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   -> Volume 7, Issue 5


Answer: Wavelets & super-resolution of 2d images (WD 7.4 #25)
 
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Bernard Cena (bernard@cs.uwa.edu.au)
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PostPosted: Sat Apr 18, 1998 7:26 am    
Subject: Answer: Wavelets & super-resolution of 2d images (WD 7.4 #25)
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#23 Answer: Wavelets & super-resolution of 2d images (WD 7.4 #25)

Nick Pelling writes:

Has anyone seen any references to wavelets' application to 2d-image
super-resolution? ie, using a wavelet decomposition of an image to
predict higher-order wavelets, and thus infer likely (but
non-existent) detail. 8^)

Answer:

I don't recall any direct references to super-resolution, but as far
as I understand you want to interpolate in-between pixels of your
image ? As you are no doubt aware, when reconstructing data with the
Discrete Wavelet Transform (DWT) if you cut out all wavelet
coefficients leaving only scaling functon coefficients you will
basically SEE the shape of the scaling functions in your
reconstruction. This is the reason why people use smooth fiters for
reconstruction of "denoised" or "compressed" images because cutting
out "details" results in showing the characteristics of the scaling
function and wavelet- if that scaling function/wavelet is very
"jagged" ie. not smooth, you will get ugly artifacts in your
reconstruction. The same applies if you try to do super-resolution in
images which is basically the same as interpolating a larger image
from a smaller one.

I am not sure what you mean by "higher order" wavelets - I presume you
meant finer scale scaling functions ? Anyway, I myself am working on
the application of wavelets to data visualisation using volume
rendering. To do that I use Wim Sweldens' Lifting Scheme to arrive at
arbitary order interpolating and average interpolating scaling
functions/wavelets. The trick in all this is to realise what
"continuous function" does your date "come from", or more precisely,
has been sampled from. If you want to achieve "super-resolution" you
just keep on iterating the reconstruction phase of the DWT which will
give you proper results if you match the scaling function to the type
of image properly. In essence if comes down to identifying a
subdivision scheme which given samples of a function will result in
the same function when the subdivision algorithm is iterated. This is
the beauty of wavelets: you can design your own wavelets for a
particular purpose.

So for example, CCD arrays (digital cameras) perform some sort of
averaging (integration) for each pixel. It would therefore make sense
to use some order of an average-interpolating subdivision scheme to
build your scaling function and associated wavelet. The lifting scheme
is ideal for this as it starts with a subdivision scheme and builds a
biorthogonal filter set with the advantages of being able to handle
boundaries, non- uniform sampling, non-Euclidean spaces (eg. wavelets
on the sphere), integer transform, in-place computation, etc.

Please refer to Wim Sweldens' bibliography at:
http://cm.bell-labs.com/who/wim/
and especially the paper "Building your own wavelets at home".
The interpolating subdivision scheme has been implemented
and can be obtained as LIFTPACK at:
http://www.cs.sc.edu/~fernande/liftpack/index.html

Good luck!
Bernard Cena

Bernard Cena (PhD student - Med Vis) _--_| tel: +61 8 9380 3778
Department of Computer Science / fax: +61 8 9380 1089
The University of Western Australia *_.--._/ home: +61 8 9337 8389
Nedlands, 6907, WA, Australia v mob: +61 417 923 534
http://www.cs.uwa.edu.au/~bernard ----- email: bernard@cs.uwa.edu.au
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