[stevebrot]

Dang! This rabbit hole went deep fast and with considerable sketchiness as well. Amazing too is that there are no example photos to back any of the claims.

Several years ago I read an article where someone good at math did a thorough work-through of the DOF equation (DOF being the sign posted at the entrance to this particular rabbit hole). The outcome was that all of the factors cancel out except for entrance pupil size and final magnification. Chew on that awhile and it should be obvious that format, focal length, and crop factor only count if one decides they are important.

Another consideration might be that as final magnification approaches infinity, DOF approaches zero. If one truly wishes to puzzle, consider infinite magnification coupled with infinitely narrow entrance pupil size. Now multiply aperture by crop factor.


Steve

[Wasp]

Here is an example of full frame (left) versus MFT (right). The focal length on the left is 100mm and 200mm on the right. Almost identical images but very different settings.



https://www.picturecorrect.com/tips/understanding-crop-factor-are-you-being-...manufacturers/


There is a good reason for the difference in ISO, but we will have to go a little deeper into the rabbit hole for that.

[StiffLegged]

This is why I'm happy to keep a whippet, a breed of dog "dedicated to ridding the world of rabbits."

[Wasp]

I like this particular rabbit hole, so here goes. For noise, divide by the square of the crop factor. In the example above the crop factor for MFT is two. The square is four. 3200 divided by 4 is 800. A full frame image taken at ISO 3200 has the same noise as an MFT ISO 800 image. Yes, really.

The reasons are complicated and this is only a general rule. Differences between sensors make it even more complicated. But there is no denying that a good big one beats a good little one every day of the week. Anyway, the square of the 1.5 number for APS-C is 2.25, making mental arithmetic difficult. ISO comes in a limited set of numbers. 100 is the same as 225, does not compute. Make it 250, or forget the whole thing. So why bother?

Now you will have to excuse me. I have the feeling that I need a change of clothes. Where is my flame suit?

---

Last edited by Wasp; 12-14-2019 at 04:12 PM.

[stevebrot]

Here is an example of full frame (left) versus MFT (right). The focal length on the left is 100mm and 200mm on the right.

I believe there is a minor fault in the article in that the perspective of both shots is is also almost identical, meaning that the front nodal points were very close to the same physical location. I suspect that the 100mm lens was on the right and the 200mm lens on the left resulting in the same equivalent focal lengths. As such both frames have the same magnification and same entrance pupil diameter.


Steve

[stevebrot]

In the example above the crop factor for MFT is two. The square is four. 3200 divided by 4 is 800. A full frame image taken at ISO 3200 has the same noise as an MFT ISO 800 image. Yes, really.

Everything else (pixel pitch and such) being the same, yes. This is an incredible solution that indicates nicely that to get equivalence with a small sensor one needs four times more light to the sensor at equivalent focal lengths and apertures. Such is cool if one can get it and not if one can't.


Steve

[Wasp]

I believe there is a minor fault in the article in that the perspective of both shots is is also almost identical, meaning that the front nodal points were very close to the same physical location. I suspect that the 100mm lens was on the right and the 200mm lens on the left resulting in the same equivalent focal lengths. As such both frames have the same magnification and same entrance pupil diameter.


Steve

Indeed, I missed that one.

[photoptimist]

I believe there is a minor fault in the article in that the perspective of both shots is is also almost identical, meaning that the front nodal points were very close to the same physical location. I suspect that the 100mm lens was on the right and the 200mm lens on the left resulting in the same equivalent focal lengths. As such both frames have the same magnification and same entrance pupil diameter.


Steve

Good catch! You are right that the 100mm is on the right on the MFT camera and the 200 is on the left on the FF camera. And, yes, the front nodal points should be at the same physical location -- that's required for equivalence. There's no zooming with the feet in equivalence -- any change in the distances of the front nodal points makes the images non-equivalent because it changes the foreground-subject-background geometry.

However, the magnification at the sensor level is different. The 200 on FF has double the magnification compared to the 100 on MFT -- the same number of inches of subject width are cast across twice the sensor width on the FF camera.

[swanlefitte]

If we talk digital pixels we really need to specify pixel size. The CoC is defined off the pixel size, usually at 1.5x so talking a k3 vs a k1 is different than a k10 and k1. for those who want to calculate dof based on pixel size here is a calculator in French. Articles PGM
Here is the article I found it in.
Depth of field and digital sensors - Artfx

and here is some math that shows the film and digital crop stop down. from reply 6 of this discussion. DoF, sensor size, and pixel pitch
"To apply this apparatus to digital, we must re-express the CoC in terms of pixels, not microns, as explained in my previous post.

