13. On the reconstruction of static and dynamic discrete structures

Abstract

We study inverse problems of reconstructing static and dynamic discrete structures from tomographic data (with a special focus on the 'classical' task of reconstructing finite point sets in R d ). The main emphasis is on recent mathematical developments and new applications, which emerge in scientific areas such as physics and materials science, but also in inner mathematical fields such as number theory, optimization, and imaging. Along with a concise introduction to the field of discrete tomography, we give pointers to related aspects of computerized tomography in order to contrast the worlds of continuous and discrete inverse problems.

On Polyatomic Tomography over Abelian Groups: Some Remarks on Consistency, Tree Packings and Complexity

“…Let r , s ∈ N with 2 ≤ r ≤ s such that c ≤ rs/(r + s − 1) and let (R, S) be balanced over G . Further, let T (1) , . .…”

“…S (1) R (1) S (2) R (2) S (3) R (3) S (4) R (4) 1 1 2 3 4 7 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 As it easily follows from [13] (see also [9, Prob. SP13 and comments]), Strong Partition is NP-complete.…”

“…. , σ 1, j = ν j for j ∈ N , and μ (1) 2, j = ν j for j ∈ [n]\N ; μ (2) 1, j = 1 for j ∈ [n]\N and μ (2) 2, j = 1 for j ∈ N ; μ ( ) , j = 1 for j ∈ [n] and ∈ {3, . .…”

“…Since then, a rich theory has been developed which is centered around the task of determining (theoretically and algorithmically) discrete structures by information about their interaction with certain query sets. For surveys see [1,10,12], and other sources quoted there. More than half a century after Ryser's work discrete tomography is a well-established field with seminal results reaching into combinatorics, geometry, geometry of numbers, commutative algebra, optimization and other fields.…”

“…Its many practical applications in materials science, physics, and various other areas (see e.g. [1]) have fueled studies in dimensions higher than 2, for X-rays in more than two directions, for other query sets-most notably hyperplanes, leading to the discrete Radon transform-with other image domains (modelling e.g. grey values in images) and for objects which are composed of different types of elements, the polyatomic or colored case.…”