larger size image, but not necessarily so).

EQUIVALENCE AND PARTIAL EQUIVALENCE:

Let's compare equivalent settings for the Canon 5D (35mm FF), Nikon D300 (1.5x), Canon 40D (1.6x), Olympus E3 (4/3).  The FM between the 5D and D300 is 1.5 (for both the same AOV and FOV, as they have the same aspect ratio of 3:2), the FM between the 5D and 40D is 1.6 (again, for both the same AOV and FOV), and the FM between the 5D and E3 is 2 (for the same AOV only as the E3 has an aspect ratio of 4:3).  Below are examples of equivalent settings (rounded to the nearest 1/3 stop) which means that for the same perspective (subject-camera distance) and output size (display dimensions), they will have the same AOV and DOF.  If the shutter speeds are also the same, they will have the same apparent exposures, as well.  However, the level of detail will depend on the pixel count of the sensor and the sharpness of the lenses used, and the level of 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 noise levels will generally be very close for equivalent settings).  So, that all said, let's take a look at some fully equivalent settings on different formats:

1)  5D at 80mm, f/8, 1/200, ISO 400
2)  D300 at 53mm, f/5, 1/200, ISO 160
3)  40D at 50mm, f/5, 1/200, ISO 160
4)  E3 at 40mm, f/4, 1/200, ISO 100

For equally efficient sensors, and resampling the images to the same output size and applying NR as needed to achieve the same level of detail, there will not be much difference between the systems.  The advantage of the larger sensor systems in this case is limited to PP (post-processing) options.  That is, it is not necessary to apply NR to match the detail level of the smaller sensor systems.  Instead, the more detailed, albeit more noisy, image may often be preferable (assuming, of course, that the larger sensor system has more pixels in addition to glass that resolves those pixels at least as well as the lenses for the smaller sensor system, which is pretty much always the case, as the larger sensor system is stopped down further than the smaller sensor systems to achieve the same DOF).

Sometimes, we can get away with a slower shutter speed, rather than a higher ISO, and thus have lower 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 to obtain a cleaner image, while still maintaining the same AOV, DOF, and exposure:

1)  5D at 80mm, f/8, 1/50, ISO 100
2)  D300 at 53mm, f/5, 1/125, ISO 100
3)  40D at 50mm, f/5, 1/125, ISO 100
4)  E3 at 40mm, f/4, 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.  This can even be taken in the opposite direction when one system has in-camera IS and/or in-lens IS that the other system does not.

Other times, we might rather use a more shallow DOF than a lower shutter speed to use a lower ISO and thus less noise, either because we prefer a more shallow DOF, or we need a fast shutter but lower 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)  5D at 80mm, f/4, 1/200, ISO 100
2)  D300 at 53mm, f/4, 1/200, ISO 100
3)  40D at 50mm, f/4, 1/200, ISO 100
4)  E3 at 40mm, f/4, 1/200, ISO 100

Some may have noticed that the D300 and 40D 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 FMs 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.

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 (subject-camera distance)
2) FOV (field of view / framing)
3) DOF (depth of field) / aperture (aperture = focal length / f-ratio)
4) shutter speed
5) output size (same number of pixels / display size)

An equivalent image is an image where all five of these parameters are the same.  Note that detail and noise are left out, along with other elements of IQ (bokeh, color, etc.).  Thus, "equivalence" neither means "equal" nor is it a mandate on how to use systems.  Instead, equivalence is a fair method of comparing the IQ of systems for the types of images that the systems being compared can all capture, and does not address many of the operational differences between systems, such as well as size, weight, price, available lenses and accessories, that often matter as much, if not more, than IQ alone.  The purpose of equivalence is to compare the IQ of systems for photos that have the most similar photographic characteristics, and not to artificially handicap one system with by comparing image with dissimilar photographic characteristics.  That said, people often choose systems on the basis of what one can do that the other cannot, and includes dissimilar photographic characteristics that one system can do, but the other cannot.  However, for images where both systems are able to match the perspective, FOV, DOF, and shutter speed, then a fair comparison requires that they usually should be compared in this manner, as well as at the same output size.

