Re: Convolution in frequency (Fourier) space

Markus:
The short answer is no. They're not exactly mathematically the same
theoretically. And I'll tell you why - it's related to what Bob
said. If you chop your image down into a bunch of tiny images, say an
array of 32 by 32 small images, you won't be able to see the low
frequencies in those tiny images that you could in your larger image,
can you? A large sinusoid that spans many small images wouldn't even
show up as a blip on your smal FT's but would in the FT of the
complete, larger image. Hopefully you can see why now, in spite of
the short explanation.

Also, I would expect to see horrible edge artifacts (the hated cross
artifact) unless you do some windowing to set the edges of the
subimages the same. But windowing (Hamming, Hanning, etc.) alters
your original data some more and there again - it's not the same as
doing one big transform. If you have bad cross artifacts, then the
top (or left) edge of the image is not the same gray level as the
bottom (or right) edge of the image and that is your clue that there
are spatial frequencies broader than your subimage width (you can also
see this just by looking at the average intensity as you look at a
profile across your small sub images).

But all is not lost. In practical terms, it might be very close to
being the same. It just depends on what kinds of low frequencies you
have in your image and their period relative to the size of your
subimages. If you don't have any low frequencies in that range or
don't plan on altering them in any way, then it might not make much
visual difference.

And, like Harris and pixel.to.life said I think you mean to
effectively convolve by multiplying the FT of your convolution kernel
by the FT of all the small images, not convolving directly in the
Fourier domain with any kind of window/kernel. (Hopefully that's what
you meant and you were just sloppy in your description, or else you
still need to hit the books and study some more.)
Regards,
ImageAnalyst

==============================================================
On Mar 26, 10:01 am, Markus Mayer <newsgr...@xxxxxxx> wrote:
Just a short question:

Given a matrix with spatial information, i.e. a photographic image. If I
split the matrix into several square blocks, Fourier transform them,
convolve each of the resulting matrices with a given kernel (in
frequency domain) and then inverse transform all blocks - would the
(combined) result be the same as if I had transformed, convolved and
inverse transformed the whole image as one?

Regards,
Markus


.

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