Sorry for the delay in responding to Will's concerns. My life has been a bit busy lately.
First, I would like to apologize for using an incorrect expression.
In my original posting, I wrote "equivalent exposures" when I should have written "equivalent images". That error led Will off in the wrong direction, and could make it difficult for anyone else to relate what I said to James's paper
. I have edited the original posting to correct the error for the benefit of later readers.
Now, to revisit the concept, here is what James writes:
Equivalent images are images from two different cameras that look as similar as they possibly can. The definition of "equivalent images" is as follows:
1) Same perspective (subject-camera distance)
2) Same FOV (field of view or framing)
3) Same DOF (depth of field)
4) Same shutter speed
5) Same output size (same number of pixels)
Why is this useful? Why does he make this definition? Well, consider an example. Suppose we take a picture of some interesting subject, say a flower being visited by a butterfly.
The perspective of the picture determines the relationship of foreground to background. It is how a viewer tells the difference between a shot taken from 1 foot away using a wideangle lens, and a shot taken from 10 feet away using a telephoto.
The FOV determines whether we see just the flower and butterfly, or a larger piece of garden.
The DOF determines whether only the butterfly's head will be sharp, or its wings and the flower also.
The shutter speed determines how blurred the butterfly's wings are, as a result of their motion.
The number of pixels determines sharpness, assuming of course that the lens quality is good enough to use all the pixels.
Changing the perspective, the FOV, the DOF, or the shutter speed will change the images in ways that a viewer can immediately recognize. The results are simply different pictures.
Sometimes it makes sense to compare such images, particularly to show what one camera can do that another cannot. An example is the wonderfully shallow DOF and widely blurred backgrounds that can be produced by large lenses, normally found only on large format cameras. Similarly, if you're doing distant landscapes or flat copy, then again, DOF is not an issue and the photographer can freely choose to use larger formats and larger lenses to get arbitrarily high resolution. God bless landscapes and flat stuff.
However, in closeup & macro work, photographers often struggle to get enough DOF. Since this is
photomacrography.net, I think it's pretty reasonable to allow ourselves, as James does, to consider DOF as being an important issue. Those who disagree need only stop reading -- I won't be offended.
OK, having established that we care about perspective, FOV, DOF, and shutter speed, let us now ask, "What aspects of the camera determine these things?"
Perspective -- what lines up with what -- is actually determined by the entrance pupil location. (Recall that the entrance pupil is just where the aperture appears to be, looking through whatever lens elements are in front of it.) James's characterization as "subject-camera distance" is a pretty good approximation. But let's be precise and say that it depends on entrance pupil location.
FOV -- the field size -- is determined by the relationship between sensor size and lens focal length. The required relationship is a bit complicated, particularly in the macro focusing range. Let's bypass the issue by simply presuming that either we have zoom lenses, or whatever two cameras we're comparing already come equipped with the proper ratios.
DOF -- depth of field -- is determined by the size of the entrance pupil, more specifically by its angular diameter as seen by the subject. With most cameras, that diameter is adjusted by changing the f-number, but thinking
in terms of f-number is a bit of a trap. That's because the f-number bundles together the entrance pupil diameter and the lens focal length. We care about the entrance pupil diameter because it determines DOF, but (for purposes of analysis here) we really do not care about lens focal length once it produces the correct FOV.
Shutter speed -- how long does the exposure last -- needs no further explanation.
Notice that sensor speed (ISO rating) does not appear in this list. That's because sensor speed is a derived requirement. Once you know the illumination level, the entrance pupil size, the lens focal length, and the shutter speed, then you can determine the sensor speed needed to produce a proper exposure.
It turns out, of course, that we can say something
about sensor speed quite easily. Assume that the illumination level is fixed, along with the entrance pupil size and the shutter speed. Under those conditions, the amount of light that goes through the entrance pupil during the exposure is constant
-- completely independent of the sensor size. But that constant amount of light gets spread over an area that depends on sensor size. The relationship is simple -- to produce a proper exposure, the sensor that is X times larger (on axis) must have an ISO rating that is X*X larger. For example, a 21 mm sensor must be rated 9X faster than a 7 mm sensor.
People who are steeped in film technology are likely to balk at this concept, and rightly so. It's difficult at best to buy films that meet this requirement, and maybe they cannnot even be made. I don't know, but in any case that difficulty is not relevant here since we're specifically talking about digital cameras in this thread. If film doesn't fit the model, then the model doesn't apply to film. Simple as that, no problem.
Digital sensors, however, have no trouble meeting the requirements about sensor speed. That's because digital sensors are essentially photon counters. When exposed to the same total amount of light, two digital sensors built using the same technology will capture the same number of photons. Assuming that they also have the same number of pixels (James's last criterion), then the number of photons captured for corresponding pixels will also be the same. That implies that the statistical uncertainty of the photon counts will also be the same.
It turns out that statistical uncertainty in the photon counts is the dominant cause of pixel noise in modern digital cameras, so all of this analysis ends up with a remarkably simple result: for equivalent images, all sizes of digital sensors that use the same technology and the same pixel counts, also have the same noise level.
Larger sensors definitely have the potential to produce less noisy images, but that potential is achieved only when they are allowed to capture more light, either by increasing the illumination level, widening the entrance pupil, increasing the exposure time, or any combination of those.
I will not take more space right now to go through the analysis, but it turns out that for equivalent images, larger and smaller sensors also suffer equally from diffraction effects. The reason is that once the FOV and entrance pupil are fixed, the f-number varies directly with the sensor size. With equivalent images, a sensor that is X times larger will be running with an f-number that is also X times larger. Because diffraction blur (size of the Airy disk) depends directly on f-number, the Airy disk will also be X times larger, retaining the same proportion with respect to the image size. Everything scales in proportion.
Hopefully this discussion has filled in some of the gaps, inconsistencies, and misconceptions raised by my first posting. Again, I apologize for using an incorrect term, which has now been corrected. But I believe that with that correction, all the other points stand as stated.
Roger N. Clark's article at http://www.clarkvision.com/photoinfo/dof_myth/
performs essentially the same analysis in more detail. Quoting briefly from the article:
Roger N. Clark wrote:...if one keeps aperture of the larger camera the same as that in the smaller camera, the two cameras record the same image with the same signal-to-noise ratio and the same depth of field with the same exposure time.
The treatment of DOF versus entrance pupil size rather than f-number is explained further in Dick Lyon's article, "DOF Outside the Box", currently available at http://www.dicklyon.com/tech/Photograph ... d-Lyon.pdf
. My life has been greatly simplified by this paper.
Dick Lyon wrote:It is not necessary to know the focal length and f-number of a camera lens to compute a depth of field, and indeed formulas that use the "outside the box" parameters, field of view and entrance pupil diameter, may be easier to understand and reason with.