Tom, thanks for your comments.
On 8/17/2009 4:16:38 PM, Tom_Gutman wrote:
>I haven't been looking at the
>details, but there is
>something I don't understand
>about the whole concept of
>resampling. Why?
>
>As I remember, the purpose was
>to calculate the convolution
>representing a pixel response.
>The procedure for resampling
>is to do a Fourier transform,
>insert additional frequence
>components with zero magnitude
>and then do the inverse
>Fourier transform. But the
>convolution is then done by
>doing a Fourier transform and
>multiplying by the transform
>of the pixel response. That
>seems to make the original
>inverse Fourier transform (the
>actual interpolation)
>redundant. The Fourier
>transform should just
>reproduce the original Fourier
>transfor with the inserted
>zeros. And since zeros don't
>contribute much to a product,
>it would seem that the whole
>calculation could just as well
>be done with the original
>Fourier transform, without the
>interpolation.
Maybe Im not the best person to answer this (Lou, Richard, Guy, Phillip!), but I think the answer is that there are two things going on.
The convolution of the psf, pixel response and bead (i.e. a non-point object) through oversampling allowed me (or Lou!) to perform the FFT. This leads to a high res domain representation of a theoretical calculation for the diffraction limit that I would expect to observe on my camera with that bead. I could:
1: Stop here (in the high res domain) and compare my real CCD image after converting to the same high res domain, or
2: Compare everything in the CCD pixellation domain. For visual comparisons, as I actually see the CCD images, I like the comparison in this domain, but for the actual measurement on size, I revert back to the high res domain.
The resample function allows me to take my "blocky" CCD iamge and convert to the high res domain.
>To Jason: You keep comparing
>the resampling to the Virenda
>fit. There is no valid
>comparison. Of course the
>resampling "fit" the
>background better -- it is an
>interpolation and simple
>reproduces the geven data. It
>is not any sort of fit. Like
>all interpolations, it makes
>no distiction whatsoever
>between signal and noise,
>cheerfully reproducing both.
>And, again like any
>interpolation, a good fit at
>the sampled points does not
>mean that it is at all
>reasonable between such points
>(see the section on ringing).
The more I explore this whole topic, the more I learn. From the original Gaussian fits to the spots, through the Bessel functions and then lastly the Virendra equations have in my opinion been one step better than the previous. I was happy with this until....Lou/Richard showed me the concept of frequency padding. And of course there is Jean's Pad fucntion. I am not using the Virendra fits to obtain any quantity, but as it was the best fucntion that allowed energy transfer into the rings, it was a great equation to use for interpolating for a complete set of thru' focus images (which suffer a shape change), and then determine the minimum sized spot and call this "best focus". The Virendra equations are what you would expect the psf to be, but strictly speaking not the convolved data, so there is a slight error in using these and they do assume rotational symmetry. Lou's method did seem to be able to represent the blocky images better.
I would rather show graphics that can replicate the blocky pixels better than whats interpolated between them. I know I will never match exactly, but when I come across another method that appears to be better than my previous one, I want to pursue it. This is how I feel about resample2D. I know Jean says its no good for the letter "A", but I'm not examining these shapes, I am only interested in circular spots.
The down sampling is a seperate issue (again its only for visual presnetations). The improc zoom Jean told me about is interesting, but I believe it does some averaging and I don't know if this really represents the capture of a continuous signal onto a discrete CCD array. I didn't really undertsand Lou's argument here, I would have though it would be the integral as a given pixel will sum all light hitting that pixel.
The convolution of the psf, pixel and bead was an attempt to more accurately estimate how far from the diffraction limit my measurements are (rather than just using the psf). But the more I experience with this, there are just too many things I just can't know to be exactly sure what the D.L is (after convolution)- for example the fill factor on the camera, the lenslet design, what is the actual "emission" shape of my bead.
The resample and high res size measurements are an attemp for me to quantify the size of my currently measured objects (irrespective of how far they are from diffraction limit). I can then vary my optics and see if they get bigger or smaller.
As with your original
>calculations, you are being
>seduced by technical
>capabilities without proper
>regard as to the applicability
>and meaning for your
>situation.
>__________________
> Tom Gutman
I think you are wrong, Im actually being seduced by the whole of Mathcad, its capabilities and the friendliness and interest of this Forum.
I'm also well aware that I have a tendency to prolong topics, so as I don't want to outstay my welcome, please let me know if you want to finish the topic.
Jason
