### Is the good ol fashion Hyperfocal distance formula accurate for LF? I think not!

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I know this forum has a lot of photographic math wizards, someone should be able to answer this one. I will lay out my question in a logical sequence so it’s easy for you to troubleshoot my thinking….

1. The Hyperfocal distance formula requires the user to know the lens fl, desired max. size circle of confusion (cc) and the f stop.

2. The formula will calculate the near (half the hyperfocal distance) and far (infinity) in which all parts of film will have a circle of confusion (cc) no larger than the cc entered in the formula. This assumes that the lens is focused at the hyperfocal distance.

3. Here comes the first part of my question… the formula tells you that all cc on the film will be no larger than the cc entered in the formula. However, I truly doubt this, and will explain why below….there is more information required to determine the cc size that will end up on film. Lens fl, f stop and desired cc is simply not enough information, specially for LF …and what’s worse, it deceives us into believing the outcome will produce certain results, but in reality it won’t? Here is why I feel this way…. Below is a real world example….

4. Lets say we are shooting 8x10 format, we use a 360 fl lens, f32 and a desired max. size cc of .02mm (which will accommodate a desired 10x enlargement, or .2mm to print which equals 1/.2 = 5 lpmm to print, (which is the final objective here). Ok, so the Hyperfocal distance is 664 ft. So we focus at this distance, and have everything from 332 ft to infinity covering the entire gg. So in theory we should get our ENTIRE final print at 5 lpmm after 10x enlargement, right? No way…here is why, and this is where my real question comes in…where does diffraction and the “systems resolving power” (film + lens) come into play?

5. At f32, the max. a lens can resolve is diffraction limited at 1500/32 = 46 lpmm. Using the formula in the Fuji handbook of a “systems resolving power” 1/R = 1/r1 + 1/r2…. Where R = the system resolving power and r1, r2 is the system components, in our case here, r1= lens and r2 = film. So using this formula, and using Provia F film in real world contrast situations (not high contrast lens test targets) the Fuji film data for Provia F says we will resolve about 55 lpmm. So therefore, using the “system resolving power” formula, the best case scenario, which is at the point of exact focus, we should get 25 lpmm to film at the point of exact focus ONLY. (using 46 lpmm lens and 55 lpmm film). Now as you move further from the point of exact focus, 332 ft, the resolution would obviously get worse… so 25 lpmm will be ONLY at the point of exact focus and everything else in the scene would be much worse, say down to 10 lpmm? (this is just a guess for arguments sake) So if our desired goal was having everything in the print at 5 lpmm, we can only achieve a 2x enlargement, not the desired 10x enlargement that the Hyperfocal or DOF formula lead us to believe. This is a real world scenario, and as you can see, these two answers are miles apart! (5x difference) Yet both seem correct in their own right? I think the problem is a deficiency in the Hyperfocal distance formula. I imagine this time tested formula would work well in 35mm world with super high resolving lenses and shooting at f stops f5.6 and below, but it falls way short of the mark for LF!

6. So it seems that just arbitrarily putting a desired cc in the DOF or Hyperfocal formulas totally ignores, the lens, the film and diffraction. It leaves us with answers that are totally misleading as I have shown above. So, is there a formula that takes all these variables into account and comes up with the real largest cc (worst resolving area) that will end up on film? Even if the formula is complex, with today’s spreadsheets and programmable calculators, everything is reduced to entering the few changing variables each time.

7. It seems logical to have such a formula because the 1) diffraction limits, 2) the film lpmm and 3) the lens resolving powers can be estimated close enough to get to the PROPER answer. I will not even approach when one shoots at f stops which are not diffraction limited, because it seems from C Perez LF lens tests that all modern LF lenses, and most older ones, are always shot at diffraction limited f stops. (he mentions this in his test summary) Using the Fuji formula and estimating the B&W film C Perez used with high contrast targets resolved about 180 - 200 lpmm, the backtracking of the math seemed to justify the lenses which were shot at f11, 16 and 22 were always producing diffraction limited values. And unless one shoots wider than f11, diffraction is always limiting the LF lenses true capabilities. If you want to backtrack the math, I used the SSXL 110 as an example at f11= 80 lpmm, f16 = 67 lpmm and f22 = 60 lpmm. The goal is not to make the backtracking perfect, but just to get it close, will prove the point…there is many issues that can effect the resolving power outside of this… consistency of film, camera shake, exposure, etc. Furthermore, considering that LF lens MTF curves show they are optimized to be shot at f16 and higher, LF users are ALWAYS confronted with this scenario! Of course if one used high resolving B&W film the diffraction limiting f stop may move upwards by a one or two stops vs. poor resolving color films. But for arguments sake lets stick to this example so we are all comparing the same thing.

