Alain Daurat

An algorithm reconstructing convex lattice sets

In this paper, we study the problem of reconstructing special lattice sets from X-rays in a finite set of prescribed directions. We present the class of “Q-convex” sets which is a new class of subsets of Z2 having a certain kind of weak connectedness. The main result of this paper is a polynomial-time algorithm solving the reconstruction problem for the “Q-convex” sets. These sets are uniquely determined by certain finite sets of directions. As a result, this algorithm can be used for reconstructing convex subsets of Z2 from their X-rays in some suitable sets of four lattice directions or in any set of seven mutually nonparallel lattice directions.

Salient and reentrant points of discrete sets

The border-salient and reentrant points of a discrete set are special points of the border of the set. When they are given with multiplicity they completely characterize the set, and without multiplicity they characterize the set if all its 8-components are 4-connected. The inner-salient and reentrant are defined similarly to the border ones, but we show that, in general, they do not characterize the set, even if this set is 4-simply connected. We also show that the genus of a set can be easily computed from the number of salient and reentrant points.

Stability in discrete tomography: some positive results

The problem of reconstructing finite subsets of the integer lattice from X-rays has been studied in discrete mathematics and applied in several fields like data security, electron microscopy, and medical imaging. In this paper, we focus on the stability of the reconstruction problem for some special lattice sets. First we prove that if the sets are additive, then a stability result holds for very small errors. Then, we study the stability of reconstructing convex sets from both an experimental and a theoretical point of view. Numerical experiments are conducted by using linear programming and they support the conjecture that convex sets are additive with respect to a set of suitable directions. Consequently, the reconstruction problem is stable. The theoretical investigation provides a stability result for convex lattice sets. This result permits to address the problem proposed by Hammer (in: Convexity, vol. VII, Proceedings of the Symposia in Pure Mathematics, American Mathematical Society, Providence, RI, 1963, pp. 498-499).

Determination of Q-convex sets by X-rays

In this paper, the problem of the determination of lattice sets from X-rays is studied. We define the class of Q-convex sets along a set D of directions which generalizes classical lattice convexity and we prove that for any D, the X-rays along D determine all the convex sets if and only if it determines all the Q-convex sets along D. As a consequence, any algorithm which reconstructs Q-convex sets from X-rays can be used to reconstruct convex lattice sets from X-rays along directions which provide uniqueness. This gives a constructive answer to the discrete version of Hammers X-ray problem.

On Local Definitions of Length of Digital Curves

In this paper we investigate the ‘local’ definitions of length of digital curves in the digital space r ℤ2 where r is the resolution of the discrete space. We prove that if μ r is any local definition of the length of digital curves in r ℤ2, then for almost all segments S of ℝ2, the measure μ r (S r ) does not converge to the length of S when the resolution r converges to 0, where S r is the Bresenham discretization of the segment S in r ℤ2. Moreover, the average errors of classical local definitions are estimated, and we define a new one which minimizes this error.

The chords' problem

The chords’ problem is a variant of an old problem of computational geometry: given a set of points of Rn, one can easily build the multiset of the distances between the points of the set but the converse construction is known, for a longtime, as to be difficult. The problem that we are going to investigate is also a converse construction with the difference that it is not one of the distances’ multisets but one of the chords’ multisets. In dimension 1, the old distances’ problem and the chords’ problem coincide with each other whereas in other dimensions, the chords’ multisets contain more information on the sets than their distances’ multisets. This paper provides, in dimension 1, two different algorithms to reconstruct the set of points according to their chords’ multiset. The first one is given for its effectiveness in spite of an uncertain complexity whereas the second one is the first polynomial algorithm solving the chords’ problem. At least, we will explain how to transform a chords’ problem in dimension n into an equivalent chords’ problem in dimension 1.

Random generation of Q-convex sets

The problem of randomly generating Q-convex sets is considered. We present two generators. The first one uses the Q-convex hull of a set of random points in order to generate a Q-convex set included in the square [0, n)2. This generator is very simple, but is not uniform and its performance is quadratic in n. The second one exploits a coding of the salient points, which generalizes the coding of a border of polyominoes. It is uniform, and is based on the method by rejection. Experimentally, this algorithm works in linear time in the length of the word coding the salient points. Besides, concerning the enumeration problem, we determine an asymptotic formula for the number of Q-convex sets according to the size of the word coding the salient points in a special case, and in general only an experimental estimation.

Medians of Discrete Sets according to a Linear Distance

Abstract. In this paper we present some results concerning the median points of a discrete set according to a distance defined by means of two directions p and q . We describe a local characterization of the median points and show how these points can be determined from the projections of the discrete set along directions p and q . We prove that the discrete sets having some connectivity properties have at most four median points according to a linear distance, and if there are four median points they form a parallelogram. Finally, we show that the 4-connected sets which are convex along the diagonal directions contain their median points along these directions.

Technical Section: About the frequencies of some patterns in digital planes. Application to area estimators

In this paper, we prove that the function giving the frequency of a class of patterns of digital planes with respect to the slopes of the plane is continuous and piecewise affine, moreover the regions of affinity are specified. It allows to prove some combinatorial properties of a class of patterns called (m,n)-cubes. This study has also some consequences on local estimators of area: we prove that the local estimators restricted to regions of plane never converge to the exact area when the resolution tends to zero for almost all slopes of plane. All the results are generalized for the regions of hyperplanes in any dimension d>=3.

Reconstruction of convex lattice sets from tomographic projections in quartic time

Filling operations are procedures which are used in Discrete Tomography for the reconstruction of lattice sets having some convexity constraints. Many algorithms have been published giving fast implementations of these operations, and the best running time [S. Brunetti, A. Daurat, A. Kuba, Fast filling operations used in the reconstruction of convex lattice sets, in: Proc. of DGCI 2006, in: Lecture Notes in Comp. Sci., vol. 4245, 2006, pp. 98-109] is O(N^2logN) time, where N is the size of projections. In this paper we improve this result by providing an implementation of the filling operations in O(N^2). As a consequence, we reduce the time-complexity of the reconstruction algorithms for many classes of lattice sets having some convexity properties. In particular, the reconstruction of convex lattice sets satisfying the conditions of Gardner-Gritzmann [R.J. Gardner, P. Gritzmann, Discrete tomography: Determination of finite sets by X-rays, Trans. Amer. Math. Soc. 349 (1997) 2271-2295] can be performed in O(N^4)-time.