Stephen Barrett wrote:

Thanks Detail Man,

Your approach is much more sophisticated than mine, which is quite crude.

I don't really know anything about the low-pass filters in cameras, or about de-mosaicing algorithms or details about Bayer arrays. What I have called "sensor resolution" also has an implicit "fudge factor" of 2" so maybe my "sensor resolution" could be considered to include some of these factors that you mention.

Combining my "lens resolution" and "sensor resolution" in quadrature has some cited precedent, but I have seen an argument ( http://www.normankoren.com/Tutorials/MTF.html ) that they should be combined linearly. For now though, I have kept the quadrature combination because it seems to match the resolutions that I see in my tests for a variety of situations (telephoto, macro & telemacro). In particular, the linear combination of factors predicts that the camera should not be able to resolve things that it can resolve, whereas the quadrature combination seems to work well. Because of the quadrature combination, only the larger of the two is noticeable when one is much larger than the other. The smaller factor only becomes noticeable when it grows to a size that is comparable to the larger one. This seems to match what people report seeing. For example, people do not report any diffraction effects at short focal lengths but, as focal length is increased so that images on the sensor are spread out over more pixels, the limits of lens resolution become apparent rather suddenly. The same thing seems to happen with change of aperture.

Have seen Koren's statement before. Here is a thread started by one of the most knowledgeable posters on these forums (Marianne Oelund), and with some others (including Falk Lumo) posting. Marianne recommends combining the variances (squares) as you already do:

http://www.dpreview.com/forums/thread/3360636

http://www.dpreview.com/forums/post/50576749

If you have not already seen it, you would probably be interested in Falk Lumo's paper here:

http://www.falklumo.com/lumolabs/articles/sharpness/

PDF format of the very same text and graphics (much easier on the eyes to read):

http://www.falklumo.com/lumolabs/articles/sharpness/ImageSharpness.pdf

See the paper's Section 2.3 (Defocus) - which shows how much more complicated things get mathematically regarding the composite MTFs if/when any focusing-error exists. Important to know.

While Lumo is describing focus-error at the film/sensor plane locations, it seems to me that the "defocus" (resulting in an estimated Circle of Confusion diameter on the film/sensor plane) which result from human visual perception limitations - at a specified viewing-size (usually 10 Inches in the largest dimension) and viewing-distance (usually 25 cm) would produce something of a similar mathematical model. Complex, because the human visual perception "contrast sensitivity function" is variable between persons, changes with aging, is light-level dependent, an represents a "band-pass" response of its own).

For more about common COC standards, read pages 1-4 of "Depth of Field in Depth" here:

http://www.largeformatphotography.info/articles/DoFinDepth.pdf

For (a bit) about the human visual "contrast sensitivity functions" effects, see:

http://www.bobatkins.com/photography/technical/mtf/mtf4.html

This thread (about human perceptions of "sharpness") and its references may also interest you:

http://www.dpreview.com/forums/thread/3135840

Most of us are probably unwilling or unable to deal with MTF functions and Bessel Functions, demosaicing algorithms etc. Is it possible to derive a simpler formula that combines several factors in order to compute resolution?

I don't think so (other than the combining of variances of the individual space-domain dimensions). The formulas for Diffraction, and for Photosite Aperture convolved with optical low-pass filters are not highly complicated. All one needs to do is to set the relevant parameters, and then multiply those two functions together (as real numbers in the spatial frequency domain).

Here is a post where those identities are shown (with constants for different "strength" AA filters):

http://www.dpreview.com/forums/post/51323901

The assumption in the above identities is that Photosite Pitch equals Photosite Aperture (Fill Factor = 100%). For making rough calls as to when Diffraction MTF "extinction" effects impact the net composite spatial frequency (MTF) response, I added some other things:

Assuming a "fudge factor" (K) for a Bayer-arrayed, CFA image-sensor (which is likely a bit wider still, due to the fact that most de-mosaicing algorithms utilize wider than 2x2 arrays in their interpolations).

I calculate the "critical" (minumum) Photosite Aperture dimension to be:

Pa = ( Pa / ( K * Pp ) ) * ( Fz ) * ( W * N )

where:

Pa is Photosite Aperture;

K is the Bayer-arrayed and de-mosaiced "fudge factor";

Pp is Photosite Pitch;

Fz is the fraction of the spatial sampling frequency (which is equal to the reciprocal of Photosite Pitch) at which the first zero magnitude response occurs in the composite Optical ("AA") Filter combined (convolved in the spatial domain, multiplied in the spatial frequency domain) with the Photosite Aperture);

W is the Wavelength;

N is the F-Ratio of the lens-system.

.

Solving for the simple case of a 100% Fill Factor (Photosite Aperture equals Photosite Pitch), setting the value of K to conservative value equal to 2, and setting the value of Fz to 1/2 (the strongest possible "AA Filter" (resulting in a zero magnitude response at the Nyquist spatial frequency), the identity presented above simplifies to the following form:

Pa = ( W * N ) / 4

Re-arranging to solve for the maximum F-Ratio (N) as a function of Wavelength (W) and Photosite Aperture (Pa):

Nmax = ( 4 * Pa ) / W

http://www.dpreview.com/forums/post/51858399

.

I use an Excel-type spreadsheet (which a friend created, and I have modified for my specific purposes) to perform the (basic) MTF calculations, and to create graphs with multiple individual plots within those graphs. I can email it to you if you want (PM me an email address if so). You would need to figure out on your own what is going on where with the individual variables used in the calculations, and then make modifications appropriate for your desired utilizations. There is no "users manual", and it has been a while since I messed around with it.

Perhaps it would have to be calibrated for each camera + lens combination. The formulas that I have proposed seem to work well for my camera, but I don't really know about other cameras. Are these formulas reasonable, even if they are crude?

They are crude. Whether they are "reasonable" depends entirely on the context it would seem ?

Can they be corrected or refined? Any insight that you have on this would be much appreciated.

One needs to be able to estimate the Photosite Aperture and the "strength" of the optical low-pass ("AA") filter. The Diffraction identity that I use is for an ideal lens (with a circular aperture opening).

Optical lens-system aberrations cannot be modelled using frequency domain multiplications (ray-tracing in the spatial domain has to be used instead), and such aberrations can only reduce the magnitude of the MTF responses.

So, the "ideal" identities are useful for comparing the interplay of diffraction, optical filtering, and photosite responses - but lens-aberrations, de-focusing, camera-motion, and de-mosaicing are all goint to (also) affect results (at the RAW image-data level, prior to further processing).

Other (also) relevant factors (at the RAW image-data level) may well be optical characteristics of the (entire) optical "filter stack" [other than the b-refringement ("AA") element], any micro-lens asemblies present, and optical properties of the semiconductor materials themselves. Have read that when Fill Factor is not equal to 100%, then optical properties of the semiconductor "diffusion layers" cannot be dealt with in the spatial frequency domain in a simple manner.

The greater our knowledge increases, the greater our ignorance unfolds.

- John F. Kennedy

DM ...