Sorry for being so lengthy, but I felt this issue was of importance to all of us who shoot LF since every shot is a battle with DOF and desired cc, and it seems to me, we are quite often confusing ourselves with these formulas that have been around forever! Any input would be helpful… Thank you all in advance..

-- Bill Glickman (bglick@pclv.com), December 18, 2000

### Answers

I know this is not the answer that you want. I have been using LF for 15 years, I stopped trying to use charts more than 10 years ago. No matter how sharp your pencil you can not take all of the variables into account, do the math and make a meaningful photograph. Even if you could figure it all out, accurate measurement is almost impossible in real world situations, at least accurate enough to justify these complicated formula. How many microns off does your film plane need to be to through everything out the window? I would suggest that you read this. I have never found the need to be any more accurate than this article suggests and it works!! Remember, the important thing is to make photos. Have fun!

-- Jeff White (jeff@jeffsphotos.com), December 18, 2000.

Bill:

Read the new section the Q posted on the LF Homepage on "How to select the f-stop" in 8/00.

This analysis, originally published by Peterson and Hansma in Phototechniques, explicitly includes both diffraction and DOF to obtain an "optimal" f-stop for a given COC. Clearly, one should not pick a COC that the film stock cannot make full use of!

On the other hand, photography is not just math. One of the main reasons for shooting 8x10 in the first place is that film and diffraction effects can essentially be ignored. You can make a tack sharp 24x30 print with only 15 lp/mm on film! Any film can do that, and you can stop down to nearly f/128 before diffraction becomes an issue. At some point, the light output of the sun becomes the limiting factor!

Diffraction is not the boogey-man waiting to destroy your images at some magic f-stop. It is a progressive deterioration that, along with all of the other factors that determine image quality, should be considered when selecting exposure settings. If DOF is the overriding factor to produce the image you want to create, select an f-stop that will give you that DOF!

I dare say that far more disappointment has occurred at the light table due to lack of DOF than softness from diffraction.

-- Glenn Kroeger (gkroeger@trinity.edu), December 18, 2000.

Jeff, I do appreciate your attitude in not getting overly tied up in math. I do realize there is a ton of variables. My goal in most photographic math is to just get in the ballpark. Anything more defined than that seems to be futile for all the reasons you mentioned. However, in this scenario, I am looking at two different formulas that are well respected that give answers that are 5x apart!!! Maybe they are even further apart, I made a conservative guess above to even determine this, I think they are closer to 10x apart. This is just too far off, and I want to know what is going wrong here. I did read Q's page, but it did not really get to the bottom of why these two very well accepted formulas conflict each other so bad! May of Q's math never introduced the lens and film potential. But it was an excellent piece...

Glen... you wrote..

This analysis, originally published by Peterson and Hansma in Phototechniques, explicitly includes both diffraction and DOF to obtain an "optimal" f-stop for a given COC. Clearly, one should not pick a COC that the film stock cannot make full use of!

Yes, but I do not see how your last sentence ties in....since it's the lens and the film together that dictactes the system resolving power as per the Fuji formula. You can't just determine a lpmm from all the other variables - and then say, the film must be able to resolve at least this amount. Alhtough that would certainly be nice and would love to see that this is true.

I understand that one can make a tack sharp print with 3x enlargement with only 15 lpmm to film. That was not my goal, as I stated in my example. As I see how the relationship between film and lens resolving power combine, I do see difraction as a boogey man who is seriously degrading the overall images. It gets worse as the format size goes up, so 8x10 gets hammered!

