Theory
Discrete tomography has strong connections with other mathematical fields, such as number theory, discrete mathematics, complexity theory and combinatorics. In fact, a number of discrete tomography problems were first discussed as combinatorial problems. In 1957, Herbert John Ryser found a necessary and sufficient condition for a pair of vectors being the two orthogonal projections of a discrete set. In the proof of his theorem, Ryser also described a reconstruction algorithm, the very first reconstruction algorithm for a general discrete set from two orthogonal projections. In the same year, David Gale found the same consistency conditions, but in connection with the network flow problem. Another result of Ryser is the definition of the switching operation by which discrete sets having the same projections can be transformed into each other.
The problem of reconstructing a binary image from a small number of projections generally leads to a large number of solutions. It is desirable to limit the class of possible solutions to only those that are typical of the class of the images which contains the image being reconstructed by using a priori information, such as convexity or connectedness.
Read more about this topic: Discrete Tomography
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