Normals estimation for digital surfaces based on convolutions

International audienceIn this paper, we present a method that we call on-surface convolution which extends the classical notion of a 2D digital filter to the case of digital surfaces (following the cuberille model). We also define an averaging mask with local support which, when applied with the iterated convolution operator, behaves like an averaging with large support. The interesting property of the latter averaging is the way the resulting weights are distributed: given a digital surface obtained by discretization of a differentiable surface of R^3 , the masks isocurves are close to the Riemannian isodistance curves from the center of the mask. We eventually use the iterated averaging followed by convolutions with differentiation masks to estimate partial derivatives and then normal vectors over a surface. The number of iterations required to achieve a good estimate is determined experimentally on digitized spheres and tori. The precision of the normal estimation is also investigated according to the digitization step

Minimal simple pairs in the cubic grid

International audiencePreserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. This paper constitutes an introduction to the study of non-trivial simple sets in the framework of cubical 3-D complexes. A simple set has the property that the homotopy type of the object in which it lies is not changed when the set is removed. The main contribution of this paper is a characterisation of the non-trivial simple sets composed of exactly two voxels, such sets being called minimal simple pairs

A concise characterization of 3d simple points

International audienceWe recall the definition of simple points which uses the digital fundamental group introduced by T.Y.Kong in [Kong89]. Then, we prove that a not less restrictive definition can be given. Indeed, we prove that there is no need of considering the fundamental group of the complement of an object in order to characterize its simple points. In order to prove this result, we do not use the fact that "the number of holes of X is equal to the number of holes in \overline{X}" which is not sufficient for our purpose but we use the linking number defined in [FoureyMalg00b]. In so doing, we formalize the proofs of several results stated without proof in the literature (Bertrand, Kong, Morgenthaler)