J. Carlos Trillo

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.

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 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.

Denoising using linear and nonlinear multiresolutions

Purpose – To provide several comparisons between linear and nonlinear approaches in denoising applications.Design/methodology/approach – The comparison is based on the peak signal noise ratio (PSNR) image quality measure. Which one of the algorithms gives higher PSNR and then denoises more the original picture is studied.Findings – Nonlinear reconstruction operators can improve the accuracy of the prediction in the vicinity of isolated singularities. A better treatment of the singularities corresponding to the image edges and, therefore, an improvement on the sparsity of the multiresolution representation of images are then expected.Research limitations/implications – In this paper the point‐value framework is considered. Other frameworks, as the cell‐average discretization, are more suitable for image processing where noise and texture appear. But, the point value schemes can be adapted to the cell‐average discretization using primitive function.Practical implications – People can use the new denoising a...

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.

Point Values Hermite Multiresolution for Non-smooth Noisy Signals

In this paper, a nonlinear interpolation is used in order to compute adaptively derivatives from the discrete information of any signal. Using these derivatives a multiresolution based on Hermite interpolation is performed for compressing the signal. The way in which the derivatives are approximated is crucial when noise or singularities appear.

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.

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

Purpose – Multiresolution 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/approach – A nonlinear multiresolution scheme within Hartens framework corresponding to a nonlinear cell‐average technique is used for color image denoising.Findings – This 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/implications – The present study provides a starting point for further research in the denoising problems using nonlinear techniques.Originality/value – Moreover, the proposed framework has proven to be useful in improving the classical linear multiresolution approaches.

Analysis of the Gibbs phenomenon in stationary subdivision schemes

Abstract In this paper sufficient conditions to determine if a stationary subdivision scheme produces Gibbs oscillations close to discontinuities are presented. It consists of the positivity of the partial sums of the values of the mask. We apply the conditions to non-negative masks and analyze (numerically when the sufficient conditions are not satisfied) the Gibbs phenomenon in classical and recent subdivision schemes like B-splines, Deslauriers and Dubuc interpolation subdivision schemes and the schemes proposed in Siddiqi and Ahmad (2008).