International audienceIn 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 non parallel lattice directions

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.

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 precised. 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 slope of plane. Actually all the results of this paper can be generalized for the regions of hyperplanes for any dimension d ≥ 3. The proofs of some results used in this article are contained in the extended version of this paper [1].

Abstract. In this paper we investigate the 'local' definitions of length of digital curves in the digital space rZ 2 where r is the resolution of the discrete space. We prove that if µr is any local definition of the length of digital curves in rZ 2 , then for almost all segments S of R 2 , the measure µr(Sr) does not converge to the length of S when the resolution r converges to 0, where Sr is the Bresenham discretization of the segment S in rZ 2 . Moreover, the average errors of classical local definitions are estimated, and we define a new one which minimizes this error.

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