Some Geometric and Combinatorial Properties of Binary Matrices Related to Discrete Tomography

Abstract. Digital tomography deals with the problem of reconstructing an image from its projections. The image may or may not be reconstructible uniquely. The effective reconstruction also depends on the kind of projections taken. We consider the simplest two-dimensional case in which we have a 2D matrix and the projections are orthogonal. The matrices which are not uniquely reconstructible are known as ambiguous. In this report we concentrate on decomposing such an ambiguous matrix into a minimum number of matrices such that each of them are unambiguous. We claim that the XOR sum of these component matrices would return the original matrix. As the component matrices can be stored just by storing the row-sum and column-sum (the horizontal and vertical projections), we can store any ambiguous matrix by storing the projections of the components. The space management highly depends on the minimum number of components which we define as XOR-dimension. We study the trend of change in XOR-dimension first for n×n matrices and then for general m× n matrices.