Binary Matrices under the Microscope: A Tomographical Problem

Details Binary Matrices under the Microscope: A Tomographical Problem Binary Matrices under the Microscope: A Tomographical Problem

2006-09-28

arxiv.org/abs/math/0609792v1

Mathematics

Combinatorics

25 pages, 15 figures, submitted

Scientific paper

A binary matrix can be scanned by moving a fixed rectangular window (submatrix) across it, rather like examining it closely under a microscope. With each viewing, a convenient measurement is the number of 1s visible in the window, which might be thought of as the luminosity of the window. The rectangular scan of the binary matrix is then the collection of these luminosities presented in matrix form. We show that, at least in the technical case of a smooth m x n binary matrix, it canbe reconstructed from its rectangular scan in polynomial time in the parameters m and n, where the degree of the polynomial depends on the size of the window of inspection. For an arbitrary binary matrix, we then extend this result by determining the entries in its rectangular scan that preclude the smoothness of the matrix.

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