Lens Equivalents

[Roy]

The article by Charles S. Johnson, Lens Equivalents, posted on the site today makes the important point that the maximum achievable DOF doesn't change with sensor size.

At the risk of starting another confusing thread on DOF, I'd like to point out that this is only true for maximum DOF. At an aperture larger than that at which maximum DOF is achieved, the smaller sensor with a lens of equivalent angle of view will give greater DOF.

For example, at f/8 (well below the APS diffraction limit of f/14 and the full-frame limit of f/22), a 35mm lens with an APS size sensor will give roughly 50% more DOF than a 50mm lens on full-frame sensor.

The point to take away: below the diffraction limit and at the same f stop, an APS sensor camera will give more DOF than a full-frame camera when both have equivalent angle of view lenses. (Equivalent angle of view (or field of view) lenses in the case of APS compared to full-frame means that the focal length of the full frame lens is 1.5 times that of the APS lens.)

Remember the definition of circle of confusion: a group of photographers discussing DOF.

[BJL]

At an aperture larger than that at which maximum DOF is achieved, the smaller sensor with a lens of equivalent angle of view will give greater DOF.

For example, at f/8 ...
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True for comparison at equal aperture ratio, as in your f/8 example. But since a larger format (larger sensor and larger focal length) generally allows using larger photo-sites (for the same amount of image detail through roughly equal pixel count) which in turn allows using a higher ISO speed while still limiting noise sufficiently. Thus a larger format can typically attain a given shutter speed at a higher aperture ratio. Then the DOF comparison is less clear.

Rough study of noise sources suggests that usable (noise limited) ISO speed is roughly proportional to pixel area, sensor area, or image area at equal pixel count, and so proportional to the square of focal length. If so, then the aperture ratio needed to get a given shutter speed varies in proportion to focal length, bringing us back to a main point of Charles S. Johnson's essay: the aperture "d" (more fully, the effective aperture diameter) needed is about the same, so DOF is about the same under the equivalent printing and viewing conditions discussed in the essay.

One more time: larger formats (larger sensors and larger focal lengths together) benefit mainly when they allow one to use larger aperture sizes "d" before aperture ratios get impractically low. That and IQ advantages from sometimes being able to use higher aperture ratios and thus lower shutter speeds at minimum ISO before highlights are blown.

[Roy]

...Thus a larger format can typically attain a given shutter speed at a higher aperture ratio. Then the DOF comparison is less clear...
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I expected this might get confusing, but I didn't expect red herrings!  

Of course larger sensors are potentially capable of better image quality and superior low light performance, but that isn't what this post is about. It is about DOF and under the conditions stated, a smaller sensor will give more DOF.

Another thread is the appropriate place to discuss the merits of large sensors.

[BJL]

I expected this might get confusing, but I didn't expect red herrings!   
It is about DOF and under the conditions stated, a smaller sensor will give more DOF.

Another thread is the appropriate place to discuss the merits of large sensors.
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I am tempted to say that a claim of one format giving more or less DOF than another is a red herring if it makes an unjustified assumption that the larger format will use an aperture that is larger in proportion to the increase in focal length used.

Here I mean "aperture" in the proper sense described in [a href=\"http://www.luminous-landscape.com/essays/Equivalent-Lenses.shtml]Charles Johnson's essay[/url]: d=f/N where f is focal length, N is aperture ratio.

Why should any user of a larger format care that using the same aperture ratio and thus a larger aperture will give less DOF if the DOF attained with the smaller format can be matched by simply choosing an appropriate aperture, meaning the same aperture, d=f/N? There is no reason why the user of a larger format would be constrained to choose the same aperture ratio as would be chosen in a smaller format unless the same shutter speed is needed and the larger format is limited to the same choice of exposure index (ISO speed). Equal limits on exposure index choices are very unlikely with different formats, so equal limits on aperture ratio choices are also unlikely.


