Thank you for this different formula. The result from with the values I first assumed would be: max system resolution = 1 /(squareroot((1/max resolution film)^2 + (1/max resolution lens)^2))=1/squareroot((1/200)^2+(1/75)^2)=70 lpm. Perez / Thalmann´s results at f/22 are slightly below that for the best lenses. So the "paradox" would be clarified.
Originally Posted by alanrockwood
I looked to find out the "best" approximation of a sytem´s resolution and found in contrary to your formula at the very in depth sites of Norman Koren (http://www.normankoren.com/Tutorials/MTF.html) the following:
Resolution of an imaging system (old definition)— Using the assumption that resolution is a frequency where MTF is 10% or less, the resolution r of a system consisting of n components, each of which has an MTF curve similar to those shown below, can be approximated by the equation, 1/r = 1/r1 + 1/r2 + ... + 1/rn (equivalently, r = 1/(1/r1 + 1/r2 + ... + 1/rn )). This equation is adequate as a first order estimate, but not as accurate as multiplying MTF's. [I verified it with a bit of mathematics, assuming a second order MTF rolloff typical of the curves below. It's not sensitive to the MTF percentage that defines r. The approximation, 1/r2 = 1/r12 + 1/r22 + ..., is not accurate.
This is the formula that I used in the beginning and that led to the contradiction between theory and the test results of Perez / Thalmann.
Bob Atkins cites your "square formula" which is also written in the high res photography bible "Image Clarity" by Williams. Atkins doubt it. He writes here http://www.bobatkins.com/photography/technical/rrs.html :
This relationship is often given as:
1/Rs^2 = 1/Rf^2 + 1/Rl^2
where Rs is the final system resolution, Rf is the film resolution and Rl is the lens resolution. This relationship is quoted in many texts, including publications by Kodak (P315 - Scientific Imaging) and in "Image Clarity" by John Williams. It is certainly a relationship which can give a decent approximation of system resolution under some conditions. However it is not based on any sound theory and can, in fact, be shown to give erroneous results when used improperly. There is no reason why the exponent should be 2. Some studies have shown a better fit with smaller exponents in the region of about 1.5, but even then, it's still empirical - it applies only under the conditions used for the experiment in which it was determined.
I´m still confused but a step farther when I see that the formulae for calculating system´s resolution provide only roughly approximated values if you don´t use the MTF of the sytem´s components. But where do I find a lense´s MTF at spatial frequencies in near of 60 lpm, maybe at 50 cycles?