So, what is meant by "usually"?  One common exception is when there is no need for the shutter speeds to be the same.  In these instances, it makes more sense to use the lowest possible ISO on each system that provides a "sufficiently fast" shutter speed.  Sometimes, that will favor larger sensor systems by allowing them to use base ISO and thus get a cleaner and more detailed image, and sometimes this will favor smaller sensor systems that have in-camera IS and can use a lower shutter speed, and thus lower ISO in low light, than larger sensor systems without in-camera IS or an available IS lens.

Another time that it is not necessary to compare fully equivalent images is when the entire scene is within the DOF.  In this scenario, it makes the most sense to use the aperture that provides the sharpest image while still yielding a sufficiently fast shutter speed.  The most common scenario for this situation is landscapes, where the scene might be fully within the DOF by f/5.6, for example, but we choose f/8 or f/11 to achieve sufficient corner sharpness, yet still have sufficient shutter speed where the ISO need not be raised.

So the point of Equivalence is to make for a fair comparison of the IQ of images that the compared systems can produce.  It's simply silly to compare corners at different DOFs.  In situations where the corners of an image matter, people do not shoot at shallow DOFs.  It is silly to compare the noise of different systems at different levels of detail, when the more detailed image can apply NR (noise reduction) to match the detail level of the image with less detail, and then compare on a more fair basis.  Since it is silly to compare prints of different sizes, it is just as silly to compare sharpness of 100% crops when one image is made of more pixels than the other.  Lastly, although this is not a postulate of equivalence, it is silly to compare hardware on the basis of in-camera jpgs.  While that may be the most appropriate method of comparing for people who do not shoot RAW, it is hardly indicative of the potential of the systems.

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.  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, and have, according to some, IQ as good as medium format for certain situations (please take a read of this article).

So, even though equivalent images are not identical, by using the definition of equivalence given, they will be as close as two images from two different systems could possibly be, and this should be used as the starting point, but not the extent, of a fair comparison.  We choose one system over another on the basis of size, weight, operation, available lenses, IQ, DOF range, and, of course, price.  How each person weighs the options will dictate their own unique choice for which system is best for their needs.

THE FIVE POSTULATES OF EQUIVALENCE:

FOV / AOV:

The FOV (field of view) is synonymous with framing.  It differs from the AOV (angle of view) in that it includes the geometry of the captured scene, usually a 3:2 rectangle for most DSLRs or a 4:3 rectangle for some DSLRs and compacts.  For the mathematically inclined, we have:

AOV = 2 x tan^-1 (half the sensor diagonal in mm / focal length in mm)

For example, the AOV for 50mm on 35mm FF is 2 x tan^-1 (21.6 mm / 50 mm) ~ 47°.  Similarly, the AOV for 25mm on 4/3 is 2 x tan^-1 (10.8 mm / 25 mm) ~ 47°.  The FOV, on the other hand, is a bit more complicated, as it gives two measurements: one for the width of the scene, and the other for the height.  Thus, to calculate the FOV, we need to know both the aspect ratio and the subject-camera distance.  We'll begin by defining some variables to make the calculations easier:

d = subject-camera distance
s = sensor diagonal (in mm)
r = sqrt (a² + b²) where the aspect ratio is a:b
FL = focal length (in mm)

First, we calculate c = (d x s) / (r x FL).  Then the FOV is (a x c) by (b x c) in the same units as d.  For example, for a 50mm lens 10 ft from the subject using a 35mm FF camera with a 3:2 aspect ratio, we have:

d = 10 ft
s ~ 43.3 mm
r = sqrt (3² + 2²) ~ 3.6
FL = 50 mm

so c = (10 ft x 43.3mm) / (3.6 x 50mm) = 2.4 ft.  Thus, the FOV is (3 x 2.4 ft) by (2 x 2.4 ft) = 7.2 ft x 4.8 ft.

