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   -> Volume 7, Issue 9

Question: Irregular remapping
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Christian Ohreneder (

PostPosted: Thu Aug 27, 1998 9:09 am    
Subject: Question: Irregular remapping
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#18 Question: Irregular remapping

I am working on surface-reconstruction from stereo images. For that
purpose I need to "transform" or "remap" an image.

f(.,.) ... original Image.

New Grid: (x,y); x = {0,1,2,3,...}, y={0,1,2,...} for every Position
(x,y) there is a position (x',y') in the original image x' =x'(x,y),
y' = y'(x,y).

(x',y') = T(x,y) T...Map

The transformed Image is given by f_new(x,y) = (f o T)(x,y) = f(x'(x,y),

Depending on the differences x'[x+1,y] - x'[x,y], etc. f is resampled
with high or low density. For regions where the sampling points
(x',y') are very sparse relative to the frequency content of f, the
transformed image f_new does not reflect the structure of f in an
apropriate way because of aliasing.

A possible solution is to filter the image before the resampling step.

f_new2(x,y) := (phi * (f o T)) (x,y)

phi...smoothing filter
* denotes convolution

If the transformation is linearly approximated at position vec{x} = (x,y)
the remapped image may be written as

f_new2(x) ~ int{ 1/|dT/dx| phi( (dT/dx)^{-1} v) f(x - v) dv }

Wherein x, and v have to be understood as vector quantities. dT/dx is
the Jacobi matrix of T at location x. This is essentially a location
variant filter.

It's essential for me to have a fast numerical implementation for that
problem. I have tried a naive approach in 1D that constructs the
samples of the transformed image by appropriatly weighting the
coefficients from different scales of an image pyramid. This gives
reasonable results. The extension to 2D does not seem to be that easy
because a simple interpolation between two pyramid levels does not
allow different filter scalings in x and y direction. A wavelet
decomposition scheme would provide an obvious advantage. Before trying
this I want to look for existing solutions and concepts. The
difficulty seems to be to have the location variant filtering and the
interpolation at the new sampling points integrated into one

My questions are:
Is this problem familiar to anyone?
Has this type of problem a name under which it may be found in the
Has anyone code for that purpose?

Many thanks for every hint you can give!
Christian Oehreneder
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