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   -> Volume 2, Issue 16


Computing line integrals of a 3D function described in wavelets?
 
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Peter Schröder (ps@princeton.edu)
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PostPosted: Fri Nov 29, 2002 3:38 pm    
Subject: Computing line integrals of a 3D function described in wavelets?
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Computing line integrals of a 3D function described in wavelets?

Question: How to compute line integrals of a 3D function described in
wavelets?

Consider a scalar function f(x,y,z)=f(vex{x}) (sufficiently smooth)
defined on a lattice [0..127]^3. Assume further that we have already
transformed it into the wavelet basis. We would now like to compute

exp(-int_{t_0}^{t_1} f(vec{x}_0+s(vec{x}_1-vec{x}_0)) ds)

or in words, the exponential of -f() integrated along some ray from a
starting point to some endpoint in the original volume. Why? Well, if f()
represents the total scattering cross section of some material the above
integral will give us optical depth in the given medium along a line of
sight. This is only a subproblem of a larger problem.

Now, obviously this is easy in the 1D case since path integration is
trivial but things become a lot less straightforward in higher D. In
particular there is no hope of somehow precomputing the above exp()
integrals for all possible choices of (x,y,z)_{t_0} (x,y,z)_{t_1} since
that is a 6 dimensional object and even if we can do the wavelet transform
quickly in one dimension the cost of doing it along all dimensions is going
to be totally overwhelming. Knowing this it seems we need some lazy scheme
to evaluate it, i.e. somehow take advantage of the sparse structure of the
original wavelet representation of f() and directly compute the above
function for a given (t_0 t_1). Of course the exp() should help *a lot* in
this task as well since the tails will contribute very little and we should
be able to take advantage of this in the multi resolution framework.

Presumably this would be helped considerably if we knew something about
objects of the form

exp(-int_{t_0}^{t_1} phi(vex{x}_0+s(vec{x}_1-vec{x}_0)))

and similarly for psi. Or in words, exp()s of path integrals through
wavelet bases (product bases) in 3D. Does anybody have any experience with
this? References? War stories?

Thanks for any help you might have!

Peter (ps@princeton.edu)
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