Discrete Tomography pp 137-162 | Cite as

# Reconstruction of Two-Valued Functions and Matrices

## Abstract

The reconstruction of a two-valued function from its two projections is considered. It is shown that this problem can be transformed into the solved problem of the reconstruction of characteristic functions (i.e., functions with values 0 and 1). Necessary and sufficient conditions are given to decide the existence and the uniqueness of a two-valued function if its projections and the two values are known. These conditions can also be applied if the two values are not given in advance. It is proved that merely the knowledge of the projections (i.e., without the two values) is not enough for the unique reconstruction of a two-valued function. However, on the basis of two given functions it is possible to decide whether they are the projections of a uniquely reconstructible two-valued function. Also the corresponding values can be determined in this way. The reconstruction of two-valued matrices from their row and column sums (projections) is also considered. It is shown that this problem can be transformed into the solved problem of the reconstruction of (0,1)-matrices. In the same way as in the case of two-valued functions, necessary and sufficient conditions are given to decide the existence and the uniqueness of a two-valued matrix if its projections and the two values are known. It is proved that, generally, there is only a finite number of solutions even if the two values are not fixed. Finally, an algorithm is given to reconstruct two-valued matrices from two projections.

## Keywords

Admissible Pair Reconstruction Problem Half Line Uniqueness Pair Vertical Projection## Preview

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