I greatly like the idea of comparing what is possible with different equipment in different formats in terms of the goal of "equivalent composition" stated in Charles Johnson's essay: equal
1. Angle of view (field of view).
2. Perspective
3. Depth-of-field (DoF)
4. Diffraction broadening
5. Shutter speed/exposure

If one compares capabilities for such equivalent compositions, the main punch line of that essay is that with a larger format and focal length one will use equal aperture size d=f/N and thus will
- adjust aperture ratio in proportion to focal length (but only if equal DOF is necessary)
and
- adjust exposure index (ISO speed) in proportion to the square of the focal length (but only if equal shutter speed is necessary, which it is not always).

An interesting practical question to me is then
- how good can the image quality be from various formats for these equivalent compositions?
When constraints on shutter speed and DOF lead to the need for higher than minimum exposure index, this leads to the next question
- does increasing the exposure index used with a larger format in proportion to the square of the increase in focal length lead to better or worse image quality from the larger format?
My guess is that it turns out roughly equal, due to the inherent advantages in high exposure index performance of larger format sensors.

P. S. Please do not take this a a "bigger is better" advocate's cynicism about claims in favor of smaller formats. I am well known in these forums as a fan of "less large" formats like 4/3".

[stever]

without the figues, i found the analysis a bit tedious to follow

my practical take was that a crop sensor camera will have a larger DOF at a given lens aperture than a full frame camera and will also become diffraction limited at a smaller larger aperture than the full frame camera

if i've got this right, it's useful to know that i've got about 1 stop more depth of field at a given aperture with the 20D than the 5D and that i will need to open up a stop to get equivalent isolation of the subject

is this correct?

[Ray]

An interesting practical question to me is then
- how good can the image quality be from various formats for these equivalent compositions?
When constraints on shutter speed and DOF lead to the need for higher than minimum exposure index, this leads to the next question
- does increasing the exposure index used with a larger format in proportion to the square of the increase in focal length lead to better or worse image quality from the larger format?
My guess is that it turns out roughly equal, due to the inherent advantages in high exposure index performance of larger format sensors.
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Can I suggest the following experiment using a 5D or preferrably a 1Ds2, 50mm lens and 400mm lens. The 400mm lens is used to mimic the performance of an 8x10" field camera with standard lens.

A 35mm format 400mm lens will not stop down to f64, but my Canon 100-400 IS zoom reaches f40 at 400mm.

For DoF equivalence between the two formats (call it 8x12" to keep the aspect ratio the same), the f stop used on the 35mm camera should be multiplied by 8 for the 8x12" camera; thus, f5 with 50mm lens should be equivalent to f40 with the 400mm lens. In both cases the physical diameter of the apetures at these f stops is the same.

If we are to use the same shutter speed with both formats (whilst maintaining the same DoF), we should take a shot with 50mm lens and f5 at ISO 100, and a shot of the same scene (but heavily cropped of course) from the same position with the 400mm lens at f40, but underexposed by -2 EV at ISO 1600. In other words, we are effectively using ISO 6400 which we would have to use with our 8x12" field camera at f40 if we want correct exposure at the same shutter speed.

Now to compare images. The shot with the 400mm lens is basically what we would get if we went to the trouble of fitting a 1Ds2 body to the back of an 8x12" field camera, which I imagine would be a bit tricky.

Since we have captured only a 1/64th part of the scene using the 35mm format camera with the 400mm lens, we should compare this entire 16mp image with a 1/64th part of the shot using 50mm lens. That is, we should crop the ISO 100 shot to 16mp/64=250kp.

Thus, we are comparing a noise-free 250kp image with a very noisy 16mp image that's been underexposed by 2 stops at ISO 1600. Both images will have the same FoV and (possibly?) the same DoF. It is assumed that we have chosen the crop of the 50mm shot to correspond exactly with the scene content of the 400mm shot.

Some interesting questions arise in my mind.