But let's first apply it to FILM cameras, which have no pixels, to the calculate the DoF's of full-frame vs APS. For concreteness, suppose we're using a full-frame 35mm SLR to photograph an object 10 meters away, using a 100mm lens (f = .1), using a CoC of 40 microns, at an arbitrary aperture N. If we work out the arithmetic (keeping track of all the zeroes) we find that T = .3N, implying a DoF of 1.2 meters at f/4, or 2.4m at f/8, etc., which is in the ballpark with the DoF scales engraved on a typical 100mm SLR lens.

Now suppose we take the same shot from the same vantage point with an an "APS" (16 x 24mm frame) body, using a wider lens (because of the "crop factor") to get the same FoV; the proper lens to use is one of focal length f' = (2/3)f = 68mm. Moreover, because the APS negative is only 2/3 as large, it will have to be enlarged more to yield the same final print size, by a factor of 3/2; hence, in order to end up with the same-sized blur spot on the print, we must take a smaller CoC, C' = (2/3)C. All other factors in the equation are the same, so the DoF for the APS shot, T', is

T' = [2*u*u*N*C']/f'*f' = [2*u*u*N*(2/3)C]/[(2/3)f*(2/3)*f] = (3/2)T = (3/2)*.3N

Thus, at the same aperture, the DoF in the APS image is 3/2 as large; to get the same DOF we would have to open up the aperture to N' = (2/3)N. Or, to put it the other way around, the DoF for a full-frame camera, using a longer lens to give the same FoV, is only 2/3 as deep, compared to the APS camera; and to get the same DoF we would have to stop down the full-frame's longer lens by the "crop factor" of 3/2: if the APS image (using the 68mm lens) were shot at f/4, we would have to stop down the 100mm lens on the full-frame camera to f/6.

This relationship, and the rule "stop down by the crop factor" is a general one, which does not depende on the numerical specifics of our example. This type of ananlysis is the basis for similar "rules", such as Alexander's "CoC =Diag/1500" rule, and Wrotniak's "M x A" rule.

But now let's apply this apparatus to digital, taking account of pixel pitch, specificaly to the issue of DoF on APS-C versus full-frame DSLRs, and using the D2X (an APS-C, 12MP camera with a pixel pitch of 5.5 microns) as a baseline.

The DoF of a 100mm lens focused at 10m, using a CoC = 30 microns, is T = .3N, as shown above (irrespective of the sensor size); so for the D2X, the relevant "CoC" measured in PIXELS--call it the DCoC--is 30/5.5 = 5.5 pixels in diameter. A 300-ppi print has about 12 pixels/mm, so the resulting blur spot would be about 420 microns in diameter, somewhat larger than the film-camera example used above; but this calculation ignores the effects of demosiacing & sharpening, and is close enough for illustrative purposes (suppose we're viewing all the digital prints at a somewhat greater viewing distance.) Thus the DoF for a D2X shot taken with a 100mm lens, focussed on an object 10 meters distant, is also given by T = .3N

Now let's compare the D2X image with one taken by a full-frame DSLR, in which the pixel pitch has been increased (by a factor of 3/2) to make the sensor large enough to fill the full 24 x 36mm frame. The pixel count is the same, so both cameras will yield the same (300-ppi) print size. To get the same-sized blur spot on the print, we must choose a DCoC 5.5 pixels in diameter, which for these larger pixels reqiuires a CoC of C' = (3/2)C. To get the same FoV, because of the "crop factor", we would use a lens of f' = (3/2)f = 150mm; so plugging these into the equation

T' = [2*u*u*N*C']/[f'*f'] = [2*u*u*N*(3/2)C]/[(3/2)f*(3/2)*f] = (2/3)T = (2/3)*.3N

This follows "stop down by the crop factor" rule, as did the film-camera example above.

But now suppose we use a full-frame camera--the "D3Z"--which uses the same pixel pitch as the D2X. Its 27MP image will yield a consdierably larger, 300-ppi print; but suppose we still view it at the same distance, using the same blur-spot size to define the in-focus field. To get the same blur-spot on the print, we must use the same CoC, C' = C = 30 microns, which is still 5.5 pixels in diameter; so, taking account of the longer lens, we have:

T" = [2*u*u*N*C']/[f'*f'] = [2*u*u*N*C]/[(3/2)f*(3/2)*f] = (4/9)T = (4/9)*.3N

There is now less than half the DoF; for the full-frame camera to get the same DoF as in the D2X shot, we must now stop down by the SQUARE of the "crop factor", in this case by (9/4). If the D2X shot were taken at f/4, we would have to stop the D3Z's 150mm lens to f/9--which is approaching the limit (f/11, according to Thom) at which D2X users have found that diffraction-induced blurring effects increase objectioanlly (basically because the diameter of the Airey disk exceeds the 2-pixel Nyquist limit)."

