Academic literature on the topic 'MULTIWAVELET'

Methods for digital image compression have been the subject of much study over the past decade. Advances in wavelet transforms and quantization methods have produced algorithms capable of surpassing the existing image compression standards like the Joint Photographic Experts Group (JPEG) algorithm. For best performance in image compression, wavelet transforms require filters that combine a number of desirable properties, such as orthogonality and symmetry. However, the design possibilities for wavelets are limited because they cannot simultaneously possess all of these desirable properties. The relatively new field of multiwavelets shows promise in removing some of the limitations of wavelets. Multiwavelets offer more design options and hence can combine all desirable transform features. The few previously published results of multiwavelet-based image compression have mostly fallen short of the performance enjoyed by the current wavelet algorithms. This thesis presents new multiwavelet transform methods and measurements that verify the potential benefits of multiwavelets. Using a zerotree quantization scheme modified to better match the unique decomposition properties of multiwavelets, it is shown that the latest multiwavelet filters can give performance equal to, or in many cases superior to, the current wavelet filters. The performance of multiwavelet packets is also explored for the first time and is shown to be competitive to that of wavelet packets in some cases. The wavelet and multiwavelet filter banks are tested on a much wider range of images than in the usual literature, providing a better analysis of the benefits and drawbacks of each. NOTE: (03/2007) An updated copy of this ETD was added after there were patron reports of problems with the file.<br>Master of Science

L’objectif principal de cette thèse est de développer une méthode hp-adaptative efficace en termes de coût et précision pour les schémas Galerkin discontinus appliqués aux équations de Navier-Stokes, en combinant flexibilité de l’adaptation a posteriori et précision de l’adaptation multi-résolution. Les performances de l’algorithme d’adaptation hp sont illustrées sur plusieurs cas d’écoulements stationnaires en une et deux dimensions. La première direction de recherche emploie une nouvelle méthodologie basée sur les multiondelettes pour estimer l’erreur de discrétisation de la solution numérique dans le contexte de simulations avec adaptation h. Les résultats démontrent clairement la viabilité de cette méthode pour atteindre un gain de calcul significatif par rapport àun raffinement de maillage uniforme. La deuxième voie de recherche aborde l’analyse et le développement d’une nouvelle stratégied’adaptation hp basée sur la décroissance du spectre des multi-ondelettes comme critère adaptation hp. Cette stratégie permet de discriminer avec succès les régions caractérisées par une grande régularité de celles contenant des phénomènes discontinus. De manière remarquable, l’algorithme d’adaptation hp est capable d’atteindre une haute précision caractéristique des solutions numériques d’ordre élevé tout en évitant les oscillations indésirables en adoptant des approximations d’ordre réduit à proximité des singularités<br>The main objective of the present thesis is to devise, construct and validate computationally efficient hp-adaptive discontinuous Galerkin schemes of the Navier-Stokes equations by bringing together the flexibility of a posteriori error driven adaptation and the accuracy of multiresolution-based adaptation. The performance of the hp-algorithm is illustrated by several steady flows in one and two dimensions.The first research direction employs a new multiwavelet-based methodology to estimate the discretization error of the numerical solution in the context of h-adaptive simulations. The results certainly demonstrate the viability of h-refinement to reach a significant computational gain with respect to uniformly refined grids. The second line of investigation addresses the analysis and development of a new hp-adaptive strategy based on the decay of the multiwavelet spectrum to drive hp-adaptive simulations. The strategy successfully discriminates between regions characterized by high regularity and discontinuous phenomena and their vicinity. Remarkably, the developed hp-adaptation algorithm is able to achieve the high accuracy characteristic of high-order numerical solutions while avoiding unwanted oscillations by adopting low-order approximations in the proximity of singularities

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.<br>Includes bibliographical references (p. 130-134).<br>In this thesis, we develop wavelet surface wavelet representations for complex surfaces, with the goal of demonstrating their potential for 3D scientific and engineering computing applications. Surface wavelets were originally developed for representing geometric objects in a multiresolution format in computer graphics. However, we further extend the construction of surface wavelets and prove the existence of a large class of multiwavelets in Rn with vanishing moments around corners that are well suited for complex geometries. These wavelets share all of the major advantages of conventional wavelets, in that they provide an analysis tool for studying data, functions and operators at different scales. However, unlike conventional wavelets, which are restricted to uniform grids, surface wavelets have the power to perform signal processing operations on complex meshes, such as those encountered in finite element modeling. This motivates the study of surface wavelets as an efficient representation for the modeling and simulation of physical processes. We show how surface wavelets can be applied to partial differential equations, cast in the integral form. We analyze and implement the wavelet approach for a model 3D potential problem using a surface wavelet basis with linear interpolating properties.<br>(cont.) We show both theoretically and experimentally that an O(h2/n) convergence rate, hn being the mesh size, can be obtained by retaining only O((logN)7/2 N) entries in the discrete operator matrix, where N is the number of unknowns. Moreover our theoretical proof of accuracy vs compression is applicable to a large class of Calderón-Zygmund integral operators. In principle, this convergence analysis may be extended to higher order wavelets with greater vanishing moment. This results in higher convergence and greater compression.<br>by Julio E. Castrillón Candás.<br>Ph.D.

The success of any transform coding technique depends on how well the basis functions represent the signal features. The discrete wavelet transform (DWT) performs a multiresolution analysis of a signal; this enables an efficient representation of smooth and detailed signal regions. Furthermore, computationally efficient algorithms exist for computing the DWT. For these reasons, recent image compression standards such as JPEG2000 use the wavelet transform. It is well known that orthogonality and symmetry are desirable transform properties in image compression applications. It is also known that the scalar wavelet transform does not possess both properties simultaneously. Multiwavelets overcome this limitation; the multiwavelet transform allows orthogonality and symmetry to co-exist. However recently reported image compression results indicate that the scalar wavelets still outperform the multiwavelets in terms of peak signal-to-noise ratio (PSNR). In a multiwavelet transform, the balancing order of the multiwavelet is indicative of its energy compaction efficiency (usually a higher balancing order implies lower mean-squared-error, MSE, in the compressed image). But a high balancing order alone does not ensure good image compression performance. Filter bank characteristics such as shift-variance, magnitude response, symmetry and phase response are important factors that also influence the MSE and perceived image quality. This thesis analyzes the impact of these multiwavelet characteristics on image compression performance. Our analysis allows us to explain---for the first time---reasons for the small performance gap between the scalar wavelets and multiwavelets. We study the characteristics of five balanced multiwavelets (and 2 unbalanced multiwavelets) and compare their image compression performance for grayscale images with the popular (9,7)-tap and (22,14)-tap biorthogonal scalar wavelets. We use the well-known SPIHT quantizer in our compression scheme and utilize PSNR and subjective quality measures to assess performance. We also study the effect of incorporating a human visual system (HVS)-based transform model in our multiwavelet compression scheme. Our results indicate those multiwavelet properties that are most important to image compression. Moreover, the PSNR and subjective quality results depict similar performance for the best scalar wavelets and multiwavelets. Our analysis also shows that the HVS-based multiwavelet transform coder considerably improves perceived image quality at low bit rates.<br>Master of Science