A 3D Sequential Thinning Scheme Based on Critical Kernels

International audienceWe propose a new generic sequential thinning scheme based on the critical kernels framework. From this scheme, we derive sequential algorithms for obtaining ultimate skeletons and curve skeletons. We prove some properties of these algorithms, and we provide the results of a quantitative evaluation that compares our algorithm for curve skeletons with both sequential and parallel ones

On parallel thinning algorithms: minimal non-simple sets, P-simple points and critical kernels

International audienceCritical kernels constitute a general framework in the category of abstract complexes for the study of parallel homotopic thinning in any dimension. In this article, we present new results linking critical kernels to minimal non-simple sets (MNS) and P-simple points, which are notions conceived to study parallel thinning in discrete grids. We show that these two previously introduced notions can be retrieved, better understood and enriched in the framework of critical kernels. In particular, we propose new characterizations which hold in dimensions 2, 3 and 4, and which lead to efficient algorithms for detecting P-simple points and minimal non-simple sets

Distance, granulometry, skeleton

In this chapter, we present a series of concepts and operators based on the notion of distance. As often with mathematical morphology, there exists more than one way to present ideas, that are simultaneously equivalent and complementary. Here, our problem is to present methods to characterize sets of points based on metric, geometry and topology considerations. An important concept is that of the skeleton, which is of fundamental importance in pattern recognition, and has many practical application

Transformations topologiques discrètes

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Powerful Parallel Symmetric 3D Thinning Schemes Based on Critical Kernels

The main contribution of the present article consists of new 3D parallel and symmetric thinning schemes which have the following qualities: - They are effective and sound, in the sense that they are guaranteed to preserve topology. This guarantee is obtained thanks to a theorem on critical kernels; - They are powerful, in the sense that they remove more points, in one iteration, than any other symmetric parallel thinning scheme; - They are versatile, as conditions for the preservation of geometrical features (e.g., curve extremities or surface borders) are independent of those accounting for topology preservation; - They are efficient: we provide in this article a small set of masks, acting in the grid Z3, that is sufficient, in addition to the classical simple point test, to straightforwardly implement them

Characterizing and Detecting Loops in n-Dimensional Discrete Toric Spaces

International audienc

Discrete bisector function and Euclidean skeleton in 2D and 3D

International audienceWe propose a new definition and an exact algorithm for the discrete bisector function, which is an important tool for analyzing and filtering Euclidean skeletons. We also introduce a new thinning algorithm which produces homotopic discrete Euclidean skeletons. These algorithms, which are valid both in 2D and 3D, are integrated in a skeletonization method which is based on exact transformations, allows the filtering of skeletons, and is computationally efficient

Homeomorphic Alignment of Weighted Trees

International audienceMotion capture, a currently active research area, needs estimation of the pose of the subject. For this purpose, we match the tree representation of the skeleton of the 3D shape to a pre-specified tree model. Unfortunately, the tree representation can contain vertices that split limbs in multiple parts, which do not allow a good match by usual methods. To solve this problem, we propose a new alignment, taking into account the homeomorphism between trees, rather than the isomorphism, as in prior works. Then, we develop several computationally efficient algorithms for reaching real-time motion capture

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

Topological monsters in Z^3: A non-exhaustive bestiary

International audienceSimple points in Z^n, and especially in Z^3, are the basis of several topology-preserving transformation methods proposed for image analysis (segmentation, skeletonisation, ...). Most of these methods rely on the assumption that the --iterative or parallel-- removal of simple points from a discrete object X necessarily leads to a globally minimal topologically equivalent sub-object of X (i.e. a subset Y which is topologically equivalent to X and which does not strictly include another set Z topologically equivalent to X). This is however false in Z^3, and more generally in Z^n. We illustrate this fact by presenting some topological monsters, i.e. some objects of Z^3 only composed of non-simple points, but which could however be reduced without altering their topology