Re: Modelling Spatial Frequency (MTF) Response Effects

Stephen Barrett wrote:

Thank you, Detail Man, for your detailed response and for the wealth of references.

It is going to take me some time to absorb all of this.

You are welcome. The bit about calculating the square-root of the sum of the squares with "Blur diameters" works for Gaussian distributions. The Standard Deviation of the convolution (which is what is happening in the spatial-domain) of two Gaussian distributions is equal to the square-root of the sum of the squares of the individual Standard Deviations (which relates to "Blur diameters") of the individusal Gaussian distributions

While a Gaussian disribution can be made to fairly closely fit the inner main-lobe of an Airy disk function (only), the side-lobes of the Airy disk extend quite a bit farther outwards. Therefore, the Bessel function of the Airy disk pattern is different enough from a Gaussian approximation that it is not a good approximation in the outer tails of the (space-domain) Point Spread Function, abbreviated as PSF (and correspondingly at low spatial frequencies in the MTF).

In fact, experiments have indicated that the some photon patterns are most closely described as a particular type of Gamma function. The deeper one goes, the more complex things appear.

The MTF of diffraction through a circular aperture is not Gaussian. It's so-called "Chinese hat" function is nearly downwardly sloping linear - except when near it's right-most position on the X-axis.

Where it comes to the MTF of a Photosite Aperture, that is a "sinc" [ sin(x)/(x) ] function. The addition of an optical low-pass ("AA") filter represents a mathematical product of two "sinc" functions that form a zero-magnitude point at some sub-multiple of the spatial sampling frequency

And it seems that any focusing error vastly complicates things still, significantly attenuating the MTF magnitude when the COC = 3 * Wavelength * F-Number, and profoundly attenuating MTF magnitude when approaching and surpassing COC = 5 * Wavelength * F-Number. Further, that function is not a Gaussian function, either, and it makes the rest look relatively trivial numerically.

What is comes down to is that radially-symmetric Gaussian distributions are so vastly easier to compute with (as compared to any other functional descriptions), that the "Gaussian" PSFs as something "close" in optical systems seems to have become inculcated in the minds of many.

At any rate, the idea of simply calculating a scaled arithmetic sum of blur-diameters seems likely to generate greater errors. My vote would be to compute in the spatial frequency (MFT) domain. The results are then directly in the "units" of interest (magnitude as a function of spatial frequency).

The arguments about "sharpness" are endless. Th overall shape of the net composite MTF response matters (much more comprehensive than a single data-point where MTF=50%, etc.), and it appears to be the integral of that MTF curve over a couple of critical "octaves" of spatial frequency that (coupled with individual perceptual CSFs in individual viewers) form our dominant impressions.

DM ...