Re: A "slanted edge" analysis program


"Lorenzo J. Lucchini" <ljlbox@xxxxxxxxxx> wrote in message news:7A91f.15406$133.4116@xxxxxxxxxxxxxxxxxxxxxxxx
SNIP
Why is my edge odd-looking?

The pixels "seem" to be sampled with different resolution horizontally/vertically. As you said earlier, you were experimenting with oversampling and downsizing, so that may be the reason. BTW, I am using the version that came with the compiled alpha 3.

SNIP
OTOH, I think I've understood the way you're trying to reconstruct the PSF; I'm not sure I like it, since as far as I can understand you're basically assuming the PSF *will be gaussian* and thus try to fit a ("3D") gaussian on it.

In fact I fit the weighted average of multiple (3) Gaussians. That allows to get a pretty close fit to the shape of the PSF. Although the PSFs of lens, film, and scanner optics+sensor all have different shapes, the combination usually resembles a Gaussian just like in many natural processes. Only defocus produces a distinctively different shape, something that could be added for even better PSF approximation in my model.

This is how the ESF of your "testedge" compares to the ESF of my model:
http://www.xs4all.nl/~bvdwolf/main/foto/Imatest/ESF.png

Now, perhaps the inexactness due to assuming a gaussian isn't really important (at least with the scanners we're using), but it still worries me a little.

That's good ;-) I also don't take things for granted without some soul searching (and web research).

Also, the book says that gaussian PSFs have gaussian LSFs with the same parameters -- i.e. that a completely simmetrical gaussian PSF is the same as any corresponding LSF.

The LSF is the one dimensional integral of the PSF. If the PSF is a Gaussian, then the LSF is also a Gaussian, but with a different shape than a simple cross-section through the maximum of the PSF.
The added benefit of a Gaussian is that it produces a separable function, the X and Y dimension can be processed separately after one an other with a smaller (1D) kernel. That will save a lot of processing time with the only drawback being a very slightly less accurate result (due to compounding rounding errors, so negligible if acurate math is used).

Our PSFs are generally not symmetrical, but they *are* near-gaussian, so what would you think about just considering the two LSFs we have as sections of the PSF?

Yes, we can approximate the true PSF shape by using the information from two orthogonal resolution measurements (with more variability in the "slow scan" direction). I'm going to enhance my (Excel) model, which now works fine for symmetrical PSFs based on a single ESF input (which may in the end turn out to be good enough given the variability in the slow scan dimension). I'll remove some redundant functions (used for double checking), and then try to circumvent some of Excel's shortcomings.

I think in the end I'll also try implementing your method, even though there is no "automatic solver" in C, so it'll be a little tougher. But perhaps some numerical library can be of help.

My current (second) version (Excel spread***) method tries to find the right weighted mix of 3 ERF() functions (http://www.library.cornell.edu/nr/bookcpdf/c6-2.pdf page 220, if not present in a function library) in order to minimize the sum of squared errors with the sub-sampled ESF. The task is to minimize that error by changing three standard deviation values.
I haven't analyzed what type of error function this gives, but I guess (I can be wrong in my assumption) that even an iterative approach (although not very efficient) is effective enough because the calculations are rather simple, and should execute quite fast. One could consider (parts of) one of the methods from chapter 10 of the above mentioned Numerical Recipes book to find the minimum error for one Standard Deviation at a time, then loop through them again for a number of iterations until a certain convergence criterion is met.

This is basically the model which is compared at the original ESF sub-sample coordinates:
ESFmodel = ( W1*(1+IF(X<0;-Erf(-X/(SD1*Sqrt(2)));Erf(X/(SD1*SQRT(2)))))/2
+ W2*(1+IF(X<0;-Erf(-X/(SD2*Sqrt(2)));Erf(X/(SD2*SQRT(2)))))/2
+ W3*(1+IF(X<0;-Erf(-X/(SD3*Sqrt(2)));Erf(X/(SD3*SQRT(2)))))/2 )
/ (W1+W2+W3)

That will ultimately give, after optimization (minimizing the error between samples and model), three Standard Deviations (SD1,SD2,SD3) which I then use to populate a two dimensional kernel with the (W1,W2,W3) weighted average of three symmetrical Gaussians. The population is done with symbolically pre-calculated (with Mathematica) functions that equal the 2D pixel integrals of 1 quadrant of the kernel. The kernel, being symmetrical, is then completed by copying/mirroring the results to the other quadrants.

That kernel population part is currently too inflexible for my taste, but I needed it to avoid some of the Excel errors with the ERF function. It should be possible to make a more flexible kernel if the ERF function is better implemented.

Bart 

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