Juan Ruiz

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.

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.

Fast Multiresolution Algorithms and Their Related Variational Problems for Image Denoising

When considering fast multiresolution techniques for image denoising problems, there are three important aspects. The first one is the choice of the specific multiresolution, the second one the choice of a proper filter function and the third one the choice of the thresholding parameter. Starting from the classical one, namely, linear wavelet algorithms with Donoho and Johnstone’s Soft-thresholding with the universal shrinkage parameter, the first aim of this paper is to improve it in the three mentioned directions. Thus, a new nonlinear approach is proposed and analyzed. On the other hand, the linear approach of Donoho and Johnstone is related with a well known variational problem. Our second aim is to find a related variational problem, more adapted to the denoising problem, for the new approach. We would like to mention that the analysis of theoretical properties in a nonlinear setting are usually notoriously more difficult. Finally, a comparison with other approaches, including linear and nonlinear multiresolution schemes, SVD-based schemes and filters with a non-multiresolution nature, is presented.

Improving the compression rate versus L1 error ratio in cell-average error control algorithms

Multiresolution representations of data are powerful tools in data compression. A common framework for applications is the cell-average setting. For a proper adaptation to singularities, it is interesting to develop nonlinear methods. Thus, one needs to control the stability of these representations. We introduce a generalization, depending on a parameter λ, of the classical cell-average error-control algorithms and we study the choice of the parameter to get the best relation quality vs. ratio of compression. It turns out that λ can be chosen nonlinearly and that for the L1 norm we can get a significant improvement over the classical error-control algorithms.

Reciprocal polynomial extrapolation vs Richardson extrapolation for singular perturbed boundary problems

The reciprocal polynomial extrapolation was introduced in Amat et al. (J Comput Math 22(1):1–10, 2004), where its accuracy and stability were studied and a linear scalar test problem was analyzed numerically. In the present work, a new step in the implementation of the reciprocal polynomial extrapolation, ensuring at least the same behavior as the Richardson extrapolation, is proposed. Looking at the reciprocal extrapolation as a Richardson extrapolation where the original data is nonlinearly modified, the improvements that we will obtain should be justified. Several theoretical analysis of the new extrapolation, including local error estimates and stability properties, are presented. A comparison between the two extrapolation techniques is performed for solving some boundary problems with perturbation controlled by a small parameter ϵ. Using two specific boundary problems, the error and the robustness of the new technique using centered divided differences in a uniform mesh are investigated numerically. They turn out to be better than those presented by the Richardson extrapolation. Finally, investigations on the accuracy when using a special non-uniform discretization mesh are presented. A numerical comparison with the Richardson extrapolation for this particular case, where we present some improvements, is also performed.

On a nonlinear mean and its application to image compression using multiresolution schemes

This paper is devoted to the compression of colour images using a new nonlinear cell-average multiresolution scheme. The aim is to obtain similar compression properties as linear multiresolution schemes but eliminating the classical Gibbs phenomenon of this type of reconstructions near the edges. The algorithm is based on a nonlinear reconstruction operator (using a nonlinear trigonometric mean). The new reconstruction is third-order accurate in smooth regions and adapted to the presence of discontinuities. The data used are always centred with optimal support. Some theoretical properties of this scheme are analysed (order of approximation, convergence, elimination of Gibbs effect and stability).

N-dimensional error control multiresolution algorithms for the cell average discretization

We present N-dimensional multiresolution algorithms with error control strategies in the cell average setting as a generalization to N dimensions of the work done in this direction. We present results proving the stability and giving explicit error bounds. We also explain how to carry out the programming and we include two numerical experiments to exemplify the utility of these algorithms.

On an New Algorithm for Function Approximation with Full Accuracy in the Presence of Discontinuities Based on the Immersed Interface Method

This paper is devoted to the construction and analysis of an adapted and nonlinear multiresolution algorithm designed for interpolation or approximation of discontinuous univariate functions. The adaption attained allows to avoid numerical artifacts that appear when using linear algorithms and, at the same time, to obtain a high order of accuracy close to the singularities. It is known that linear algorithms are stable and convergent for smooth functions, but diffusion and Gibbs effect appear if the functions are piecewise continuous. Our aim is to develop an algorithm for function approximation with full accuracy that is capable to adapt to corners (kinks) and jump discontinuities, that uses a centered stencil and that does not use extrapolation. In order to reach this goal, we will need some information about the jumps in the function that we want to approximate and its derivatives. If this information is available, the algorithm is the most compact possible in the sense that the stencil is fixed and we do not need a stencil selection procedure as other algorithms do, such as ENO subcell resolution (ENO-SR). If the information about the jumps is not available, we will show a technique to approximate it. The algorithm is based on linear interpolation plus correction terms that provide the desired accuracy close to corners or jump discontinuities.