bjs wrote:
In any case, you are probably on the wrong track with convolution, deconvolvingAliasing may result in a random, or even chaotic output, but the process is entirely linear. It's simply the result of taking samples of a signal at regular intervals with no fixed phase relationship between the signal and the sampling frquency. For any sinusoidal signal, if we knew the phase relationship, then we could easily reconstruct the amplitude of the original signal, and if we knew the amplitude, then obtaining the phase would present little problem.
and Fourier transforms since those assume a linear system and aliasing is a
non-linear transformation.
I guess that what you are saying is that the aliased output won't present us with a set of fourier transforms that are in any way representative of the original signal.
This would be true only if we knew absolutely nothing about the original.
In the case of signals such as aliased audio, it's not really possible to estimate anything about the original signal, so Fourier analysis won't get us very far.
In the case of image information, there is a great deal that we know about the original.
1) The sample is not an instantaneous 'snapshot' of the waveform at the sample frequency. It's an integrated sample, of intensity over area, or distance, if we consider only one axis.
2) The grain of any particular film will have a characteristic shape
in cross section. Usually a nucleus, surrounded by a more nebulous area
in the case of colour film, or simply a sharply defined area for B&W.
Each layer of colour film will most likely exhibit a different grain 'shape'.
This grain shape will show a regular and characteristic harmonic series
of spatial frequencies. Thus if we detect the fundamental, we can fill
in the harmonics by interpolation.
3) For any of the higher spatial frequencies that we look at, the grain
will have a fixed amplitude. The peak density of individual grains
will not vary, simply their size.
Since grains, by definition, have 100% contrast, then the amplitude
of the spatial frquencies that they generate in the image will simply be
the MTF of the system.
4) The samples do not exist in isolation. We have a cross reference
of x axis samples, and y axis samples, at the linear spatial frequency
of the pixels.
Not only that, but if we assume square shaped CCD elements (and I think
we can), we have two diagonal axes to look at as well.
These diagonal axes are effectively sampled at 0.707 times the x and
y axis frequency.
This gives us another set of aliasing artefacts with very little phase
correlation to our horizontal and vertical axes.
So, in view of the above clues to the true identity of the original signal, I don't think it at all far fetched that a great deal of the original could be reconstructed from deconvolving, fourier analysis, and transformation.
Anyway: Whether it proves of any use in tackling aliasing or not, isn't
it high time that image manipulators availed themselves of the powerful
tool of Fourier transformation?
(I'm sure that certain dark agencies have been using it on images for
years.)
Audio engineers use it as a regular, everyday part of life.
Take a look at an audio processing package like 'CoolEdit Pro', to
see just how easily you can clean up a signal using FFTs.
FFT noise reduction can be done almost in real time on audio signals,
using even my modestly powerful 400MHz PC.
Image data should be no sweat.
Conventional matrix type image filters only take a limited amount of image data into account. Fourier transformation would work on the image as a whole, and take in the big picture. (Quite frankly, I don't care if you pardon the pun or not.)
Regards, Photoscientia.
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