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

Convolutions on digital surfaces: on the way iterated convolutions behave and preliminary results about curvature estimation

In [FoureyMalgouyres09] the authors present a generalized convolution operator for functions defined on digital surfaces. We provide here some extra material related to this notion. Some about the relative isotropy of the way a convolution kernel (or mask) grows when the convolution operator is iterated. We also provide preliminary results about a way to estimate curvatures on a digital surface, using the same convolution operator

Efficient Encoding of n-D Combinatorial Pyramids

International audienceCombinatorial maps define a general framework which allows to encode any subdivision of an n-D orientable quasi-manifold with or without boundaries. Combinatorial pyramids are defined as stacks of successively reduced combinatorial maps. Such pyramids provide a rich framework which allows to encode fine properties of objects (either shapes or partitions). Combinatorial pyramids have first been defined in 2D, then extended using n-D generalized combinatorial maps. We motivate and present here an implicit and efficient way to encode pyramids of n-D combinatorial maps

Nombre d'intersection et d'entrelacement de courbes discrètes et application à la caractérisation de la préservation de la topologie en imagerie

Topology preservation problems raise in the writing of image processing applications, particularly for pattern recognition purposes. In this thesis, we are interested in the characterization of topology preservation in two kinds of digital spaces. The first one is the space of objects made of points of Z3 which are identified with unit cubes: the voxels. Several authors have given local but also global characterizations of the fact that a point can be removed from a subset of Z3 without changing its topological properties. This kind of point is usually called a simple point. One of these characterizations uses the digital fundamental group, structure of algebraic topology adapted to digital topology and which relies on the set of equivalence classes of closed paths in an object. Here, we define the linking number between discrete closed paths which allows us to prove a new and more concise global characterization of 3D simple points in term of digital fundamental group isomorphism. The other space which has been considered is constituted by the surface of an object of Z3, surface made of the « visible » faces of the voxels of this object: the surfels. Such a surface is also commonly called a digital boundary. In this framework, we define here another topological invariant, the intersection number between two surfel paths, whose invariance property by homotopic deformation of the paths is proved. This intersection number is used in order to prove a new Jordan theorem in the space of digital boundaries, but also to establish a new global characterization of the fact that a part of a surface can be obtained by a sequential deletion of simple surfels in a set in which it is included.La préservation de la topologie est un problème qui apparaît au cours du développement d'applications d'analyse d'images, plus particulièrement dans le domaine de la reconnaissance des formes. On s'intéresse dans cette thèse à la caractérisation de la préservation de la topologie dans deux types d'espaces discrets. Le premier est celui des objets composés de cubes élémentaires, les voxels, auxquels sont identifés les points de Z3. Divers auteurs ont caractérisé localement mais aussi de façon globale le fait qu'un point puisse être supprimé d'une partie de Z3 sans en modifier les caractéristiques topologiques. Ce type de point est communément appelé un point simple. L'une des caractérisations données utilise le groupe fondamental discret, structure issue de la topologie algébrique et basée sur l'ensemble des classes d'équivalences de chemins fermés dans un objet. On définit ici un nouvel invariant, le nombre d'entrelacement entre courbes fermées discrètes. Celui-ci permet de démontrer une nouvelle caractérisation globale plus concise des points simples 3D en terme d'isomorphisme de groupe fondamental discret. L'autre espace discret considéré dans cette thèse est constitué par la surface d'un objet de Z3, surface composée des facettes « visibles » des voxels de cet objet : les surfels. Une telle surface est aussi appelée une surface de surfels. Dans ce cadre, on définit ici un autre invariant topologique, le nombre d'intersection entre deux chemins, dont la propriété d'invariance par déformation des chemins est démontrée. Ce nombre est utilisé pour prouver un nouveau théorème de Jordan dans l'espace des surfaces de surfels, mais aussi pour établir une nouvelle caractérisation globale du fait qu'une partie de surface puisse être obtenue par suppression séquentielle de surfels dans une autre partie qui la contient

Nombre d'intersection et d'entrelacement de courbes discrètes et application à la caractérisation de la préservation de la topologie en imagerie

