19 research outputs found
Scalar and Hermite subdivision schemes
AbstractA criterion of convergence for stationary nonuniform subdivision schemes is provided. For periodic subdivision schemes, this criterion is optimal and can be applied to Hermite subdivision schemes which are not necessarily interpolatory. For the Merrien family of Hermite subdivision schemes which involve two parameters, we are able to describe explicitly the values of the parameters for which the Hermite subdivision scheme is convergent
Hermite Subdivision Schemes and Taylor Polynomials
International audienceWe propose a general study of the convergence of a Hermite subdivision scheme of degree in dimension 1. This is done by linking Hermite subdivision schemes and Taylor polynomials and by associating a so-called Taylor subdivision (vector) scheme . The main point of investigation is a spectral condition. If the subdivision scheme of the finite differences of is contractive, then is and is . We apply this result to two families of Hermite subdivision schemes, the first one is interpolatory, the second one is a kind of corner cutting, both of them use Obreshkov interpolation polynomial
Level-dependent interpolatory Hermite subdivision schemes and wavelets
We study many properties of level-dependent Hermite subdivision, focusing on
schemes preserving polynomial and exponential data. We specifically consider
interpolatory schemes, which give rise to level-dependent multiresolution
analyses through a prediction-correction approach. A result on the decay of the
associated multiwavelet coefficients, corresponding to a uniformly continuous
and differentiable function, is derived. It makes use of the approximation of
any such function with a generalized Taylor formula expressed in terms of
polynomials and exponentials
Dual Hermite subdivision schemes of de Rham-type
International audienceThough a Hermite subdivision scheme is non-stationary by nature, its non-stationarity can be of two types, making useful the distinction between Inherently Stationary and Inherently Non-Stationary Hermite subdivision schemes. This paper focuses on the class of inherently stationary, dual non-interpolatory Hermite subdivision schemes that can be obtained from known Hermite interpolatory ones, by applying a generalization of the de Rham corner cutting strategy. Exploiting specific tools for the analysis of inherently stationary Hermite subdivision schemes we show that, giving up the interpolation condition, the smoothness of the associated basic limit function can be increased by one, while its support width is only enlarged by one. To accomplish the analysis of de Rham-type Hermite subdivision schemes two new theoretical results are derived and the new notion of HC-convergence is introduced. It allows the construction of Hermite-type subdivision schemes of order d + 1 with the first element of the vector valued limit function having regularity â„ d
PARALLEL â3-SUBDIVISION with ANIMATION in CONSIDERATION of GEOMETRIC COMPLEXITY
We look at the broader field of geometric subdivision and the emerging field of parallel computing for the purpose of creating higher visual fidelity at an efficient pace. Primarily, we present a parallel algorithm for â3-Subdivision. When considering animation, we find that it is possible to do subdivision by providing only one variable input, with the rest being considered static. This reduces the amount of data transfer required to continually update a subdividing mesh. We can support recursive subdivision by applying the technique in passes. As a basis for analysis, we look at performance in an OpenCL implementation that utilizes a local graphics processing unit (GPU) and a parallel CPU. By overcoming current hardware limitations, we present an environment where general GPU computation of â3-Subdivision can be practical
Extended Hermite Subdivision Schemes
International audienceSubdivision schemes are efficient tools for building curves and surfaces. For vector subdivision schemes, it is not so straightforward to prove more than the Hölder regularity of the limit function. On the other hand, Hermite subdivision schemes produce function vectors that consist of derivatives of a certain function, so that the notion of convergence automatically includes regularity of the limit. In this paper, we establish an equivalence betweena spectral condition and operator factorizations, then we study how such schemes with smooth limit functions can be extended into ones with higher regularity. We conclude by pointing out this new approach applied to cardinal splines