Re: Existence of unique blur measure
3
J A C S wrote:
alanr0 wrote:
J A C S wrote:
bobn2 wrote:
fvdbergh2501 wrote:
The paper itself does not provide much evidence that this is an good way to measure "sharpness" subjectively. I mean, the experiment involved only five images, filtered with three filters (one of which was a Gaussian), judged by six persons!
This is somewhat typical of engineers' approach to perceptual issues. Any amount of rigour in the maths, then stick a finger in the air when it comes to perceptual impact.
Actually, this is one of the most mathematical papers I have ever seen posted here. There is even a theorem and a proof which looks correct to me.
The derivation of the proof may be correct, but a key starting assumption is an unjustified leap of faith.
Not really.
Uniqueness of blur measures section 2.1
The transform of a point through an imaging chain is expected to have essentially a finite size. Thus we can assume that the kernel has finite moments of all orders.
I suggested elsewhere that this is invalid for diffractionlimited images. More generally this extends to cases where the point spread function is determined by a combination of diffraction and aberrations.
The point spread function of an optical system is the Fourier transform of the optical transfer function.
The optical transfer function is the autocorrelation of the pupil function. (Goodman, Introduction to Fourier Optics 2nd ed. eq. 6.28, p 155, or H. Gross slide 31)
If the field amplitude near the pupil boundary is finite, and there is a step drop in intensity at the pupil boundary, then the first derivative of the OTF will be discontinuous at zero spatial frequency. It follows that there is a singularity in the Laplacian of the OTF at the origin, and the second moment of the PSF is not finite.
It follows that Buzzi & Guichard's Theorem 1 does not apply to optical imaging systems where a sharp pupil boundary is illuminated, and where edge diffraction has a significant impact on the optical transfer function.
Have you noticed that the sampling theorem requires your function to be analytic (and no, the diffraction does not do that, see below) and it requires infinitely many samples? We still use it though and it can be explained why it works (approximately).

Now, about diffraction  The formula you referred to is valid only approximately for a small observation angle. This means not too far from the center.
You can always bruteforce rayleigh sommerfeld, or do angular spectrum. Chirp Z transform, too.
In practice, unless the system has an extremely large PSF, a Fresnel transform from a pupil to an image plane is the best choice, and the PSF drops to negligible values by the edge of the FFT window. If the PSF is very large, you likely have aliasing in the phase on the pupil plane anyway because you don't have enough samples.
The discontinuity of the OTF at the origin and infinite support of the PSF is, IMO much more severe than the limits of the Fresnel transform approximation.