(1) Can we assume that a 35mm format 400mm lens used at f40 will have approximately the same resolving power as an 8x10" format 400mm lens at f40? It would seem to me that we probably can assume that because both lenses are almost certainly going to be fully diffraction limited at such f stops, bearing in mind that diffraction limitation is not a sudden occurrence but a gradual transition.

(2) However, our 50mm lens at f5 is certainly not diffraction limited, so the question arises, on the basis of everything else being equal (or at an equal level of development), does the format which achieves the desired DoF through use of a lens at its diffraction limit have an inherent advantage over a smaller format which is unable to achieve diffraction limitation at the equivalent f stop for the same DoF?

(3) 64x16 megapixels is certainly good enough for 8x12" format (a slight overkill here, I would say   ), but is 16mp good enough for 35mm format?

[Roy]

without the figues, i found the analysis a bit tedious to follow

my practical take was that a crop sensor camera will have a larger DOF at a given lens aperture than a full frame camera and will also become diffraction limited at a smaller larger aperture than the full frame camera

if i've got this right, it's useful to know that i've got about 1 stop more depth of field at a given aperture with the 20D than the 5D and that i will need to open up a stop to get equivalent isolation of the subject

is this correct?
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That's a good summary, and one stop is very close to the difference for the same DOF.

I think by "smaller larger" you mean that the crop sensor camera will become diffraction limited at a larger aperture (i.e. f/14) than the full-frame camera (f/21). I just round this to: don't stop down beyond f/16 with an APS size sensor or f/22 with a full-frame sensor.

[CSidney]

I'm glad to see all the discussion.  Too bad the images didn't work with IE at first.  I think that is all fixed now.

Maybe I didn't make it sufficiently clear that both the focal length and the f-number must be scaled, i.e. they must both be proportional to the sensor size, to get the same field of view and depth of field.  In that way the aperture will be held constant regardless of the focal length.  Of course, the approximate DoF equation will not hold for closeup work where the distance from the lens to the sensor (q) is much different than the focal length.  By the way, don't forget that cropping an image is the same as using a smaller sensor!

I certainly didn't mean to suggest that small sensors are better for all purposes.  The f-number range for point-and-shoot cameras is severely limited by diffraction at the high end and technology at the low end.  I should say technology and economics.

However, if you need to take existing light photos indoors with a large depth of field, a little Fuji F30 does a good job.  As with most small cameras it does not offer raw format, and one is stuck with in camera sharpening and enhanced saturation.

Finally, it is true that the essay is another DoF article.  I did want to emphasize that all cameras can easily be stopped down to the point where diffraction becomes significant, and that stopping down past that point is a losing proposition.

[BJL]

my practical take was that a crop sensor camera will have a larger DOF at a given lens aperture than a full frame camera and will also become diffraction limited at a smaller larger aperture than the full frame camera
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I agree with one change: when you say aperture, you mean aperture RATIO. The essay makes it clear that the distinction between aperture ratio "N" and aperture "d" is important. To avoid confusion and ambiguity, I propose using the jargon
- aperture RATIO for the quantity N mentioned in the essay, also know as f-stop.
- aperture SIZE for the quantity d mentioned in the essay, d=f/N.
Also, the phrases "crop sensor" and "full frame" are red herrings, especially when the ideas apply from everything from compact digicams (no cropping involved) to formats larger than 35mm.

Then I would restate the above summary as follows.

A. a smaller format camera will have a larger DOF at a given lens aperture RATIO than a larger format camera and will also become diffraction limited at a lower aperture RATIO than a larger format camera.

B. cameras in any format used at a given lens aperture SIZE will have about equal DOF and equal diffraction limitation.

(At least if one assumes that the detail that the viewer can see is limited by print size, viewing conditions and the viewers visual acuity, not by sensor resolution or such.)


Moral: for equivalent results, aperture ratio and focal length both vary in proportion to format size.