Here is a simple example to prove my point. In 8x10 format it is very common to shoot at f45. Lets use f32 to be conservative... so that is 20 lpmm at the point of exact focus. So therefore the moving away from the point of exact focus this 20 lpmm degrades... so at the edges of DOF, assuming non excessive DOF, the lpmm would probably drop to 5 lpmm. So this chrome will allow for a 1x enlargement if your goal is a min. 5 lpmm on the print??

I just happened to get an email from Chris Perez on this subject. Chris works so hard to help so many of us.... He seems to remember someone discovering the Fuji formula is wrong! He is reasearching this further. Has anyone else heard the formula Fuji posts in their Data Guide is wrong? It sure would be nice if this is true! It would certainly explain the inconsistencies here. Then one could just do as Glen suggested above, be sure your film is capable of acheiving the cc you are entering into your DOF calcs.! This would not be the first time I have seen a major company produce flawed formulas!

-- Bill Glickman (bglick@pclv.com), December 18, 2000.

I regularly use the maximum f-stop which my shutter allows. My 90/f8 has f64 for the maximum. I don't worry about it being soft due to diffraction at all. I worry about not getting the near/far in focus because of the distances involved. Forget diffraction limits. In LF the 4 or 6 times magnification is really too small to worry about it. James

-- lumberjack (james_mickelson@hotmail.com), December 18, 2000.

Bill:

The Fuji formula is, at best, a worst case scenario. Actual resolutions are much higher than you are calculating. First, you need to use the high contrast resolution, not the low contrast number. I think Provia is over 100 lp/mm at that contrast. Just look at Chris'es test numbers. He gets over 60 lp/mm at f/22 all the time. I shoot lens tests on Velvia at f/22 and clearly get over 50 lp/mm, so even at f/45, you can get over 25 lp/mm, even on color transparency film.

I can't see where you figure it gets worse with larger format. Essentially its a wash, since all the numbers are linear. Yes you need smaller f-stops, but you can tolerate proportionally lower lp/mm. What you gain is still small grain and better tonality, but not necessarily significantly better on print resolution.

There are a number of careful analyses online, one I think at Photodo.com, that show that due to DOF and diffraction, essentially all formats are equal... except for film which favors larger formats.

-- Glenn Kroeger (gkroeger@trinity.edu), December 18, 2000.

The Fuji formula is a crude reciprocal approximation based on multiplying MTF values of lens and film. Since resolution is NOT REALLY A QUANTITATIVE value, no equation that uses it can be "exact". Resolution is a perceived limit to MTF contrast at a given spatial frequency. Even the diffraction limits we toss around are just approximations using the Rayleigh criterion for resolution of point diffractors. It is not at all clear that that is "exactly" applicable either to arrays of dark and light lines, or worse, to real world spatial frequencies. But, it is a reasonable approximation. So assume Provia can resolve high contrast at about 100 lp/mm (not the 140 at super high contrast) and assume about 22 lp/mm at f/64. Even using the Fuji formula you still get about 18 lp/mm on film which will make a critically crisp 30x40 inch print. Anybody viewing larger prints at this distance is nuts anyway.

-- Glenn Kroeger (gkroeger@trinity.edu), December 18, 2000.

The optics that I outlined in the article previously mentioned give you the aerial definition r_lens. Applying the 1/R = 1/r_lens + 1/r_film formula would give the actual definition on film. While it seems that this latter would be more relevant, I ignored this point in the article because when r_lens becomes critical, the effect or r_film becomes quite neglectible. In theory, you'd have to convolute MTF's, and what you'd likely find is that the formula 1/R = 1/r_lens + 1/r_film wouldn't be a good approximation. In practice, several people, including Paul Hansma and Chris Perez have measured on-film resolutions pretty similar to aerial resolutions.

-- Q.-Tuan Luong (qtl@ai.sri.com), December 18, 2000.

Glen..you wrote.. I shoot lens tests on Velvia at f/22 and clearly get over 50 lp/mm, so even at f/45, you can get over 25 lp/mm, even on color transparency film.

I think this supports what C Perez mentioned, that the Fuji formula is just plain wrong. Q seems to support this also in his response...assuming I understood him correctly.

