Algebraic Discrete Tomography
Lajos Hajdu, Rob Tijdeman · Birkhäuser Boston eBooks · 2008
In this chapter we present an algebraic theory of patterns that can be applied in discrete tomography for any dimension. We use that the difference of two such patterns yields a configuration with vanishing line sums. We show by introducing generating polynomials and applying elementary properties of polynomials that such so-called switching configurations form a linear space. We give a basis of this linear space in terms of the so-called switching atom, the smallest nontrivial switching configuration. We do so both in case that the material does not absorb light and absorbs light homogeneously. In the former case we also show that a configuration can be constructed with the same line sums as the original and with entries of about the same size, and we provide a formula for the number of linear dependencies between the line sums. In the final section we deal with the case that the transmitted light does not follow straight lines.