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cshimokita
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Using the "Zeiss formula" to understand the circle of confusion

How I learned to live with the technical details of photography and still keep my sense of humor - or not.  We all have our own circles of confusion, but today we take a brief look at the CoC as it pertains to taking pictures.

You are about to enter another dimension, a dimension not only of sight but of mind. A journey into a wondrous land of imagination.  Witness if you will a dungeon, made out of mountains, salt flats and sand that stretch to infinity. The dungeon has an inmate standing next to a tripod...

Depth of Field has been explained in multiple ways, using formulae or drawings of light passing through an imaginary lens to focus or not on the surface of the film or sensor... A certain amount of math is to be expected and this next bit is a little technical but it's the background to Hyperfocal distance which is key to understanding DOF so lets jump in...

"Photography for Students of Physics and Chemistry" by Louis Derr, Norwood Press (1906)

Hyperfocal Distance. - It is not difficult to calculate, for any diameter and focal length of lens, how far distant a sharply focused object must be in order that all beyond it may be defined with a distinctness according to the  above limit. 

In Fig. 49, let F' be the image of the point whose distance p from the lens is to be such that the circle  of confusion ab shall not exceed a diameter d, ab being the blurred image of an infinitely distant point whose sharp image is at the principal focus F.  Also, let

Then, from the similar triangles of the figure, having their common vertex at F,

Combining this with equation (4),

in order to eliminate p', and reducing, gives the result,

This distance p, beyond which all is in focus, is sometimes called the Hyperfocal distance.

Certainly a little math from 1906 was not too difficult... using formula (19) from the text by Derr we can simplify: the aperture diameter (D) is the ratio of the focal length (f) to the numerical aperture (N) and the length ab=d is the diameter of the circle of confusion (c), which gives us the present day formula as follows:

Hyperfocal distance (H) is equal to the lens focal length (f) plus f² divided by the f-stop (N) x our good friend CoC.

For example: a 50 mm lens shooting at f/5.6 gives us the following results (note: an accepted CoC value for full frame is 0.03 mm - more on that later)...

Hyperfocal distance=50 mm + 50² / 5.6 x 0.03=14.9 meters

From the Hyperfocal distance all else comes clear...  the near and far limits for a given distance to subject (s)... and the total depth of field.

It's an easy spreadsheet calculation and there are online calculators to make it simple...

Pause to catch our breath and then lets take a look at circles of confusion (CoC)...

One rule-of-thumb is that the final image, the print, would be viewed at "perspective-correct" distance such that the angle of view would be the same as that of the original image.  This would imply that the "optimal" viewing distance=focal length of the lens × enlargement.  So for example a 360 x 240 mm print (size 4P, a 10x enlargement) taken with a 50 mm lens would be viewed at a distance of 50 cm... I just checked and it feels about right, WOW.

With the so-called "Zeiss formula", the circle of confusion is calculated as d/1730 where d is the diagonal measure of the original negative (or sensor size). A full-frame format of 24×36 mm has a 43 mm diagonal that gives us a CoC of 0.025 mm.  On the other hand, the industry standard more or less agreed on a full frame CoC of 0.030 to 0.033 mm.  A CoC of 0.03 mm is about 33 lp/mm (line pairs per mm, where one line pair consists of one black and one white line).

The international standard for an "acceptable" level of blur is defined as 1/1000 of the camera format diagonal or a full frame CoC of 43 μm (0.043 mm), about 23 lp/mm...  why is the international standard always so conservative when compared to real-world logic.

You will love this bit (note sense of humor): The "Zeiss formula" is apocryphal... it has become well known by repeated use on the internet.  There is no official source, no tangible connection to the Carl Zeiss Company, and no usage within the industry... it's a web thing.

Summary and a word about DOF, CoC, and format dependence...  So far we have been talking about optics and to some extent 'media granularity' as it relates to the viewing experience.  With digital some of the underlying assumptions and 'standards' are shifting... but the last time I looked there was still a physical lens (and all that implies) at the front end of the process... and as long as that's the case, Hyperfocal distance and Circles of Confusion (with roots that are 70+ years old) are still in the game...  The big changes in recent years have been with the technology of the capture (new films post WWII and then digital) and how we view the image (wet prints, digital printing, 4K & 8K screens). To me it seems that we are ready to see a tidal change to the optics component and then we can find new ways to talk about blur.

They took the credit for your second symphony
Rewritten by machine on new technology
And now I understand the problems you can see
Oh a oh

"Video Killed the Radio Star", written by Trevor Horn, Geoff Downes and Bruce Woolley in 1977 and first recorded by Bruce Woolley and The Camera Club.

... because every thread needs one interesting photo (DOF=47.5 cm)