Nonlinear Multiresolution Scheme Research Articles

On some new variational problems for image denoising

This paper is devoted to image denoising problems using multiresolution schemes related to variational problems. We start with the linear approach of Donoho and Johnstone, that is related to a well known diffusion‐type variational problem. In order to improve the behavior of this approach, we propose some new nonlinear variational problems more adapted to the problem of denoising. Moreover, the discretization is performed using nonlinear multiresolution schemes. In particular, we obtain some fast and well adapted schemes for the considered problem of denoising.

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.

Denoising using linear and nonlinear multiresolutions II: cell‐average framework and color images

PurposeMultiresolution representations of data are classical tools in image processing applications. The purpose of this paper is to discuss a particular problem, obtaining good reconstructions of noise images.Design/methodology/approachA nonlinear multiresolution scheme within Harten's framework corresponding to a nonlinear cell‐average technique is used for color image denoising.FindingsThis paper finds it is possible, for example, to apply the theoretical framework to case studies in internationally operating companies delivering a mix of goods and services.Research limitations/implicationsThe present study provides a starting point for further research in the denoising problems using nonlinear techniques.Originality/valueMoreover, the proposed framework has proven to be useful in improving the classical linear multiresolution approaches.

A fully adaptive multiresolution scheme for image processing

A nonlinear multiresolution scheme within Harten’s 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 multiresolution schemes using a stencil selection procedure: applications to ENO schemes

This paper is devoted to multiresolution schemes that use a stencil selection procedure in order to obtain adaptation to the presence of edges in the images. Since non adapted schemes, based on a centered stencil, are less affected by the presence of texture, we propose the introduction of some weight that leads to a more frequent use of the centered stencil in regions without edges. In these regions the different stencils have similar weights and therefore the selection becomes an ill-posed problem with high risk of instabilities. In particular, numerical artifacts appear in the decompressed images. Our attention is centered in ENO schemes, but similar ideas can be developed for other multiresolution schemes. A nonlinear multiresolution scheme corresponding to a nonlinear interpolatory technique is analyzed. It is based on a modification of classical ENO schemes. As the original ENO stencil selection, our algorithm chooses the stencil within a region of smoothness of the interpolated function if the jump discontinuity is sufficiently big. The scheme is tested, allowing to compare its performances with other linear and nonlinear schemes. The algorithm gives results that are at least competitive in all the analyzed cases. The problems of the original ENO interpolation with the texture of real images seem solved in our numerical experiments. Our modified ENO multiresolution will lead to a reconstructed image free of numerical artifacts or blurred regions, obtaining similar results than WENO schemes. Similar ideas can be used in multiresolution schemes based in other stencil selection algorithms.

Stability Through Synchronization in Nonlinear Multiscale Transformations

Nonlinear analogues of classical wavelet transformations have been designed within spatially oriented multiresolution frameworks such as Harten’s framework [J. Appl. Numer. Math., 12 (1993), pp. 153-193] and the lifting framework [R. L. Claypoole e, IEEE Trans. Image Process., 12 (2003), pp. 1449-1459]. In designing and using nonlinear multiscale transformations for data compression purposes, the main issue is that of ensuring stability. In [J. Appl. Numer. Math., 12 (1993), pp. 153-193], Harten ensures stability by designing a special error-control mechanism that provides estimates for the compression error. In [R. L. Claypoole e, IEEE Trans. Image Process., 12 (2003), pp. 1449-1459] no theoretical results are provided, but the authors argue that stability is a consequence of synchronization between the nonlinear decisions made by the prediction operators in the direct and inverse transformations. By interpreting the nonlinear multiscale transformation of R. L. Claypoole e within Harten’s cell-average framework, we are able to provide stability bounds for the scheme of R. L. Claypoole e and establish a theoretical basis for comparison between both nonlinear multiresolution schemes.

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

A nonlinear multiresolution scheme within Harten's 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.

Nonlinear multiscale decompositions: The approach of A. Harten

Data‐dependent interpolatory techniques can be used in the reconstruction step of a multiresolution scheme designed “a la Harten”. In this paper we carefully analyze the class of Essentially Non‐Oscillatory (ENO) interpolatory techniques described in [11] and their potential to improve the compression capabilities of multiresolution schemes. When dealing with nonlinear multiresolution schemes the issue of stability also needs to be carefully considered.

M-band nonlinear subband decompositions with perfect reconstruction

We investigate nonlinear multirate filterbanks with maximal decimation and perfect reconstruction. Definitions of the desired properties of such structures are given for general nonlinear filterbanks. We then consider a triangular representation of linear filterbanks and see that it may be easily extended to the nonlinear case. Furthermore, general nonlinear filterbanks are presented, for which perfect reconstruction is either inherently guaranteed or ensured subject to an easily verified condition. Extensions to bidimensional filters are also discussed and an application for nonlinear multiresolution schemes to feature sieves is shown.

Continuous Wavelet Decompositions, Multiresolution, and Contrast Analysis

A continuous version of multiresolution analysis is described, starting from usual continuous wavelet decompositions. Scale discretization leads to decomposition into functions of arbitrary bandwidth, satisfying QMF-like conditions. Finally, a nonlinear multiresolution scheme is described, providing multiplicative reconstruction formulas.