Low Bit-rate Color Video Compression using Multiwavelets in Three Dimensions

In recent years, wavelet-based video compressions have become a major focus of research because of the advantages that it provides. More recently, a growing thrust of studies explored the use of multiple scaling functions and multiple wavelets with desirable properties in various fields, from image de-noising to compression. In term of data compression, multiple scaling functions and wavelets offer a greater flexibility in coefficient quantization at high compression ratio than a comparable single wavelet. The purpose of this research is to investigate the possible improvement of scalable wavelet-based color video compression at low bit-rates by using three-dimensional multiwavelets. The first part of this work included the development of the spatio-temporal decomposition process for multiwavelets and the implementation of an efficient 3-D SPIHT encoder/decoder as a common platform for performance evaluation of two well-known multiwavelet systems against a comparable single wavelet in low bitrate color video compression. The second part involved the development of a motion-compensated 3-D compression codec and a modified SPIHT algorithm designed specifically for this codec by incorporating an advantage in the design of 2D SPIHT into the 3D SPIHT coder. In an experiment that compared their performances, the 3D motion-compensated codec with unmodified 3D SPIHT had gains of 0.3dB to 4.88dB over regular 2D wavelet-based motion-compensated codec using 2D SPIHT in the coding of 19 endoscopy sequences at 1/40 compression ratio. The effectiveness of the modified SPIHT algorithm was verified by the results of a second experiment in which it was used to re-encode 4 of the 19 sequences with lowest performance gains and improved them by 0.5dB to 1.0dB. The last part of the investigation examined the effect of multiwavelet packet on 3-D video compression as well as the effects of coding multiwavelet packets based on the frequency order and energy content of individual subbands

The Application of Multi-Wavelet Theory in Deformation Monitoring Data Processing

ABSTRACT: With wavelet technology used more widely in deformation analysis, the paper will talk multi-wavelet (the second generation wavelet) theory used for deformation monitoring data analysis. The paper studies signal adopting different preprocessing method, makes a study of the selection problem in optima multi-wavelet preprocessing method. The deformation monitoring signal is disposed using different multi-wavelet which adopts optima preprocessing method, and the paper makes a comparison to the conventional odd wavelet. The result confirms: multi-wavelet is more superiority than conventional wavelet, which decreases RMSE, advances SNR, obtains higher analytic precision, conforms the validity and practicability in physical problem, and offers a new road for deformation monitoring signal process

Theory and applications of the multiwavelets for compression of boundary integral operators

In general the numerical solution of boundary integral equations leads to full coefficientmatrices. The discrete system can be solved in O(N2) operations by iterative solvers ofthe Conjugate Gradient type. Therefore, we are interested in fast methods such as fastmultipole and wavelets, that reduce the computational cost to O(N lnp N).In this thesis we are concerned with wavelet methods. They have proved to be veryefficient and effective basis functions due to the fact that the coefficients of a wavelet expansiondecay rapidly for a large class of functions. Due to the multiresolution propertyof wavelets they provide accurate local descriptions of functions efficiently. For examplein the presence of corners and edges, the functions can still be approximated with a linearcombination of just a few basis functions. Wavelets are attractive for the numericalsolution of integral equations because their vanishing moments property leads to operatorcompression. However, to obtain wavelets with compact support and high order of vanishingmoments, the length of the support increases as the order of the vanishingmomentsincreases. This causes difficulties with the practical use of wavelets particularly at edgesand corners. However, with multiwavelets, an increase in the order of vanishing momentsis obtained not by increasing the support but by increasing the number of mother wavelets.In chapter 2 we review the methods and techniques required for these reformulations,we also discuss how these boundary integral equations may be discretised by a boundaryelement method. In chapter 3, we discuss wavelet and multiwavelet bases. In chapter4, we consider two boundary element methods, namely, the standard and non-standardGalerkin methods with multiwavelet basis functions. For both methods compressionstrategies are developed which only require the computation of the significant matrix elements.We show that they are O(N logp N) such significant elements. In chapters 5 and6 we apply the standard and non-standard Galerkin methods to several test problems

Spatially adaptive multiwavelet representations on unstructured grids with applications to multidimensional computational modeling

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.Includes bibliographical references (p. 130-134).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.(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.by Julio E. Castrillón Candás.Ph.D

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Piecewise Linear Wavelet Collocation on Triangular Grids, Approximation of the Boundary Manifold and Quadrature

In this paper we consider a piecewise linear collocation method for the solution of a pseudo-differential equations of order r = 0,-1 over a closed and smooth boundary manifold. The trial space is the space of all continuous and piecewise linear functions defined over a uniform triangular grid and the collocation points are the grid points. For the wavelet basis in the trial space we choose the three-point hierarchical basis together with a slight modification near the boundary points of the global patches of parametrization. We choose three, four, and six term linear combinations of Dirac delta functionals as wavelet basis in the space of test functionals. Though not all wavelets have vanishing moments, we derive the usual compression results, i.e. we prove that, for N degrees of freedom, the fully populated stiffness matrix of N2 entries can be approximated by a sparse matrix with no more than O(N [log N]2.25) non-zero entries. The main topic of the present paper, however, is to show that the parametrization can be approximated by low order piecewise polynomial interpolation and that the integrals in the stiffness matrix can be computed by quadrature, where the quadrature rules are combinations of product integration applied to non analytic factors of the integrand and of high order Gau{\ss} rules applied to the analytic parts. The whole algorithm for the assembling of the matrix requires no more than O(N [log N]4.25) arithmetic operations, and the error of the collocation approximation, including the compression, the approximative parametrization, and the quadratures, is less than O(N-1[log N]2). Note that, in contrast to well-known algorithms by v.Petersdorff, Schwab, and Schneider, only a finite degree of smoothness is required

Multidimensional Wavelets and Computer Vision

This report deals with the construction and the mathematical analysis of multidimensional nonseparable wavelets and their efficient application in computer vision. In the first part, the fundamental principles and ideas of multidimensional wavelet filter design such as the question for the existence of good scaling matrices and sensible design criteria are presented and extended in various directions. Afterwards, the analytical properties of these wavelets are investigated in some detail. It will turn out that they are especially well-suited to represent (discretized) data as well as large classes of operators in a sparse form - a property that directly yields efficient numerical algorithms. The final part of this work is dedicated to the application of the developed methods to the typical computer vision problems of nonlinear image regularization and the computation of optical flow in image sequences. It is demonstrated how the wavelet framework leads to stable and reliable results for these problems of generally ill-posed nature. Furthermore, all the algorithms are of order O(n) leading to fast processing