YOu mentioned..There are a number of careful analyses online, one I think at Photodo.com, that show that due to DOF and diffraction, essentially all formats are equal...

I do agree with this, provided the Fuji system formula is false, and as I mentioned I now conclude that it is false... I too have gotten 70 lpmm to Velvia and a SSXL 150 at f22. Even though this was at the focus plane, not at the near and far DOF points. But even so, the Fuji formula says the best on film resolution under this scenario is 48 lpmm. I am convinced I got 70 lpmm. It took a 15x loupe till I saw black and white lines getting fuzzy where they adjoin. So this further supports the fact the Fuji formula must be incorrect!

Q, you wrote...In practice, several people, including Paul Hansma and Chris Perez have measured on-film resolutions pretty similar to aerial resolutions.

So what are you concluding in this statement? I don't want to speculate what your intent is...

It seems that most of the test data supports chris Perez's information that the Fuji formula is not accurate! I wish someone would confirm this. I did speak to Fuji, and they stand by the formula, but I am sure somewhere inside Fuji there would be doubters! This field seems to be filled with bad information. And when it comes from such reliable sources as Fuji, it can really confuse people. Sure glad this forum exist...where else can we bounce these ideas off the best minds in photography!

-- Bill Glickman (bglick@pclv.com), December 18, 2000.

Bill, if I might add another comment: I've never seen anything definitive on the 1/R = 1/r1 + 1/r2… equation, other than something like "this formula is commonly used".

In John Williams' book "Image Clarity…", he uses the resolution numbers squared, ie, 1/R^2 = 1/r1^2 + 1/r2^2…etc. I have heard (third hand?) of other powers between 1 and 2 being used; I think it is somewhat empirical.

FWIW, I believe the book "Basic Photographic Materials and Processes" also uses the same formula for system resolution that you (and Fuji) have, so I wouldn't condemn it so quickly. I think (just guessing) that spread functions of the film (ie, characteristics of how a point of light blurs) might have something to do with the particular formula needed.

Regarding your large discrepancy, "…and as you can see, these two answers are miles apart! (5x difference)", is this just based on your guess as to resolution at the DOF limits? If so, I would question how legitimate that guess is; res might not fall off near as badly as you might think.

-- Bill C (bcarriel@cpicorp.com), December 18, 2000.

Bill, thanks for the input... you may have just rekindled the Fuji formula... to answer your question about the 5x discrepancy, see item #5 in the original post, it clearly esplains who I derived at it.. the only estimating I did was how much less resolution is at the near and far points of DOF vs. the plane of sharp focus which is what the formula calculates at.

BTW, all those A's eith the rounded top hats in my original post are a result of cutting and pasting the text from Microsoft Word into this forum. I was not aware this happened until after I did it...sorry about that...

-- Bill Glickman (bglick@pclv.com), December 18, 2000.

Bill: The Fuji formula isn't really wrong, it is just an approximation. The derivation depends on convolving the response of the film and optical system. The approximation is most valid when both the film and lens are being used at spatial frequencies with very low contrast, ie. near their resolution limits. Thus the formula is pretty good for 35mm work where lenses and films are operating near their limits. In LF work, we are usually interested in frequencies (lp/mm) where the film is not near its limit. This is particularly true with 8x10 where 20 lp/mm is way short of the films limits. This is where Q is correct, that you can essentially neglect the film, and consider only the lens limits. In this situation, the Fuji formula is a poor approximation. As to whether the frequencies need to be squared or not, it depends on assumptions about the shape of the response curves of film and lens, but again, in the case of 8x10 it is a poor approximation in either case.

-- Glenn Kroeger (gkroeger@trinity.edu), December 18, 2000.

Glen, this is very well stated and more clear than before. Maybe the Fuji formula is a good fit for 35mm only, all other formats stay clear of it!

It seems the best approach is to use the standard DOf formulas, but be sure to select a desired cc which is less than what the film is capable of recording at a given contrast level. Then one should be safe.

BTW, I do like the article, View camera focussing by Hansma referenced on Q's page. It offers a simple approach to DOF...I will have to test it out...

-- Bill Glickman (bglick@pclv.com), December 18, 2000.

