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What is the correct way to estimate the minimum magnification (maximum subject distance) at which features of a certain size are resolved?

My thinking is that given a system capable of resolving a certain number of line pairs/mm you should be able to estimate the point where certain features are no longer resolved by doing the following:

1 / (lp/mm) = minimum feature width on sensor

minimum feature width on sensor / desired object space feature width = minimum magnification

As an example resolving individual hairs on a Canon 7D where the sensor is the limiting factor:

1/(104 lp/mm) = 9.62um at the sensor

9.62um / 100um (avg human hair) = 0.096x minimum magnification

Applying this to the approx sensor size (22.5mm x 15mm) we can state that we shouldn't expect to see individual hairs in an image with a object space field of view greater than approximately 235mm x 156mm.

Am I missing something? Or is my reasoning generally correct?

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I think I get what you are asking - we can assume the sensor limits resolution, and if we know magnification of the system we can then relate this to an object space resolution limit.

Your approach is going in the right direction, but you should use the rayleigh criterion as the definition of resolution on the sensor (this assumes diffraction limit. Aberration limit would be much harder to calculate without actual lens data). This is proportional to F/#, so as noted in one of the other answers focal length does come into this. We set it so that the radius of the airy disk that results from diffraction covers 2 pixels. If you know magnification you can relate this to object space information, or you can use h=f*tan(HFOV) where h is the height on the detector (2 pixels), f is the focal length and HFOV is the half field of view. This will give you an angular limit that two points can approach and still be resolved. From this angle you can find separation if you know distance, and vice versa.

Please let me know if you have any questions, this can be a difficult subject to follow.

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Richard, IMHO you mix a little bit things. You need magnification of the lens. And most of the kit lens have magnification 1:5 which is 0.2 and it's enough to see the hair on the image. I find in one old answer here, in StackExchange formula you can use to calculate how big will be in sense of pixels one real object: How do I calculate the distance of an object in a photo?

  • Thanks, I appreciate the answer perhaps my question is poorly phrased. I want to know how approximately how far away (magnification) you can be from an object before features of a certain size are no longer resolvable for a given system (camera + lens). I'm trying to do this independent of focal length hence my use of actual magnification. The linked question looks promising however. – Richard Smith Dec 5 '14 at 7:21
  • Richard, I am afraid you can't do this calculation w/o focal length. Think about next example: zoom lens, fixed distance to object with short FL you get some elements/details of the object unregistered by sensor because the size of these elements on the sensor are under one pixel (or registered as one pixel only). Same lens and distance, but zoom out you will "see" just part of the object, but these elements/details will be projected to the sensor on much bigger size so they will be visible on the image. In the above formula just set object size (pixels) to 0.9 and calculate the distance – Romeo Ninov Dec 5 '14 at 8:01
  • P.S. Sorry for my English, but this is not my mother tongue :) – Romeo Ninov Dec 5 '14 at 8:02
  • No problem about the English. The zoom lens example is changing the magnification as you change the focal length. Obviously it can never be entirely independent of focal length as magnification is (f/d-f). I suspect I just need to do some algebra with the equations in the linked question. – Richard Smith Dec 5 '14 at 10:27
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It seems that you are looking for the resolution of a given optical system. If this is true then such estimation is rather complex as it involves both the resolution of the lens and the resolution of the sensor. These computations take into account basically the lens quality and the apperture, the size and density of the sensor and more. In addition, to be accurate one has to account for atmospheric and other factors, but these could be eliminated if tests are guaranteed to be conducted under the exact same conditions.

Since this is a complex situation, there has been a practical approach to determine the resolution of an optical system, which dates back to the 1950s. Take a look at the 1951 USAF resolution test chart. It comprises of a test pattern carefully designed so as to provide a good estimate of the resolution of an optical system. One has to digitize this pattern with the optical system at hand (in our case to photograph it), notice in which cases the pattern remains clear and distinguishable and then make a calculation using a specific formula. You might also be aware of (or find) other, more modern, patterns available out there that are based on similar notions. Typical professional scanning systems come with a variaty of such test patterns.

Once you are able to get an estimate of your systems' resolution (using one of the above charts) you are able to design your own test setup, i.e. to fix distances, cameras, lights, conditions, etc and starting experimenting and comparing...

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