The Zeiss 135mm API, f/2 and Depth-of-Field

[Bart_van_der_Wolf]

Nice chart, Bart.  May I ask how you determined the blur radius readings?  And did you do it off a raw single channel or demosaiced data?

Hi Jack,

I produce such charts for my lenses with the procedure outlined here.

For additional precision, I take 10 readings (evaluate, 1 line down and approx. 10 pixels to the right, repeat) along the slanted edge, in order to avoid phase errors that the approx. 10x oversampling may produce. This is especially important to eliminate the effects of aliasing. See the attached overview of such a data collection effort. I base the readings on the only White balanced and Raw converted (unsharpened) image (no geometric distortion correction, since that will blur parts of the image, but CA correction is allowed because it improves sharpness). That allows to incorporate the effect that a Raw converter has on the resolution of base image for Capture sharpening. Expressing these results in Gaussian Sigma blur units makes it easier to relate to practical consequences than a LW/PH metric would offer.

Right.  The main difference in my mind being that in theory one cannot undo defocus (= also axial spherical and chromatic aberrations in this context) in post while, always in theory, one can undo diffraction.

Well, it depends. Defocus can be reversed with deconvolution, but it depends on the specifics (e.g. noise and blur quality) how successful one will be. The Chromatic errors that will diminish with narrower apertures may be harder to correct at wider apertures. But for focus stacking one can choose the optimum aperture, or allow a certain (known) level of diffraction blur.

Hence if one does not need the shallower DOF one would most likely be better off shooting the D3X+ApoT*135 at around f/7 and your 1DsIII+EF100 at f/8 instead of wide open: same linear spatial resolution, fewer aberrations, most likely 'sharper' result once properly rendered.

Yes, one can easily see the trade-offs with a chart like that. Residual aberrations with a wider aperture may produce the same amount of blur as stopping down beyond the optimum, but the cause of the blur may be harder to correct at the wide end than at the diffracted end. So if a certain level of blur is allowed, I'd err on the side of diffraction.

Cheers,
Bart

[Jack Hogan]

Hi Jack,

I produce such charts for my lenses with the procedure outlined here.

For additional precision, I take 10 readings (evaluate, 1 line down and approx. 10 pixels to the right, repeat) along the slanted edge, in order to avoid phase errors that the approx. 10x oversampling may produce. This is especially important to eliminate the effects of aliasing. See the attached overview of such a data collection effort. I base the readings on the only White balanced and Raw converted (unsharpened) image (no geometric distortion correction, since that will blur parts of the image, but CA correction is allowed because it improves sharpness). That allows to incorporate the effect that a Raw converter has on the resolution of base image for Capture sharpening. Expressing these results in Gaussian Sigma blur units makes it easier to relate to practical consequences than a LW/PH metric would offer.

Well, it depends. Defocus can be reversed with deconvolution, but it depends on the specifics (e.g. noise and blur quality) how successful one will be. The Chromatic errors that will diminish with narrower apertures may be harder to correct at wider apertures. But for focus stacking one can choose the optimum aperture, or allow a certain (known) level of diffraction blur.

Yes, one can easily see the trade-offs with a chart like that. Residual aberrations with a wider aperture may produce the same amount of blur as stopping down beyond the optimum, but the cause of the blur may be harder to correct at the wide end than at the diffracted end. So if a certain level of blur is allowed, I'd err on the side of diffraction.

Cheers,
Bart

Very nice, Bart.  Out of curiosity, what's your criterion for the edge of the blur circle?

[Jack Hogan]

So, there is resolution and acutance.

On acutance.

[NancyP]

luscious looking cyclamen! Very nice indeed.

[Michael Erlewine]

I am working on an illustrated album of what I would call my “kit,” the lenses I regularly like to use, not that I carry them all around in a bag. It is funny. Once I get up into the stratosphere of highly-corrected APO lenses, there are still decisions to make as to which lenses I admire most. Many of these highly-corrected APO lenses are very flat, what I sometimes call “forensic” or “copy” lenses. Nothing wrong with that; in fact, that is what I have been striving for. Then I kind of look a gift-horse in the face and say, now, which of you lenses have character?

In other words, a lens can be highly corrected (like the Coastal Optics APO Macro f/4) and still not have character, which is a little unfair because first I want all aberrations, etc. (character) taken out of the lens, and then I rebuke it for just that. After all, I pay for expensive forensic-style lenses not to have any character whatsoever, to be perfectly transparent, and then I demand character. There is a disconnect here.

One of the lenses that is most highly corrected, not only in the visible spectrum, but in the near ultra-violet and near-infrared as well is the enlarger lens APO El Nikkor 105mm f/5.6, not to be confused with the same El Nikkor which is not APO.

This indeed is an incredible lens, very highly corrected, but one that also has all kinds of character. Go figure. Here is a stacked image (Zerene Stacker) I took with this lens probably taken on the Nikon D800E. You can see the almost 3D quality. It still needs some retouching, I know. Do you see what I am getting at here about character?

