Date: Fri, 8 Dec 95 13:30:59 CST
From: rusin (Dave Rusin)
To: phil@hermes.univ-poitiers.fr
Subject: Re: RECONSTRUCTION PROCESS (tomography)
Newsgroups: sci.math.research

In article <49l0nb$1qi@hermes.univ-poitiers.fr> you write:
>Hello,
>
>Is it possible to reconstruct a continuous function equal to zero outside a
>compact set when all of the integrals over set of constant radius balls are
>known?
>
In dimension 1, this is true: if you know  F(x)=integral(f(t), t=x-c...x)
for all  x, and you know  f(x)=0  for sufficiently negative  x, (say for
x < x0) then  f(x)  may be computed as the derivative of 
integral(f(t), t=x0..x) = F(x) + F(x-c) + F(x-2c) + ...  Near any given
x, there are only finitely many terms which will affect the sum, and so
the derivative  f  may be computed locally. (Notice that it is not necessary
that  f  vanish outside a _compact_ set, only that its support be bounded 
below. If you do have known compact support, then  f  is in fact the derivative
of a finite sum of the  F(x-nc). )

In higher dimensions I don't know. I'm not even sure I have an intuitive idea.
Did you get any responses?

dave

==============================================================================

Date: Fri, 6 Feb 1998 00:20:25 GMT
From: Robert Israel <israel@math.ubc.ca>
To: rusin@math.niu.edu

[some deletia]

I noticed [...] the question wasn't answered in dimensions > 1.  
Actually the answer is yes, as is easily seen using Fourier transforms.  
Let V(x) be the indicator function of a ball of the given radius.  Then the
question is equivalent to: if f is a continuous function of compact support
and the convolution V * f = 0, is f = 0?  Now F(V*f) = F(V) F(f) (where F is
Fourier transform).  F(V) can be computed explicitly; there are no sets of 
positive measure on which F(V)=0, so F(f) = 0 almost everywhere, and 
therefore f = 0 by the uniqueness theorem for Fourier transforms.
Of course, we have more than just a uniqueness theorem: an algorithm
to recover f from V * f (i.e. Fourier transform the data, divide by the Fourier
transform of V and take the inverse Fourier transform).  This is the basic
idea behind much work in the field of image reconstruction.

Cheers,
Robert Israel