I am not one to get into all the theory here, but the number of posts caught my eye. Wasn't there a website that demonstrated a format comparison with unaltered scans of Velvia in 3 formats at varying apertures? I can't recall where I found the link. Could've been a year ago.

Anyone know/remember what I'm talking about? Granted viewing results on the net is NOT what one would call a final arbiter but it was interesting, nonetheless.

-- Sean yates (yatescats@yahoo.com), December 19, 2000.

Bill: Even in 6x9 cm, color film is being used near its limit for enlargements >16x20. Most color emulsions really don't have much contrast above 50 lp/mm. From my experience, the film's MTF (or resolution) stops being a significant factor in 4x5 and larger formats.

-- Glenn Kroeger (gkroeger@trinity.edu), December 19, 2000.

In applications like photogrammetry and machine vision where you really do need to know the resolution of your system, lens manufacturers often supply a graph which shows how the MTF at some high spatial frequency dies away as you move from the point of best focus. It almost always forms a peak which is narrower than the spread between the usual depth of focus points. Edmund Scientific sell a suitable target if you want to estimate this at home.

Diffraction is one cause, as explained by Bob Atkins here:

http://bobatkins.com/photography/technical/dofdiff.html

Another cause is residual aberrations. The usual derivation of the DOF draws a diagram in which two rays from the outer edge of the lens' aperture (ok, exit pupil) cross at the focal plane. Aberrations turn this into a spray of lines with an outer envelope of sharpness which looks a bit like those 70's string and nail pictures. This can not only shift the point of best focus (the waist of the ray bundle isn't necessarily the classical focus point) but can also change how the light spot looks and spreads as you move away from the focal plane.

This can be, and is, handled mathematically with the use of optical transfer functions (a slightly more involved version of the MTF), and the blur on the resulting image treated like a convolution of errors. Quantifying the amount of blur at each individual stage and using a formula to find the total is conceptually simpler but because the blur from lenses and film follows non-gaussian statistics, the familiar root-mean-square combination is incorrect (though it'll usually get yu in the ballpark). Empirical and theoretical considerations of how film and lenses blur a point source leads to the 1/Rtot = 1/Rlens + 1/Rfilm formula. For more about this, try the following thread at Deja news, particularly Michael Gudzinowicz's contributions:

http://x72.deja.com/[ST_rn=ps]/viewthread.xp?AN=665426227

So it *can* be done, but in practical photographic use it usually turns out that the common trick of conservatively using the front and back hyperfocal distances for the next f-number more than covers the errors.

-- Struan Gray (struan.gray@sljus.lu.se), December 19, 2000.

Struan you wrote... So it *can* be done, but in practical photographic use it usually turns out that the common trick of conservatively using the front and back hyperfocal distances for the next f-number more than covers the errors.

What exactly does this mean? What is the front and back hyperfocal distances?

-- Bill Glickman (bglick@pclv.com), December 19, 2000.

"What exactly does this mean?"

I'm not sure myself - probably that I'd drunk too much Skånerost coffee before writing :-)

In smaller formats, where the lenses are usually mounted in a focussing helicoid and kind manufacturers put DOF markings on the barrel, it's quite common for people to use the markings for, say, f11 when photographing at f8. Other common tricks (in all formats) are to focus beyond the true hyperfocal distance to ensure that infinity is sharper than just 'acceptably blurred', and to stop down a bit more than a formal calculation suggests.

All of these things essentially hide the excess errors introduced by diffraction and finite film resolution, especially for large format film printed at normal sizes, where both effects are much less of a worry than accurate focussing and placement of your depth of field.

-- Struan Gray (struan.gray@sljus.lu.se), December 20, 2000.

Thank you all for your contributions.... rethinking this, I have done more research.... I now beleive there is much more credence to 1/R formula...here is a Mamiya 7 lens tested by Chris Perez. His results track 1/R perfectly...

I hope this post correctly..

