An Algebraic Solution for Discrete Tomography
Discrete tomography is the problem of reconstructing a binary image defined on a discrete lattice of points from its projections at only a few angles. It has applications in X-ray crystallography, in which the projections are the number of atoms in the crystal along a given line,and nondestructive testing. The 2D version of this problem is fairly well understood, and several algorithms for solving it are known, most of which involve discrete mathematics or network theory. However, the 3D problem is much harder to solve. This chapter shows how the problem can be recast in a purely algebraic form. This results in: (1) new insight into the number of projection angles needed for an almost surely unique solution; (2) non-obvious dependencies in projection data; and (3) new algorithms for solving it. We then present an explicit formula for reconstructing a finite-support object from a finite number of its discrete projections over a limited range of angles, again making extensive use of the discrete Fourier transform in doing so. We compute the object directly as a linear combination of the projections. The well-known ill-posedness of the limited-angle tomography problem manifests itself in some very large coefficients in these linear combinations; these coefficients,which are computed off-line,provide a direct sensitivity measure of the reconstruction samples to the projections samples. The discrete nature of the problem implies that the projections must also take on integer values; this means noise can be rejected. This makes the formula practical.
KeywordsDiscrete Fourier Transform Projection Data View Angle Tomography Problem Phase Retrieval
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