A New Formalism for Nonlinear and Non-Separable Multi-scale Representation

In this paper, we present a new formalism for nonlinear and non-separable multi-scale representations. We first show that most of the one-dimensional nonlinear multi-scale representations described in the literature are based on prediction operators which are the sum of a linear prediction operator and a perturbation defined using finite differences. We then extend this point of view to the multi-dimensional case where the scaling factor is replaced by a non-diagonal dilation matrix $M$. The new formalism we propose brings about similarities between existing nonlinear multi-scale representations and also enables us to alleviate the classical hypotheses made to prove the convergence of the multi-scale representations

Atlas-Based Character Skinning with Automatic Mesh Decomposition

Skinning is the most tedious part in the character animation process. Using standard methods, joint weights must be attached to each vertex of the character's mesh, which is often time-consuming if an accurate animation is required. We propose a new modeling of the skinning process, inspired by the notion of atlas of charts. Starting from the character's animation skeleton, we first automatically decompose the mesh into anatomically meaningful overlapping regions. Regions are then blended in their overlapping parts using continuous transition functions. This leads to a simple yet efficient skinning process for which the weights are automatically defined and do not depend on the Euclidean distance but on the distance on the surface.Le skinning est l'étape la plus fastidieuse du processus d'animation d'un personnage. Dans les méthodes classiques, un poids associé à chaque articulation doit être attaché à chaque sommet du maillage du personnage, ce qui est souvent très coûteux en temps lorsqu'une animation précise est exigée. Nous proposons une nouvelle modélisation du processus de skinning, s'inspirant de la notion d'atlas de cartes. A partir du squelette d'animation du personnage, nous décomposons d'abord automatiquement le maillage en régions anatomiquement significatives et qui se chevauchent. Ces régions sont ensuite fusionnées dans leurs zones de chevauchement grˆace à l'utilisation de fonctions de transition continues. Ceci conduit à un processus de skinning simple mais néanmoins efficace, pour lequel les poids sont automatiquement définis et ne dépendent pas de la distance euclidienne entre sommets, mais de la distance sur la surface

Elementary factorisation of Box spline subdivision

International audienceWhen a subdivision scheme is factorised into lifting steps, it admits an in–place and invertible implementation, and it can be the predictor of many multiresolution biorthogonal wavelet transforms. In the regular setting where the underlying lattice hierarchy is defined by Z s and a dilation matrix M, such a factorisation should deal with every vertex of each subset in Z s /M Z s in the same way. We define a subdivision scheme which admits such a factorisation as being uniformly elementary factorable. We prove a necessary and sufficient condition on the directions of the Box spline and the arity of the subdivision for the scheme to admit such a factorisation, and recall some known keys to construct it in practice

Multiresolution Framework on the Interval based on Fibonacci Tiling, B–splines and Lifting Scheme

Rapport interne de GIPSA-labIn this paper, we introduce a multiresolution analysis on the interval based on non–uniform B– splines defined on the Fibonacci tiling. The construction of the multiscale structure based on the substitution rules of an L–system allows the derivation of a known framework from the regular dyadic setting to a non–uniform setting while limiting the number of different filters to a few and keeping a similar stability. After having explained how our approach fits into a biorthogonal framework, we detail how to build analysis wavelet functions in the B–spline setting. Then the emphasis is put on the definition of boundary scaling and wavelet functions by means of scaling equations. Our implementation of the multiresolution structure is done in such a way that the computation is carried out in place. Finally, a numerical analysis of the stability of the proposed scheme shows its similar behavior to the same multiresolution analysis that would be derived on a dyadic sampling

Bivariate non-uniform subdivision schemes based on L-systems

L–systems have been used to describe non-uniform, univariate, subdivision schemes, which offer more flexible refinement processes than the uniform schemes, while at the same time are easier to analyse than the geometry driven non-uniform schemes. In this paper, we extend L–system based nonuniform subdivision to the bivariate setting. We study the properties that an L–system should have to be the suitable descriptor of a subdivision refinement process. We derive subdivision masks to construct the regular parts of the subdivision surface as cubic B-spline patches. Finally we describe stencils for the extraordinary vertices, which after a few steps become stationary, so that the scheme can be studied through simple eigenanalysis. The proposed method is illustrated through two new subdivision schemes, a Binary-Ternary, and a Fibonacci scheme with average refinement rate below two

Antagonism between Extraordinary Vertex and its Neighbourhood for Defining Nested Box-Splines

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Nonlinear and non-separable multiscale representations based on Lipschitz perturbation

International audienceIn this Note, we present a new formalism for nonlinear and non-separable multiscale representations. The new formalism we propose brings about similarities between existing nonlinear multiscale representations and also allows us to alleviate the classical hypotheses made to prove the convergence of the multiscale representations

A Topological Lattice Refinement Descriptor for Subdivision Scheme

International audienceThe topological step of a subdivision scheme can be described as a refinement on regular tiling lattices, or more generally as a local transformation descriptor. The former classifies all the regular lattice transformations which give rise to other regular lattices. These regular lattice descriptors are limited by the type of the control mesh faces, the subdivided mesh must be composed by faces of the same type. The latter category describes some local topological transformations as the insertion of vertices in each face, followed by the description of a connectivity; such a descriptor is called meta-scheme. But these meta-schemes cannot describe a large number of regular refinements. We propose a topological descriptor that generalizes meta-schemes. Our descriptor is locally defined by a triple of integers which describes the number of inserted vertices relatively to the components of each face: vertices, edges and the face center. It is combined with a flexible connectivity descriptor which enhances modeling capability. Our descriptor can build the schemes commonly used and it can describe a variety of others, including many regular rotative schemes. We ensure the conservation of the control mesh global topology. The subdivision operators described here can be concatenated, leading to more complex topological descriptions. In addition, we propose a general geometric smoothing step

Smoothing the Antagonism between Extraordinary Vertex and Ordinary Neighbourhood on Subdivision Surfaces

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