Wavelets For Computer Graphics

This introduction to wavelets provides computer graphics professionals and researchers with the mathematical foundations for understanding and applying this powerful tool.

From Design Methods and Generation Schemes to State-of-the-Art Applications Wavelets are powerful tools for functional analysis and geometry processing, enabling researchers to determine the structure of data and analyze 3D shapes. Suitable for researchers in computer graphics, computer vision, visualization, medical imaging, and geometric modeling as well as graduate and senior undergraduate students in computer science, Diffusion-Driven Wavelet Design for Shape Analysis presents recent research results in wavelet designs on 3D shapes and their applications in shape analysis. It explains how to apply the design methods to various types of 3D data, such as polygonal meshes, point clouds, manifolds, and volumetric images. Extensions of Wavelet Generation on Volumetric and Manifold Data The first part of the book introduces design methods of wavelets on manifold data, incorporating interdisciplinary knowledge from differential geometry, functional analysis, Fourier transform, spectral graph theory, and stochastic processes. The authors show how wavelets are purely determined by the shape geometry and how wavelet transforms are computed as inner products of wavelet kernels and input functions. Wavelets for Solving Computer Graphics Problems The second part presents applications in shape analysis/representation. The book looks at wavelets as spectral tools for geometry processing with filters in a joint space-frequency domain and examines wavelets as detail extractors for shape feature definition and detection. Going beyond these fundamental applications, the book also covers middle- and high-level applications, including shape matching, shape registration, and shape retrieval. Easy-to-Understand Implementations and Algorithms Unlike many other wavelet books, this one does not involve complicated mathematics. Instead, the book uses simplified formulations and illustrative examples to explain deep theories. Code and other materials are available on a supplementary website.

The subject of wavelet analysis has been around for less than 15 years, yet it seems to be popping up everywhere. The major reason for this is its usefulness in applications. Wavelets reduce the data requirements in fields such as signal processing and approximation through their localization property: A function that is zero in an interval has a wavelet expansion whose partial sums are small or even zero. Since most of the information in a signal is in the region where it changes, only the local data corresponding to it needs to be considered. Wavelets do this much better than classical methods because of their “zoom-in” and “zoom-out” capabilities. Hence, wavelets appeal to scientists and engineers in any field that requires analyzing large amounts of data.This is an introductory, self-contained book on wavelet analysis and its applications, focusing on computer graphics and economics. The only prerequisite is a basic knowledge of calculus and linear algebra. For this reason, preliminary material on functional analysis, function theory, and Fourier analysis are covered in Chapter 1. The driving force of the development of wavelet analysis is described in Chapter 2 using step functions and Haar wavelets. Integral wavelet transform, discrete wavelet transform, classification of wavelets, multiresolution analysis, and wavelet algorithms for the decomposition and reconstruction of functions are presented in Chapter 3 of the book. The applications of wavelet transform in computer graphics and in economics are discussed in Chapters 4 and 5, respectively. Finally, Chapter 6 discusses some advanced material such as biorthogonal wavelets, two-dimensional wavelets, nonseparable multidimensional wavelets, and wavelet packets.

Second Generation Wavelets and Applications introduces "second generation wavelets" and the lifting transform that can be used to apply the traditional benefits of wavelets into a wide range of new areas in signal processing, data processing and computer graphics. This book details the mathematical fundamentals of the lifting transform and illustrates the latest applications of the transform in signal and image processing, numerical analysis, scattering data smoothing and rendering of computer images.

This book combining wavelets and the world of the spectrum focuses on recent developments in wavelet theory, emphasizing fundamental and relatively timeless techniques that have a geometric and spectral-theoretic flavor. The exposition is clearly motivated and unfolds systematically, aided by numerous graphics. This self-contained book deals with important applications to signal processing, communications engineering, computer graphics algorithms, qubit algorithms and chaos theory, and is aimed at a broad readership of graduate students, practitioners, and researchers in applied mathematics and engineering. The book is also useful for other mathematicians with an interest in the interface between mathematics and communication theory.

Overview Historically, the concept of "ondelettes" or "wavelets" originated from the study of time-frequency signal analysis, wave propagation, and sampling theory. One of the main reasons for the discovery of wavelets and wavelet transforms is that the Fourier transform analysis does not contain the local information of signals. So the Fourier transform cannot be used for analyzing signals in a joint time and frequency domain. In 1982, Jean MorIet, in collaboration with a group of French engineers, first introduced the idea of wavelets as a family of functions constructed by using translation and dilation of a single function, called the mother wavelet, for the analysis of nonstationary signals. However, this new concept can be viewed as the synthesis of various ideas originating from different disciplines including mathematics (Calder6n-Zygmund operators and Littlewood-Paley theory), physics (coherent states in quantum mechanics and the renormalization group), and engineering (quadratic mirror filters, sideband coding in signal processing, and pyramidal algorithms in image processing). Wavelet analysis is an exciting new method for solving difficult problems in mathematics, physics, and engineering, with modern applications as diverse as wave propagation, data compression, image processing, pattern recognition, computer graphics, the detection of aircraft and submarines, and improvement in CAT scans and other medical image technology. Wavelets allow complex information such as music, speech, images, and patterns to be decomposed into elementary forms, called the fundamental building blocks, at different positions and scales and subsequently reconstructed with high precision.

Pacific Graphics is an international conference on computer graphics and applications. The conference will provide a forum for researchers, developers, and practitioners to exchange ideas and discuss future directions of computer graphics. The past two conferences were held in Korea (1993) and China (1994), and future conferences are planned in Taiwan (1996), Korea (1997), and Singapore (1998).