Let's repeat the example for the same AOV and perspective on a 4/3 camera:

d = 10 ft
s ~ 21.6 mm
r = sqrt (4² + 3²) = 5
FL = 25 mm

so c = (10 ft x 21.6mm) / (5 x 25mm) = 1.728 ft.  Thus, the FOV is 4 x 1.728 ft by 3 x 1.728 ft ~ 6.9 ft x 5.2 ft, which is 4% narrower and 8% taller (4% more area) than a 3:2 image with the same perspective and AOV.

For the same AOV, the FM (focal multiplier -- often called the "crop factor") of the system with the larger sensor is computed by dividing the diagonal of the larger sensor by the diagonal of the smaller sensor (if the aspect ratios are the same, we can use the ratio of the widths or heights of the sensors instead).  For the same FOV, the FM is computed by using the ratio of the sides of the sensors, rather than the diagonals (if the two systems have the same aspect ratio, then we can still use the ratio of the diagonals).  Often, it is convenient to give this ratio in stops to adjust for the f-ratio and ISO:  FM (in stops) = 2 log₂ FM.  Typically, we round this value to the nearest 1/3 stop (for convenience, it's good to know that 1/3 ~ 0.33 and 2/3 ~ 0.67).

If the aspect ratios, perspective, and AOV are the same for two systems, then the FOVs will also be the same.  But if the aspect ratios are different, then to get the same FOV we must frame wider and crop to the desired FOV.  Most digital cameras use an aspect ratio of either 3:2 or 4:3, which differ in area by only 4% for the same AOV.  Hence, for equivalent images, it is often more convenient to compare at the same AOV rather than FOV as the differences between 4:3 and 3:2 aspect ratios are so small and have a negligible impact on the principles of equivalence.

For the same perspective and AOV, a 4:3 image will be slightly more narrow and slightly taller in framing than the 3:2 image.  To get the same FOV with 3:2 as with 4:3, it is necessary to frame 8% wider (in terms of 35mm FF EFL) and crop away 1/18 (5.5%) of the image from the left and right sides.  To frame and crop a 4:3 image at 3:2, we must also frame 8% wider (in terms of 35mm FF EFL) as well as once again crop off 1/18 of the image, but this time from the top and bottom.   In each case, this cropping causes us to lose 1/9 of the pixels.  Thus, crops more square than 4:3 favor 4/3 by 11%, and crops more elongated than 3:2 favor the 3:2 aspect ratio by the same amount.  For aspect ratios in between, of course, the differences will be less.

For example, the dimensions of the Canon 5D sensor are 35.8mm x 23.9mm which gives a diagonal of 43.0mm, and the dimensions of the Olympus E3 sensor are 17.3mm x 13.0mm (the actual sensor dimensions are 18.0mm x 13.5mm, but only a 17.3mm x 13.0mm section of the sensor area is used for capturing the image) which gives a diagonal of 21.6mm.  This means that the FM for the same AOV between the two cameras is 43.0mm / 21.6mm = 1.99.  However, if we frame and crop the 5D image to the 4:3 aspect ratio of the E3, we get a FM of 23.9mm / 13.0mm = 1.84.  On the other hand, if we frame and crop the E3 image to the 3:2 aspect ratio of the 5D, we get a FM of 35.8mm / 17.3mm = 2.07.  These three FMs, to the nearest 1/3 stop, all round to 2 stops, thus the reason that I said in the opening paragraph in this section that, for the purposes of comparing equivalent images, the differences between 4:3 and 3:2 aspect ratios is negligible.

Sometimes, it may also be of interest to crop the image with the larger pixel count to the same number of pixels and aspect ratio of the image with the smaller pixel count to determine the max EFL that can be achieved with the same detail with the sensor that has the larger pixel count.  To do this, we multiply the FL of the system with the larger pixel count by the ratio of the diagonals (in pixels) of its pic with the diagonal of the image with the smaller pixel count.  For example, the Canon 5D produces a 4368 x 2912 (3:2 aspect ratio) pixel image which has a diagonal of 5250 pixels.  The Olympus E3 produces a 3648 x 2912 (4:3 aspect ratio) image with a diagonal of 4560 pixels.  The ratio of these diagonals is 5250 / 4560 = 1.15.  Hence, the Canon 5D can squeeze an extra 15% more "digital zoom" by cropping while still maintaining the same pixel count and aspect ratio as the E3, and thus reducing the FM for FOV by the same amount, from 2 to 1.73.

