MTF and "The Sharper Image"  What matters most ?
Re: MTF and "The Sharper Image"  Defocusing effects on MTF/PSF
In reply to Detail Man,
May 27, 2013

Detail Man wrote:
... it looks like defocusing provides a link to a "more Gaussianlike" shaped MTF (which implies a "more Gaussianlike" shaped PSF) than the MTFs of diffraction, diffraction convolved with a (square shaped) photosite aperture, or diffraction convolved with AA filter responses convolved with a (square shaped) photosite aperture appear to suggest.
Have a look at Sections 2.3 of Falk Lumo's:
http://www.falklumo.com/lumolabs/articles/sharpness/ImageSharpness.pdf
from: http://www.falklumo.com/lumolabs/articles/sharpness/
"C" is the CoC of a defocused (and radially symmetrical) blurspot divided by the quantity that is product of wavelength and Fnumber [ equal to CoC / (λN) ].
Have a look at the family of curves in Figure 11 (MTF for defocus of C = 1, 2, 3, 5, 10, 20, and 50).
Have a look at the shape of the MTFs shown in Figure 11 at low spatial frequencies near zero for values of C < 5. Note how much more it resembles a "Gaussianlike" shape that the aforementioned composite system MTF responses that I have previously calculated, graphed, and posted.
So (given certain amounts of defocusing taking place  by virtue of focusing errors and/or finite depth of field), it does appear that the PSF (the Fourier transform of the MTF) can be said to (in some cases) take on a shape that appears as more "Gaussianlike" than the PSF resulting from the spatial convolution of diffraction, AA filters, and photosite aperture alone. Thus, it seems that a Gaussian PSF might be a reasonable choice as a "catch all" kind of PSF shape.
For a more concrete example, consider the Nikon D800E (which may not have no AA filtering assembly, but hopefully gets us close to a system with no AA filter). DxOMark lists the photosite pitch at 4.7 Microns. The photosite aperture will be somewhat smaller than 4.7 Microns. Let's use a photosite aperture of 4.44 Microns for the purposes of this example.
Consider a Wavelength multiplied by FNumber product of 555 nM times 8.0 equaling 4.44 Microns. Let's use the Leica Full Frame COC diameter of 25 Microns (for 8"x10" viewingsize from a 25cm viewingdistance with 20/20 visual acuity).
In this case, "C" (in Falk Lumo's Figure 11 below, equal to the COC diameter from Defocusing divided by the Wavelength multiplied by FNumber product) is approximately equal to a value of 5.
Look at the plot in Figure 11 for C=5 (the 4th redcolored plot counting downward from the top).
In order to envision the composite system MTF it is necessary to multiply the redcolored (Defocusing/Diffraction) plots by the value of the bluecolored (Diffraction) plot. It is also necessary to multiply the redcolored (Defocus) plots by the value of the MTF of the Photosite Aperture. The product of the MTFs of the Diffraction (bluecolored plot) and the photosite aperture (of 4.44 Microns) is shown below in the redcolored plot (Wavelength of 555 nM and FNumber of 8.0):
Performing the above describe multiplications reduces the value of the redcolored (C=5) plot where it crosses MTF=50% around F=0.4 to around MTF=20%. However, the multiplications have a more minimal effect upon the shape of the composite MTF at low spatial frequencies (below F=0.1). At F=0.1, the multiplications reduce the value of the redcolored (C=5) plot by only around 15%.
At low spatial frequencies (less than around F=0.1) the effects of Defocusing (due to the COC diameter resulting from Defocusing) in this situation described above tend to dominate over the the effects of Diffraction and the effects of Photosite Aperture where it comes to the overall shape of the composite MTF response. As a result, the shape of the composite MTF curve (and to some extent, its Fourier transform, the PSF) might be said to be somewhat closer to that of a Gaussian function (as compared to the MTF due to Diffraction and Photosite Aperture only).
The direct correspondence between the shape of the MTF and PSF functions is exact only in the case where they are Gaussian functions. However, it is here presumed that as the shape of the composite MTF response becomes somewhat more "Gaussianlike" in shape at low spatial frequencies), the PSF (its Fourier transform) will also become more "Gaussianlike" at low spatial locations near the center of the PSF.
This is all rough and crude, but seems to possibly deliver a bit of the "Gaussianlike" character of the MTF and PSF that it seems the validity of the "BxU" units assumes and depends upon  allowing the BxU unit to be related to the MTF response using mathematically simple relationships such as:
BxU = (0.035115253) * ( 1 / MTF(50) )^2
[where MTF(50) is in units of Cycles/Pixel]
sylvatica, Reply # 67 on: May 10, 2013, 0:17:31
Note that the MTF(50) being in units of Cycles/Pixel means that the results of evaluating the above identity will scale with the square of the photosite dimension (which is the same as the area in the simple, but probably not accurate, case of a squareshaped photosite aperture).
It appears that DxO Labs normalizes the results so that they are scaled by per unit area in a standard sized output image (see the "Normalizing the BxU" section in Guichard's paper here):
http://perso.telecomparistech.fr/~gousseau/ProjMVA/Flou/blur_long.pdf
Performing such a normalization would appear to result in similar results for dissimilar photosite sizes.
.
It appears to me that the composite system MTF of Diffraction, (as well as any AA Filter assembly, if included), and Photosite Aperture does not appear to have much of "Gaussianlike" shape at low spatial frequencies. When the effects of Defocusing (whether as a result of focusing errors, or as a result of a COC diameter associated with DOF calculations performed with assumptions made about human visual acuity) are added, the overall shape of the composite MTF response appears to gain something of a more "Gaussianlike" shape at low spatial frequencies.
If one adds in such effects (which is not discussed in Guichard's paper regarding the derivation of the BxU metric), then the assumption of a Gaussian MTF response makes somewhat more sense.
However, the idea that the 2nd derivative of the MTF response at zero spatial frequency corresponds with the "sharpness" of viewed images seems to contradict all characterizations of the human Contrast Sensitivity Function that I am able to find, such as this one shown below:
From: http://www.cg.tuwien.ac.at/research/theses/matkovic/node20.html
... where the spatial frequency response of human vision is show as being attenuated by a factor of 10 (and in some other cases, even more so) as spatial frequency approaches zero value.
Since (some variation of the) human Contrast Sensitivity Function appears to be used by DxO Labs in their recent "Perceptual Megapixel" metrics  and if they are using a weighting function that attenuates to such a high degree at low spatial frequencies approaching zero  then it would seem that either the "BxU" assumptions are valid, or the "Perceptual Megapixel" assumptions are valid, but both cannot seem to be valid (as they would appear to contradict each other).
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