[Jack Hogan]

In other words, a lens can be highly corrected (like the Coastal Optics APO Macro f/4) and still not have character, which is a little unfair because first I want all aberrations, etc. (character) taken out of the lens, and then I rebuke it for just that. After all, I pay for expensive forensic-style lenses not to have any character whatsoever, to be perfectly transparent, and then I demand character. There is a disconnect here.

Reminds me of discussions around high end audio equipment.

This indeed is an incredible lens, very highly corrected, but one that also has all kinds of character. Go figure. Here is a stacked image (Zerene Stacker) I took with this lens probably taken on the Nikon D800E. You can see the almost 3D quality. It still needs some retouching, I know. Do you see what I am getting at here about character?

Beautiful.  Although, not to diminish the contribution of your outstanding lenses, I have a feeling that what we are perceiving more of is the character of your virtuoso performance in Zerene Stacker and PP

[Michael Erlewine]

Reminds me of discussions around high end audio equipment.

Beautiful.  Although, not to diminish the contribution of your outstanding lenses, I have a feeling that what we are perceiving more of is the character of your virtuoso performance in Zerene Stacker and PP

I wish, but lens experts agree with my view of the APO El Nikkor 105mm. For example, here is a quote from macro-lens expert Klaus D. Schmitt, author of the very fine Macro Lens Collection Database, that was just exchanged yesterday. We were discussing the APO El Nikkors, in particular the 105mm’s big brother, the 210mm APO El Nikor, which weighs a couple of pounds!

http://www.macrolenses.de/

He writes:

“Yes, the APO El 105mm is indeed a fantastic lens. I consulted a friend doing macro and close-up shots of sparkling jewelry, and he’s blown away about the results he now gets. Very different than the Printing Nikkors, the latter being fantastic lenses themselves, but quite different in rendering, the AENAPO105 is more “alive,” hard to express indeed.”

So, I do believe that even very flat lenses that, because they are highly corrected, might be similarly transparent as far as character go, definitely can have real character INDEPENDENT of the skill of the photographer. Here is another shot, this one taken with the Nikon D810 and the APO El Nikkor 105mm enlarger lens, mounted on the PB-4 Bellows. Stacked with Zerene Stacker.

[Bart_van_der_Wolf]

Very nice, Bart.  Out of curiosity, what's your criterion for the edge of the blur circle? And (bonus rhetorical question) what's the PSF of defocus?

Jack, I'll try and not hijack Michael's thread, but the question is relevant.

The Point spread functions of lens aberrations, defocus, and diffraction have different shapes. Therefore, if we were to look at them in isolation (e.g. for construction of a mathematical model), we would probably need something shaped like a (perhaps elliptical) Gaussian PSF, a disc shaped PSF, and an Airy pattern shaped PSF. We could then weigh in their contribution proportionally, although that will be computationally expensive because it requires integration to allow of positive and negative contributions to the overall MTF. And then there is the influence of the sensel aperture, and the demosaicing, and gamma pre-compensation for display.

As it happens to turn out, the composite of all that, looks almost perfectly like a Gaussian shaped PSF. When adding different PSFs, one very quickly get a Gaussian average, and the empirical results prove that to be true. An edge transition, like that of a Slanted edge target, takes on the shape of a normal Cumulative Distribution Function (CDF). We can therefore fit a CDF model to the edge transition, and the blur sigma is the result.

This blur sigma does not have a fixed diameter because a Gaussian has an infinite extent, so we will have to draw an arbitrary boundary at which we declare it a no longer acceptable amount of blur, e.g. the Circle of Confusion. One possible metric is the 10 to 90% rise of the edge profile which can be expressed in a radial distance in pixels, and for focus stacking one could define one's personal COC and thus pick an aperture that satisfies that condition. Since we are using a Gaussian shaped PSF model, it's easy to translate the 'blur sigma' to a 10-90% edge profile rise, and restate the y-axis values of the chart. A 10-90% edge profile rise (the difference between the 10th and 90th percentile points) of a CDF is approx. equal to 2.5631 x sigma, in pixels.

So, if we find a 2 pixel diameter to be our limit of acceptable blur on my above tested lens, then we should draw the limit at a maximum blur of 2 / 2.5631= 0.78 sigma, or f/6.3, and choose our stacking distance interval accordingly. It also shows that at optimum performance, at aperture f/4.5 or f/5.0, we will have a COC of 0.72 x 2.5631 =1.85 pixels for that specific lens and camera and converter combination. That suggests we need deconvolution sharpening for restoration of full resolution if we want uncompromised resolution at the pixel level. In the case of Michael who seems to do reduced size output, we can scale those requirements up accordingly.

Cheers,
Bart

[Michael Erlewine]

Bart: You are not high jacking the thread. I find your blogs amazing, even confounding. Of course, I can’t claim to understand it all. I would have to go and study, and for a long time, to do that. I am glad such explanations exist.

When you say “Michael who seems to do reduced size output,” what does that mean in layman terms? It would help folks like me if, after such an exposition, at the bottom, you had a “conclusion,” where you summed up for non-technical folks like myself, what you write so eloquently is actually all about.