[stevebrot]

However, the magnification at the sensor level is different. The 200 on FF has double the magnification compared to the 100 on MFT -- the same number of inches of subject width are cast across twice the sensor width on the FF camera.

It is the final magnification and distance to the end viewer that counts for DOF.

In regards to pixel pitch, it is relevant when doing a comparison, but only to the extent that it might require an upsample or downsample of one image and not the other and then only because of image degradation, not a change in DOF.* For purposes of theory, it useful to consider a fully continuous-tone image with no dots or pixels and "average" visual acuity.


Steve

(...magnification defeats DOF...)

* Blur is independent of and still exists even with infinite DOF.

[Kobie]

Nobody in here has both a K1 and crop Pentax?
I tried doing an absolute comparison from my point and shoot super zoom but the issue is that the field of view on that camera was actually wider than the the K-3 at basically the same focal legth. The super zoom has a 1.2/5 sensor. And an apparent 38mm wide focal length. But that did not measure up at all to the field of view on my K-3 at 40mm (there was no 38 mm). The only way to get an answer that will satisfy is to use two cameras with different sensor sizes with the the same same lens set at the same focal legth.

[Ronald Oakes]

Nobody in here has both a K1 and crop Pentax?

Anyone with a K-1 already has a crop sensor ……..built in crop mode !

[Just1MoreDave]

Nobody in here has both a K1 and crop Pentax?

No, this thread talked me out of it. If I ever get a K1 or successor, I'll throw my APS-C camera off a cliff and skip the math.

[BruceBanner]

Wow... ok... this thread really brought the nerds out

I'm sure you'll all be aware that my academic intelligence is limited, so when I see equations I just skip on by

Really tho, the point of my entire thread was not to be mathematically accurate about the comparison between crop and full frame, it was more for me to get an idea of what I am seeing when swapping lens around between crop and ff bodies, what the hypothetical equivalent would be like. Not exact... like.

All this time I have been thinking my FA77 on a KP is what a 115/1.8 would be like on my K-1, but that would be slightly incorrect, its more like a 115/2.7, so that the depth of field cannot be stopped down as much as what a real 115/1.8 FF lens would actually achieve. This is important to me, not for maths or anything, but for feedback for future lens purchases. The FA77 on a KP perhaps gives a better DoF than a DFA 100/2.8 on the K-1. I get essentially 15mm more DoF effect with similar aperture... it's just a case of whether I need 24mp (KP+FA77) or the full 36mp (K-1+DFA100). You see where I'm coming from? I get an idea of a FF 135/2 (Samyang) or if I can 'contend' with a F135/2.8 (which might be a lot like a FA77 on a KP but with more mp.

This is all I really wanted to know, and also... I was thinking that a DA 12-24/4, set at 20mm and f4 on a KP would be like a 13/2.6 shot on a K-1... but now I am thinking perhaps not? And that the same DA 12-24 at 20mm f4 on the K-1 is actually 20/4, and that it's the KP when using the same lens at 20/4 is behaving like a 30/6 FF equivalent. And this is one of the reasons I enjoy the HD DA 20-40/2.8-4 and DA 12-24/4 on the K-1 so much, because it feels like I am experiencing the 'real' aperture, and not the crop aperture (which feels too stopped down)...

Does that make sense?

[Ian Stuart Forsyth]

There is now less than half the DoF; for the full-frame camera to get the same DoF as in the D2X shot, we must now stop down by the SQUARE of the "crop factor", in this case by (9/4). If the D2X shot were taken at f/4, we would have to stop the D3Z's 150mm lens to f/9--

You should only have to stop down the lens to F6 not the F9.

which is approaching the limit (f/11, according to Thom) at which D2X users have found that diffraction-induced blurring effects increase objectioanlly (basically because the diameter of the Airey disk exceeds the 2-pixel Nyquist limit)."

While you may start to see the diffraction start to creep in around that ƒ/11 we still can capture more resolution when shooting at ƒ16 with a 36 mp camera than a 12mp camera shot at its sharpest ƒstop. While shooting at those ƒstops you will start to see the signs of diffraction start to soften your image from its sharpest ƒ stop it would be in you best interest to stop down if you want to increase your DOF with a higher resolution camera.