Jacques Liandrat

Analysis of a New Nonlinear Subdivision Scheme. Applications in Image Processing

A nonlinear multiresolution scheme within Hartens framework is presented, based on a new nonlinear, centered piecewise polynomial interpolation technique. Analytical properties of the resulting subdivision scheme, such as convergence, smoothness, and stability, are studied. The stability and the compression properties of the associated multiresolution transform are demonstrated on several numerical experiments on images.

Wavelet algorithms for numerical resolution of partial differential equations

Abstract Numerical algorithms for the approximation of non linear partial differential equations are presented. On one hand they are based on adaptive spaces of the approximation provided by wavelets and on the other hand on efficient approximations of evolution operators on these spaces. Numerical experiments are described on 1D test problems.

A fully adaptive multiresolution scheme for image processing

A nonlinear multiresolution scheme within Hartens framework [A. Harten, Discrete multiresolution analysis and generalized wavelets, J. Appl. Numer. Math. 12 (1993) 153-192; A. Harten, Multiresolution representation of data II, SIAM J. Numer. Anal. 33 (3) (1996) 1205-1256] is presented. It is based on a centered piecewise polynomial interpolation fully adapted to discontinuities. Compression properties of the multiresolution scheme are studied on various numerical experiments on images.

On a nonlinear subdivision scheme avoiding Gibbs oscillations and converging towards C s functions with s > 1.

This paper presents a new nonlinear dyadic subdivision scheme eliminating the Gibbs oscillations close to discontinuities. Its convergence , stability and order of approximation are analyzed. It is proved that this scheme converges towards limit functions Holder continuous with exponent larger than 1.299. Numerical estimates provide a Holder exponent of 2.438. This subdivision scheme is the first one that simultaneously achieves the control of the Gibbs phenomenon and has limit functions with Holder exponent larger than 1.

On the fast approximation of some nonlinear operators in nonregular wavelet spaces

We focus our attention on the approximation of some nonlinear operators in adapted wavelet spaces. We show the interest of the construction of scaling functions with a large number of zero moments. We present the convergence estimate of an algorithm based on paraproducts for the approximation of nonlinear operators using wavelets connected to scaling functions with zero moments. Numerical tests are performed on univariate examples.

A family of stable nonlinear nonseparable multiresolution schemes in 2D

Multiresolution representations of data are powerful tools in data compression. For a proper adaptation to the edges, a good strategy is to consider a nonlinear approach. Thus, one needs to control the stability of these representations. In this paper, 2D multiresolution processing algorithms that ensure this stability are introduced. A prescribed accuracy is ensured by these strategies.

On a compact non-extrapolating scheme for adaptive image interpolation

Abstract The motivation of this paper is to improve the results of ENO subcell resolution, that already obtains really good results but using extrapolation, as was stated by F. Arandiga, R. Donat and P. Mulet [Adaptive Interpolation of Images, Signal Processing 83(2) (2003) 459–464]. We present a new method for image interpolation which combines a new nonlinear cell-average interpolatory technique that uses a trigonometric mean and subcell resolution. The scheme uses a fully compact reconstruction and avoid any step of extrapolation. The experiments presented validate the theoretical results obtained.

On a class of L1-stable nonlinear cell-average multiresolution schemes

This paper is devoted to the analysis of a general family of nonlinear cell-average multiresolution schemes. The L^1-stability of this multiresolution transform is analyzed. Some examples are presented.

Analysis of some bivariate non-linear interpolatory subdivision schemes

This paper is devoted to the convergence analysis of a class of bivariate subdivision schemes that can be defined as a specific perturbation of a linear subdivision scheme. We study successively the univariate and bivariate case and apply the analysis to the so called Powerp scheme (Serna and Marquina, J Comput Phys 194:632–658, 2004).

On a nonlinear cell-average multiresolution scheme for image compression

The aim of this paper is to present a new nonlinear cell-average multiresolution scheme and its application to the compression of color images. The objective is to obtain an algorithm competing with linear multiresolution transforms in smooth regions but exhibiting a better behaviour (compression, Gibbs phenomenon reduction, …) in nonsmooth regions. Several desirable features can be associated to this new algorithm: the reconstruction operator is third-order accurate in smooth regions, the stencil used is always centred with optimal support and it is adapted to the presence of discontinuities. Monotony preservation, order of approximation, convergence of the associated subdivision scheme, elimination of Gibbs effects and stability are analysed. This paper can be considered as the second part of the paper by Amat, Donat, Liandrat and Trillo [Foundations of Computational Mathematics, 6 (2), 193–225, (2006)] where the point-value framework was considered.