[Jack Hogan]

Jack, I'll try and not hijack Michael's thread, but the question is relevant.

The Point spread functions of lens aberrations, defocus, and diffraction have different shapes. Therefore, if we were to look at them in isolation (e.g. for construction of a mathematical model), we would probably need something shaped like a (perhaps elliptical) Gaussian PSF, a disc shaped PSF, and an Airy pattern shaped PSF. We could then weigh in their contribution proportionally, although that will be computationally expensive because it requires integration to allow of positive and negative contributions to the overall MTF. And then there is the influence of the sensel aperture, and the demosaicing, and gamma pre-compensation for display.

As it happens to turn out, the composite of all that, looks almost perfectly like a Gaussian shaped PSF. When adding different PSFs, one very quickly get a Gaussian average, and the empirical results prove that to be true. An edge transition, like that of a Slanted edge target, takes on the shape of a normal Cumulative Distribution Function (CDF). We can therefore fit a CDF model to the edge transition, and the blur sigma is the result.

This blur sigma does not have a fixed diameter because a Gaussian has an infinite extent, so we will have to draw an arbitrary boundary at which we declare it a no longer acceptable amount of blur, e.g. the Circle of Confusion. One possible metric is the 10 to 90% rise of the edge profile which can be expressed in a radial distance in pixels, and for focus stacking one could define one's personal COC and thus pick an aperture that satisfies that condition. Since we are using a Gaussian shaped PSF model, it's easy to translate the 'blur sigma' to a 10-90% edge profile rise, and restate the y-axis values of the chart. A 10-90% edge profile rise (the difference between the 10th and 90th percentile points) of a CDF is approx. equal to 2.5631 x sigma, in pixels.

So, if we find a 2 pixel diameter to be our limit of acceptable blur on my above tested lens, then we should draw the limit at a maximum blur of 2 / 2.5631= 0.78 sigma, or f/6.3, and choose our stacking distance interval accordingly. It also shows that at optimum performance, at aperture f/4.5 or f/5.0, we will have a COC of 0.72 x 2.5631 =1.85 pixels for that specific lens and camera and converter combination. That suggests we need deconvolution sharpening for restoration of full resolution if we want uncompromised resolution at the pixel level. In the case of Michael who seems to do reduced size output, we can scale those requirements up accordingly.

Cheers,
Bart

Excellent explanation, thank you Bart.

[Bart_van_der_Wolf]

Bart: You are not high jacking the thread. I find your blogs amazing, even confounding. Of course, I can’t claim to understand it all. I would have to go and study, and for a long time, to do that. I am glad such explanations exist.

When you say “Michael who seems to do reduced size output,” what does that mean in layman terms? It would help folks like me if, after such an exposition, at the bottom, you had a “conclusion,” where you summed up for non-technical folks like myself, what you write so eloquently is actually all about.

Michael, from what I've seen you mostly publish very nice images in PDF documents, and on the internet. I do not know if you also print large format output. If you only need reduced size output, then you can relax the resolution requirements a lot.

When one technically determines the performance of e.g. your Otus lenses at various apertures, it then becomes much easier to draw lines as to what's still discernible at certain output sizes, and what's not. When, as in my example, a 2 pixel blur is used but you only output to 50% of the original capture size, there will be no blur in the resulting image. In a 25% of maximum output size, a 4 pixel blur (blur sigma 1.56, or in my lens case f/32) should not be much of a problem.

We do need to also account for magnification factor or focus distance, so we get a narrower effective aperture the closer we shoot. The formula for that is; Effective aperture = Aperture * (1 + (imageMagnification / pupilFactor)), so we need to divide the aperture we read the chart with by (1 + (imageMagnification / pupilFactor)).

That's the technical approach to it, but one can of course also use a more empirical approach and guess that e.g. f/16 will produce acceptable results when output size doesn't need to be huge.

Cheers,
Bart

[Michael Erlewine]

Bart, I don’t print photos and never have, although I am actually thinking about doing so for a show. All of my knowledge is empirical, just from experience. I would not want to cut any corners for reduced-size output because the main reason I do this is the “see” the results in as perfect a form as I can on-screen. Of course, what I see from a 36MP shot on the D810 in Photoshop or Lightroom (I use both) cannot be represented fairly here at sRGB format for the web. It is a sad approximation IMO.  This last piece you wrote is more understandable for me. Thanks!

[NancyP]

You should see the radiology monitors. These have replaced film for many years, and the highest grade mammography/breast ultrasound monitors now do both B and W and color images at 8 MP. I have seen a lot of film mammograms over the years, and the new digital monitors do seem to be just as good, but with the added benefit of being able to alter the "development" on the fly.

[swisscheese]

Thank you so much for sharing your immense knowledge. One thing I noticed in your comparison with the Two Dollar bills (VH8AA): The left side of the bill is quite a bit greenish @ f2, and pretty much neutral @ f16. Did anything change (light, processing etc) between the 2 shots, or does it have something to do with diffraction?
You do beautiful work!
Markus