An algebraic solution to the 3-D discrete tomography problem (original) (raw)

On the computational complexity of reconstructing lattice sets from their X-rays

Discrete Mathematics, 1999

We study the computational complexity of various inverse problems in discrete tomography. These questions are motivated by demands from the material sciences for the reconstruction of crystalline structures from images produced by quantitative high resolution transmission electron microscopy. We completely settle the complexity status of the basic problems of existence (data consistency), uniqueness (determination), and reconstruction of finite subsets of the d-dimensional integer lattice Zd that are only accessible via their line sums (discrete X-rays) in some prescribed finite set of lattice directions. Roughly speaking, it buns out that for all d 22 and for a prescribed but arbitrary set of m > 2 pairwise nonparallel lattice directions, the problems are solvable in polynomial time if m = 2 and are N P-complete

Algebraic aspects of discrete tomography

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

Let A be a finite subset of Z l . We consider the problem of reconstructing a function f : A → {0, 1} if all the line sums in finitely many directions are given. In Theorem 1 we give a complete characterization of the switching components, the functions g : A → Z having zero line sums in these directions. Theorem 2 provides an algorithm for finding a function g : A → Z having the same line sums as f in these directions such that |f (x) − g(x)| is bounded on average by a number depending only on the number of directions.

An Algebraic Framework for Discrete Tomography: Revealing the Structure of Dependencies

SIAM Journal on Discrete Mathematics, 2010

Discrete tomography is concerned with the reconstruction of images that are defined on a discrete set of lattice points from their projections in several directions. The range of values that can be assigned to each lattice point is typically a small discrete set. In this paper we present a framework for studying these problems from an algebraic perspective, based on Ring Theory and Commutative Algebra. A principal advantage of this abstract setting is that a vast body of existing theory becomes accessible for solving Discrete Tomography problems. We provide proofs of several new results on the structure of dependencies between projections, including a discrete analogon of the well-known Helgason-Ludwig consistency conditions from continuous tomography.

An algorithm for discrete tomography

Linear Algebra and its Applications, 2001

There are many algorithms in the literature for the approximating reconstruction of a binary matrix from its line sums. In this paper we provide an algorithm which starts from the line sums of an unknown binary matrix f , and outputs an integer matrix S with small entries in absolute values such that the line sums of f and S coincide. We also give the results of some experiments with the algorithm.

Linear time reconstruction by discrete tomography in three dimensions

2020

The goal of discrete tomography is to reconstruct an unknown function f via a given set of line sums. In addition to requiring accurate reconstructions, it is favourable to be able to perform the task in a timely manner. This is complicated by the presence of switching functions, or ghosts, which allow many solutions to exist in general. In this paper we consider the case of a function f : A→ R where A is a finite grid in Z3. Previous work has shown that in the two-dimensional case it is possible to determine all solutions in parameterized form in linear time (with respect to the number of directions and the grid size) regardless of whether the solution is unique. In this work, we show that a similar linear method exists in three dimensions under the condition of nonproportionality. This is achieved by viewing the threedimensional grid along each 2D coordinate plane, effectively solving the problem with a series of 2D linear algorithms. We show that the condition of nonproportionali...

Algorithms for linear time reconstruction by discrete tomography in three dimensions

2020

The goal of discrete tomography is to reconstruct an unknown function f via a given set of line sums. In addition to requiring accurate reconstructions, it is favourable to be able to perform the task in a timely manner. This is complicated by the presence of switching functions, or ghosts, which allow many solutions to exist in general. Previous work has shown that it is possible to determine all solutions in linear time (with respect to the number of directions and grid size) regardless of whether the solution is unique. In this work, we show that a similar linear algorithm exists in three dimensions. This is achieved by viewing the three-dimensional grid along each 2D coordinate plane, effectively solving the problem with a series of 2D linear algorithms. By that, it is possible to solve the problem of 3D discrete tomography in linear time.

Algorithms for linear time reconstruction by discrete tomography II

2021

The reconstruction of an unknown function f from its line sums is the aim of discrete tomography. However, two main aspects prevent reconstruction from being an easy task. In general, many solutions are allowed due to the presence of the switching functions. Even when uniqueness conditions are available, results about the NP-hardness of reconstruction algorithms make their implementation inefficient when the values of f are in certain sets. We show that this is not the case when f takes values in a field or a unique factorization domain, such as or . We present a linear time reconstruction algorithm (in the number of directions and in the size of the grid), which outputs the original function values for all points outside of the switching domains. Freely chosen values are assigned to the other points, namely, those with ambiguities. Examples are provided.