Topology preservation problems raise in the writing of image processing applications, particularly for pattern recognition purposes. In this thesis, we are interested in the characterization of topology preservation in two kinds of digital spaces. The first one is the space of objects made of points of Z3 which are identified with unit cubes: the voxels. Several authors have given local but also global characterizations of the fact that a point can be removed from a subset of Z3 without changing its topological properties. This kind of point is usually called a simple point. One of these characterizations uses the digital fundamental group, structure of algebraic topology adapted to digital topology and which relies on the set of equivalence classes of closed paths in an object. Here, we define the linking number between discrete closed paths which allows us to prove a new and more concise global characterization of 3D simple points in term of digital fundamental group isomorphism. The other space which has been considered is constituted by the surface of an object of Z3, surface made of the « visible » faces of the voxels of this object: the surfels. Such a surface is also commonly called a digital boundary. In this framework, we define here another topological invariant, the intersection number between two surfel paths, whose invariance property by homotopic deformation of the paths is proved. This intersection number is used in order to prove a new Jordan theorem in the space of digital boundaries, but also to establish a new global characterization of the fact that a part of a surface can be obtained by a sequential deletion of simple surfels in a set in which it is included.La préservation de la topologie est un problème qui apparaît au cours du développement d'applications d'analyse d'images, plus particulièrement dans le domaine de la reconnaissance des formes. On s'intéresse dans cette thèse à la caractérisation de la préservation de la topologie dans deux types d'espaces discrets. Le premier est celui des objets composés de cubes élémentaires, les voxels, auxquels sont identifés les points de Z3. Divers auteurs ont caractérisé localement mais aussi de façon globale le fait qu'un point puisse être supprimé d'une partie de Z3 sans en modifier les caractéristiques topologiques. Ce type de point est communément appelé un point simple. L'une des caractérisations données utilise le groupe fondamental discret, structure issue de la topologie algébrique et basée sur l'ensemble des classes d'équivalences de chemins fermés dans un objet. On définit ici un nouvel invariant, le nombre d'entrelacement entre courbes fermées discrètes. Celui-ci permet de démontrer une nouvelle caractérisation globale plus concise des points simples 3D en terme d'isomorphisme de groupe fondamental discret. L'autre espace discret considéré dans cette thèse est constitué par la surface d'un objet de Z3, surface composée des facettes « visibles » des voxels de cet objet : les surfels. Une telle surface est aussi appelée une surface de surfels. Dans ce cadre, on définit ici un autre invariant topologique, le nombre d'intersection entre deux chemins, dont la propriété d'invariance par déformation des chemins est démontrée. Ce nombre est utilisé pour prouver un nouveau théorème de Jordan dans l'espace des surfaces de surfels, mais aussi pour établir une nouvelle caractérisation globale du fait qu'une partie de surface puisse être obtenue par suppression séquentielle de surfels dans une autre partie qui la contient

Connecting walks and connecting dart sequences for n-D combinatorial pyramids

International audienceCombinatorial maps define a general framework which allows to encode any subdivision of an n-D orientable quasi-manifold with or without boundaries. Combinatorial pyramids are defined as stacks of successively reduced combinatorial maps. Such pyramids provide a rich framework which allows to encode fine properties of objects (either shapes or partitions). Combinatorial pyramids have first been defined in 2D. This first work has later been extended to pyramids of n-D generalized combinatorial maps. Such pyramids allow to encode stacks of non orientable partitions but at the price of a twice bigger pyramid. These pyramids are also not designed to capture efficiently the properties connected with orientation. This work presents the design of pyramids of n-D combinatorial maps and important notions for their encoding and processing

Connecting walks and connecting dart sequences for n-D combinatorial pyramids

International audienceCombinatorial maps define a general framework which allows to encode any subdivision of an n-D orientable quasi-manifold with or without boundaries. Combinatorial pyramids are defined as stacks of successively reduced combinatorial maps. Such pyramids provide a rich framework which allows to encode fine properties of objects (either shapes or partitions). Combinatorial pyramids have first been defined in 2D. This first work has later been extended to pyramids of n-D generalized combinatorial maps. Such pyramids allow to encode stacks of non orientable partitions but at the price of a twice bigger pyramid. These pyramids are also not designed to capture efficiently the properties connected with orientation. This work presents the design of pyramids of n-D combinatorial maps and important notions for their encoding and processing

Intersection number and topology preservation within digital surfaces

International audienc

Connecting walks and connecting dart sequences for n-D combinatorial pyramids

Combinatorial maps define a general framework which allows to encode any subdivision of an n-D orientable quasi-manifold with or without boundaries. Combinatorial pyramids are defined as stacks of successively reduced combinatorial maps. Such pyramids provide a rich framework which allows to encode fine properties of objects (either shapes or partitions). Combinatorial pyramids have first been defined in 2D. This first work has later been extended to pyramids of n-D generalized combinatorial maps. Such pyramids allow to encode stacks of non orientable partitions but at the price of a twice bigger pyramid. These pyramids are also not designed to capture efficiently the properties connected with orientation. This work presents the design of pyramids of n-D combinatorial maps and important notions for their encoding and processing