To get the same FOV / AOV for the same perspective with cameras that have the same aspect ratio, you multiply the FL of the smaller sensor camera by the FM.   For example, 30mm on 1.6x gives the same FOV as 50mm on 35mm FF, since 30mm x 1.6 = 50mm.  For cameras with different aspect ratios, you simply multiply by the appropriate FM (discussed above) and crop.  For example, to get the same FOV as 7mm on an Olympus E3 using a Canon 5D, we would use 7mm x 1.84 = 13mm and then crop the image to 4:3.

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.

Listed below are tables of common FMs in relation to 35mm FF for images using the same AOV.  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.
 

Compacts:
 

 
DSLRs:

 
Regardless, as was demonstrated above, we can compute a FM between any two systems using the lengths of their respective sensors, or, more simply, either divide the FMs of the respective systems or subtract their FMs when using stops.  For example, the FM between a Canon 40D and Olympus E3 can be computed (for the same AOV) as 2.00 / 1.62 = 1.23 = 0.61 stops ~ 2/3 of a stop, or, more simply:  2 stops - 1 1/3 stops = 2/3 of a stop.
 

DOF / aperture:

The DOF (depth of field) is how much depth from the focal plane of the image is considered to be in critical focus.  It is a subjective measure, to be sure, but is not arbitrary.  The subjective nature of DOF is accommodated in the DOF formula(s) with the CoC (circle of confusion).  The value of the CoC is a function of print size, viewing distance, visual acuity, and magnification (the ratio of the final display dimensions to the dimensions of the recording media -- sensor).  So, while the actual value for the DOF in a particular photo is subjective, whether or not two photos have the same DOF is objective, and can be determined both visually and mathematically.  For any two images that have the same perspective, framing, output size, and aperture, the DOFs will also be the same, regardless of the native pixel count and size.

In discussions involving DOF, it is critical to distinguish between "aperture" and "f-ratio".  These two terms are used interchangeably but are very different quantities.  The aperture is the apparent diameter of the opening in a lens when looking through the front element, and can be calculated as the quotient of the focal length and the f-ratio.  For example, the aperture at 80mm f/8 is 80mm / 8 = 10mm.  The aperture is thus related to both the DOF of an image and the total amount of light that makes up an image.  A simple way to think of these concepts when comparing systems is the following:

1) For the same perspective, framing, f-ratio, shutter speed, and output size, larger sensors yield a more shallow DOF than smaller sensors, have the same intensity of light, and put more total light on the sensor.

2) For the same perspective, framing, aperture, shutter speed, and output size, all systems have the same DOF and put the same total amount of light on the sensor.

A larger sensor means that you need to use a longer focal length for the same perspective and framing, which means you will also need to use a larger f-ratio to get the same aperture.  For example, 40mm f/4 on 4/3, 50mm f/5 on 1.6x, and 80mm f/8 on 35mm FF will all have the same AOV for the same perspective, and also have the same aperture since 40mm / 4 = 50mm / 5 = 80mm / 8 = 10mm.  Hence, all three systems will produce the same DOF at the aforementioned settings for the same perspective.  Basically, a larger sensor system requires the use of a longer focal length for the same perspective and framing, which in turn requires the use of a higher f-ratio for the same aperture, which in turn requires a lower shutter speed or higher ISO for the same apparent exposure.

