Relaxation Methods for Image Reconstruction

The problem of recovering an image (a function
of two variables) from experimentally available 
integrals of its grayness over thin strips is of great
importance in a large number of scientific areas. 
 An important version of the problem in medicine is
that of obtaining the exact density distribution 
within the human body from X-ray projectionsne approach
that has been taken to solve this problem 
consists of translating the available information into
a system of linear inequalities.  The size and 
the sparsity of the resulting system (typically, 25,000
inequalities with fewer than 1 percent of the 
coefficients nonzero) makes methods using successive
relaxations computationally attractive, as compared 
to other ways of solving systems of inequalities. 
In this paper, it is shown that, for a consistent 
system of linear inequalities, any sequence of relaxarion parameters
lying strictly between 0 and 2 generates 
a sequence of vectors which converges to a solution.
 Under the same assumptions, for a system of linear 
equations, the relaxation method converges to the minimum
norm solution.  Previously proposed techniques
are shown to be special cases of our procedure with
different choices of relaxation parameters.  The 
practical consequences for image reconstruction of the
choice of the relaxation parameters are discussed.

CACM February, 1978

Herman, G.
Lent, A.
Lutz, P.

biomedical image processing, image reconstruction,
X-ray tomography, mathematical programming, 
linear inequalities, relaxation techniques

3.12 3.17 3.34 3.63 5.14 5.18 5.41

CA780208 JB March 28, 1978  2:13 PM

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