Catálogo de publicaciones - libros

We consider the two-projection tomography problem, assuming a priori known prohibited region. We show that a modification of Ryser’s reconstruction algorithm gives a solution. We then study the relation of the switching graph for the solution sets with and without the prohibited region. Finally, we apply our idea to get a better reconstruction figure imposing prohibited region artificially.

The Mojette transform is an entirely discrete form of the Radon transform developed in 1995. It is exactly invertible with both the forward and inverse transforms requiring only the addition operation. Over the last 10 years it has found many applications including image watermarking and encryption, tomographic reconstruction, robust data transmission and distributed data storage. This paper presents an elegant and efficient algorithm to directly apply the inverse Mojette transform. The method is derived from the inter-dependance of the “rational” projection vectors ( p _ i , q _ i ) which define the direction of projection over the parallel set of lines b = p _ i l – q _ i k . Projection values are acquired by summing the value of image pixels, f ( k , l ), centered on these lines. The new inversion is up to 5 times faster than previously proposed methods and solves the redundancy issues of these methods.

A quantum mechanics based method is presented to generate sets of digital angles that may be well suited to describe projections on discrete grids. The resulting angle sets are an alternative to those derived using the Farey fractions from number theory. The Farey angles arise naturally through the definitions of the Mojette and Finite Radon Transforms. Often a subset of the Farey angles needs to be selected when reconstructing images from a limited number of views. The digital angles that result from the quantisation of angular momentum (QAM) vectors may provide an alternative way to select angle subsets. This paper seeks first to identify the important properties of digital angles sets and second to demonstrate that the QAM vectors are indeed a candidate set that fulfils these requirements. Of particular note is the rare occurrence of degeneracy in the QAM angles, particularly for the half-integral angular momenta angle sets.

Discrete tomography concerns the reconstruction of functions with a finite number of values from few projections. For a number of important real-world problems, this tomography problem involves thousands of variables. Applicability and performance of discrete tomography therefore largely depend on the criteria used for reconstruction and the optimization algorithm applied. From this viewpoint, we evaluate two major optimization strategies, simulated annealing and convex-concave regularization, for the case of binary-valued functions using various data sets. Extensive numerical experiments show that despite being quite different from the viewpoint of optimization, both strategies show similar reconstruction performance as well as robustness to noise.

Switching components play an important role investigating uniqueness of problems in discrete tomography. General projections and additive projections as well as switching components w.r.t. these projections are defined. Switching components are derived by combining other switching components. The composition of switching components into minimal ones in case of additive projections is proved. We also prove, that the product of minimal switching components is also minimal.

The concepts of weak component and simple 1 are generalizations, to binary images on the n -cells of n -dimensional cell complexes, of the standard concepts of “26-component” and “26-simple” 1 in binary images on the 3-cells of a 3D cubical complex; the concepts of strong component and cosimple 1 are generalizations of the concepts of “6-component” and “6-simple” 1. Over the past 20 years, the problems of determining just which sets of 1’s can be minimal non-simple, just which sets can be minimal non-cosimple, and just which sets can be minimal non-simple (minimal non-cosimple) without being a weak (strong) foreground component have been solved for the 2D cubical and hexagonal, 3D cubical and face-centered-cubical, and 4D cubical complexes. This paper solves these problems in much greater generality, for a very large class of cell complexes of dimension ≤4.

In this paper we define the notion of gap in an arbitrary digital picture S in a digital space of arbitrary dimension. As a main result, we obtain an explicit formula for the number of gaps in S of maximal dimension. We also derive a combinatorial relation for a digital curve.

In this paper, algorithms for computing integer (co)homology of a simplicial complex of any dimension are designed, extending the work done in [1,2,3]. For doing this, the homology of the object is encoded in an algebraic-topological format (that we call AM-model). Moreover, in the case of 3D binary digital images, having as input AM-models for the images I and J , we design fast algorithms for computing the integer homology of I ∪ J , I ∩ J and I ∖ J .

In this paper we investigate the lattice properties of several special, but important subsets of S _ n , the set of n D octagonal neighborhood sequences in ℤ^n, with respect to two ordering relations ${\sqsupseteq}$ ^* and $\sqsupseteq$ . Both orderings have some natural meaning, especially ${\sqsupseteq}$ ^* compares the ”speed” how neighborhood sequences spread in ℤ^n. We summarize our and the previous related results in a table. In particular, our theorems can be considered as extensions of some results from [1,2,3].

While connected arithmetic discrete lines are entirely characterized by their arithmetic thickness, only partial results exist for arithmetic discrete hyperplanes in any dimension. In the present paper, we focus on 0-connected rational arithmetic discrete planes in ℤ^3. Thanks to an arithmetic reduction on a given integer vector n , we provide an algorithm which computes the thickness of the thinnest 0-connected arithmetic plane with normal vector n .