Discrete Applied Mathematics 157 (2009) 3447–3456
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Discrete Applied Mathematics
journal homepage: www.elsevier.com/locate/dam
A benchmark set for the reconstruction of hv-convex discrete sets
Péter Balázs
∗
Department of Image Processing and Computer Graphics, University of Szeged, Árpád tér 2., H-6720 Szeged, Hungary
a r t i c l e i n f o
Article history:
Received 1 September 2008
Received in revised form 26 January 2009
Accepted 20 February 2009
Available online 2 April 2009
Keywords:
Discrete tomography
hv-convex discrete set
Random generation
Reconstruction
Analysis of algorithms
a b s t r a c t
In this paper we summarize the most important generation methods developed for the
subclasses of hv-convex discrete sets. We also present some new generation techniques to
complement the former ones thus making it possible to design a complete benchmark set
for testing the performance of reconstruction algorithms on the class of hv-convex discrete
sets and its subclasses. By using this benchmark set the paper also collects several statistics
on hv-convex discrete sets, which are of great importance in the analysis of algorithms for
reconstructing such kinds of discrete sets.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction
The goal of Discrete Tomography (DT) [22,23] is to reconstruct discrete sets (finite subsets of the 2D integer lattice
defined up to translation) from the number of its elements lying on parallel lattice lines along several (usually horizontal,
vertical, diagonal, and antidiagonal) directions, called projections. It has several applications in pattern recognition, image
processing, electron microscopy, angiography, non-destructive testing, and so on. The main challenge in DT is that practical
limitations usually reduce the number of available projections to at most about four—which results in a possibly extremely
large number of solutions of the same reconstruction task. This can cause the reconstructed discrete set to be quite different
from the original one. In addition, the reconstruction problem can be NP-hard, depending on the number and directions of
the projections. In certain cases one can facilitate the reconstruction task by supposing that the set to be reconstructed has
some geometrical properties. Thus, the search space of the possible solutions can be reduced which can yield fast and less
ambiguous reconstructions.
A common problem in Discrete Tomography arises in comparing reconstruction methods from the viewpoint of speed,
accuracy, noise sensitivity, etc. In the past 15–20 years many reconstruction algorithms have been developed for solving the
reconstruction problem by using different techniques. The average performance of those reconstruction algorithms were
often tested on certain subclasses of hv-convex discrete sets. The reason of this is that the reconstruction in those classes
has a well-developed theory including heuristics and exact reconstruction algorithms, as well as some important results
regarding the complexity and ambiguity of the reconstruction. As an example, the reconstruction of hv-convex discrete
sets from two projections is known to be NP-complete while it can be solved in polynomial time with the additional
condition that the set is connected in the same time. The key to obtain an exact comparison of the average performance
of different reconstruction algorithms is to develop uniform random generators for the studied classes. Unfortunately, for
some subclasses of the hv-convex discrete sets no efficient method was known to generate elements of those classes by
using uniform random distributions. In addition, even if there was a uniform generator for a certain class of discrete sets,
∗
Tel.: +36 62 546396; fax: +36 62 546397.
E-mail address: pbalazs@inf.u-szeged.hu.
0166-218X/$ – see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.dam.2009.02.019