Reconstruction of binary images from a small number of their projections
A discrete tomography reconstruction problem for two vertical and horizontal directions (left), together with its (non-unique) solution (right). The task is to color some of the white points black so that the number of black points in the rows and columns match the blue numbers.
Discrete tomography[1][2] focuses on the problem of reconstruction of binary images (or finite subsets of the integer lattice) from a small number of their projections.
In general, tomography deals with the problem of determining shape and dimensional information of an object from a set of projections. From the mathematical point of view, the object corresponds to a function and the problem posed is to reconstruct this function from its integrals or sums over subsets of its domain. In general, the tomographic inversion problem may be continuous or discrete. In continuous tomography both the
domain and the range of the function are continuous and line integrals are used. In discrete tomography the domain of the function may be either discrete or continuous, and the range of the function is a finite set of real, usually nonnegative numbers. In continuous tomography when a large number of projections is available, accurate reconstructions can be made by many different algorithms.
It is typical for discrete tomography that only a few projections (line sums) are used. In this case, conventional techniques all fail. A special case of discrete tomography deals with the problem of the reconstruction of
a binary image from a small number of projections. The name discrete tomography is due to Larry Shepp, who organized the first meeting devoted to this topic (DIMACS Mini-Symposium on Discrete Tomography, September 19, 1994, Rutgers University).
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Herman, G. T. and Kuba, A., Discrete Tomography: Foundations, Algorithms, and Applications, Birkhäuser Boston, 1999
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Herman, G. T. and Kuba, A., Advances in Discrete Tomography and Its Applications, Birkhäuser Boston, 2007
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