The reason that DOF is so important, even if DOF, per se, is not an issue to the photographer, is that it is also intimately connected with  sharpness, diffraction softening, and vignetting.  Many times, for example, people will stop down for no other reason than to increase sharpness, regardless of other considerations.  For case #1 above, larger sensors will have softer corners, less diffraction softening, and more vignetting. For case #2, the differences between systems will be at a minimum, with the larger sensor systems usually coming out ahead for the overall image, but sometimes taking a hit in the extreme corners.  Of course, we must also compare at the same output size as well -- a 100% crop from a 21 MP image will look softer than a 100% crop from a 12 MP image.  Hence, we need to resample both images to the same output size, and then compare.

There is a third situation which should be discussed:  when you use the same perspective and focal length on both systems and crop the image from the larger sensor system to the same framing as the smaller sensor system.  This scenario comes up when we are focal length limited and a TC is not an acceptable option.  For example, a 400 / 5.6L on a 40D (1.6x) has the same AOV/DOF as a 640 / 9 on 35mm FF (since 400mm x 1.6 = 640mm and f/5.6 x 1.6 = f/9), but no such lens exists.  To get the same framing with a 5D, we have a few choices:

1)  Use another lens with a 640mm FL.
2)  Use a TC.
3)  Get closer.
4)  Crop to the same framing.

But let's take a moment to discuss diffraction and sharpness since both are intimately related to DOF.  We'll begin by considering the performance of the Canon 35 / 1.4L on the 20D (1.6x) and the 5D (35mm FF):

http://www.slrgear.com/reviews/showproduct.php/product/148/cat/10

We can see that center sharpness is greatest at f/5.6 on FF, and f/4 on 1.6x. The corners are at their best one more stop down for each. What's going on is that as you stop the lens down, the aberrations lessen. But you eventually reach a point where the effects of diffraction softening kick in. You then have a battle between lessening aberrations and increasing diffraction softening.  So, per the example, we see that lens aberrations are the primary culprit for softness below f/5.6 on 35mm FF (f/4 on 1.6x), and diffraction softening is the primary culprit beyond f/16 on 35mm FF (f/11 on 1.6x), with the lessening aberrations pretty much offsetting the increasing diffraction softening in between.  As a side, this also illustrates that since a FF sensor has 1 1/3 stops (2.56x) more area than a 1.6x sensor, the effect of diffraction softening occurs 1 1/3 stops earlier on 1.6x than on FF.  Of course, it's not as simple as this, as the design of the lens will dictate when the effects of diffraction begin to overtake the effects of aberrations as primary source of softening.

It is commonly believed that diffraction softening is related to the f-ratio and the pixel size.  This is true, but not the entire story.  The pixel size, independently of the sensor size, sets the maximum f-ratio that can be used before diffraction softening begins to degrade the image.  However, the detail of the system with the greater pixel count will never degrade to less than the detail of the system with the smaller pixel count, even for the same DOF, since the system with the larger pixel count had more detail to begin with.  We can think of the situation as two cups of water set to cool on a table, one at 90 degrees, and the other at 80 degrees.  The hotter water will cool faster than the cooler water, but it will always be hotter, until both bottom out at the equilibrium temperature of the surroundings.

To that end, let us now consider the Canon 20D (1.6x) and Canon 1DsIII (35mm FF) since they have the same size pixels.  If we assume that the 20D begins to suffer from diffraction softening at f/5.6, then the 1DsIII will be able to render at least the same amount of detail at f/9 as the 20D at f/5.6 (since f/5.6 x 1.6 = f/9).  That is, if we resampled the 1DsIII image to the 8.2 MP of the 20D image, or upsampled the 20D image to the 21 MP of the 1DsIII image, we would find that the 1DsIII image would have at least as much detail as the 20D image for the same DOF.

However, to maximize the detail of its 21 MP sensor, the 1DsIII would also begin to suffer diffraction softening at f/5.6 just as the 20D, since its pixels are the same size.  But the 1DsIII will always render at least as much detail as the 20D for the same DOF.  But to maximize detail of its 21 MP, it will sometimes be better to use a smaller DOF, provided the effects of diffraction softening are stronger than the degrading effects of lens aberrations, and the subject lies within the more shallow DOF, at the wider apertures.