Editor: if you want to make sure that the line breaks are preserved (such as in tabular data like below), enclose the section which you don't want reformated between the tags <pre> and </pre>

```	to film		lens		 1/R vs. actual
f stop	lpmm	Difr	Estimate   1/R	 % difference

4	76	375	180	   95	  20%
5.6	95	268	180	   95	  0%
8	95	188	180	   95	  0%
11	85	136	136	   81	  -5%
16	67	94	94	   64	  -5%
```

As you can see above, 1/R tracks Chris's findings perfectly! This assumes about 200 lpmm for the Tmax film used. (I backtracked into this value) At f4 it makes sense that the lens is not shooting at its best, so therefore it is slightly off vs. 1/R, the rest are right on the money.

This is just one example, however, I now do beleive that in more than 95% of the tests I looked at, 1/R is the max. the system can resolve. So Glen, when you suggest just pick a cc that the film can resolve, I would now suggest it makes much better sense to pick a cc that 1/R is capable of resolving. When a film like Provia F is displaced by the above, and LF f stops are used, the figures change to this...

```f stop	diff	film max.	1/R

16	94	60	       36
22	68	60	       32
32	47	60	       27
45	33	60	       21
64	23	60	       16
```

I now beleive that shooting at f32 and hoping for 47 lpmm to film would not be practical - the best I will be able to resolve is 27 lpmm to film. This is off by almost 100%, this is why I now suggest using 1/R as the ceiling, not the films resolving power.

This 1/R would have been almost perfect after applying it to all C Perez's tests, however there was a few super lenses, like SSXL 110mm, and Fuji A series in which they defied the 1/R a bit. But since these represented less than 5% of the lenses tested I think the 95% is more compelling evidence that 1/R is the max that can be resolved to film.

Does anyone see any holes in this theory? I am starting to think the lack of providing 1/R as a ceiling for cc selection in DOF calc. has been long fooling people into beleiving they are getting more enlargement potential than actualy exist! Also this effect shortens the DOF...whereas had one chosen the ceiling value of 1/R, it would have provided much greater DOF in the shot. LF shooters beg for more DOF, that is why I see this as such great importance.

-- Bill Glickman (bglick@pclv.com), December 21, 2000.

The "hole in this theory" was already pointed out to by Glenn and myself. What your first table (experimental) indicates, is that the 1/r_final = 1/r_lens + 1/r_film works well for *high frequencies and small f-numbers*. Glenn and myself don't disagree on that. However, what we were saying about low frequencies, implies that your second table (calculated) might not be that accurate.

Also, a "super lens" cannot do better than the diffraction limit.

-- Q.-Tuan Luong (qtl@ai.sri.com), December 21, 2000.

Tuan, thank you for the response...I fully understand, however, when one goes through all the lens tests on C Perez page you will notice that up to f32, 1/R still is accurate... so what's remaining that you are questioning, f45?

I understand that these "super lenses" can not yield better than diffraction limited resolutions...I don't have a good answer why these few lenses seem to out perform difraction limited lpmm on film?? But as I mentioned, they represented less than 5% of the total bunch tested, so their is still an overwhelming amount of evidence that 1/R is accurate. right?

-- Bill Glickman (bglick@pclv.com), December 21, 2000.

OK, I want to close this thread out...sorry for belaboring this point... it was quite confusing getting detailed informaiton on this, and sometimes the responses and the test data did not coincide. Glen, you wrote..

Thus the formula is pretty good for 35mm work where lenses and films are operating near their limits. In LF work, we are usually interested in frequencies (lp/mm) where the film is not near its limit. This is particularly true with 8x10 where 20 lp/mm is way short of the films limits. This is where Q is correct, that you can essentially neglect the film, and consider only the lens limits.

Glen, I think this is by far the best explanation of 1/R purpose in photography. From studying this and the test results, there is a compromise between the dif. limited lens resolving power and 1/R for fstops f22 and f32. This compromise at f22 leans more towards 1/R and at f32 leans more towards the lenses resolving power. My guess is, at f45 and above, the difraction limited reloving power goes straight to film and is no longer compromised by 1/R. (I never saw any tests at f45 or >)

So in summary, I would say 1/R is ceiling to be used and respected up to and including f16, then at f22 and f32 it's a mix, and at f45 and above, forget 1/R ever existed! Thank you all for your input... What a battle to get to this conclusion!

-- Bill Glickman (bglick@pclv.com), December 21, 2000.

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