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The Lifting Scheme: A Construction Of Second Generation Wavelets
, 1997
"... . We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to ..."
Abstract

Cited by 377 (16 self)
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. We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, inplace calculation of the wavelet transform. Several examples are included. Key words. wavelet, multiresolution, second generation wavelet, lifting scheme AMS subject classifications. 42C15 1. Introduction. Wavelets form a versatile tool for representing general functions or data sets. Essentially we can think of them as data building blocks. Their fundamental property is that they allow for representations which are efficient and which can be computed fast. In other words, wavelets are capable of quickly capturing the essence of a data set with only a small set of coefficients. This is based on the fact that most data sets have correlation both in time (or space) and frequenc...
Building Your Own Wavelets at Home
"... Wavelets have been making an appearance in many pure and applied areas of science and engineering. Computer graphics with its many and varied computational problems has been no exception to this rule. In these notes we will attempt to motivate and explain the basic ideas behind wavelets and what mak ..."
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Cited by 127 (13 self)
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Wavelets have been making an appearance in many pure and applied areas of science and engineering. Computer graphics with its many and varied computational problems has been no exception to this rule. In these notes we will attempt to motivate and explain the basic ideas behind wavelets and what makes them so successful in application areas. The main
Regularity Of Irregular Subdivision
, 1998
"... . We study the smoothness of the limit function for one dimensional unequally spaced interpolating subdivision schemes. The new grid points introduced at every level can lie in irregularly spaced locations between old, adjacent grid points and not only midway as is usually the case. For the natural ..."
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Cited by 30 (5 self)
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. We study the smoothness of the limit function for one dimensional unequally spaced interpolating subdivision schemes. The new grid points introduced at every level can lie in irregularly spaced locations between old, adjacent grid points and not only midway as is usually the case. For the natural generalization of the four point scheme introduced by Dubuc and Dyn, Levin, and Gregory, we show that, under some geometric restrictions, the limit function is always C 1 ; under slightly stronger restrictions we show that the limit function is almost C 2 , the same regularity as in the regularly spaced case. 1. Introduction Subdivision is a powerful mechanism for the construction of smooth curves and surfaces. The main idea behind subdivision is to iterate upsampling and local averaging to build complex geometrical shapes. Originally such schemes were studied in the context of corner cutting [13, 5] as well as for building piecewise polynomial curves, e.g., the de Casteljau algorithm f...
Wavelets in Computer Graphics
 PROCEEDINGS OF THE IEEE, TO APPEAR
"... One of the perennial goals in computer graphics is realism in realtime. Handling geometrically complex scenes and physically faithful descriptions of their appearance and behavior, clashes with the requirement of multiple frame per second update rates. It is no surprise then that hierarchical modeli ..."
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Cited by 8 (1 self)
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One of the perennial goals in computer graphics is realism in realtime. Handling geometrically complex scenes and physically faithful descriptions of their appearance and behavior, clashes with the requirement of multiple frame per second update rates. It is no surprise then that hierarchical modeling and simulation have already enjoyed a long history in computer graphics. Most recently these ideas have received a significant boost as wavelet based algorithms have entered many areas in computer graphics. We give an overview of some of the areas in which wavelets have already had an impact on the state of the art.
Wavelet Sampling Techniques
 In 1993 Proceedings of the Statistical Computing Section
, 1993
"... In this paper we present several techniques to calculate the wavelet coefficients of a function from its samples. Interpolation, quadrature formulae and filtering methods are discussed and compared. 1 Introduction 1.1 Multiresolution analysis We will first briefly review wavelets and multiresolut ..."
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Cited by 7 (0 self)
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In this paper we present several techniques to calculate the wavelet coefficients of a function from its samples. Interpolation, quadrature formulae and filtering methods are discussed and compared. 1 Introduction 1.1 Multiresolution analysis We will first briefly review wavelets and multiresolution analysis. For more detailed treatments, one can consult [9, 15, 24, 26, 28]. A multiresolution analysis of L 2 (IR) is defined as a set of closed subspaces V j with j 2 ZZ that exhibit the following properties: 1. V j ae V j+1 , 2. v(x) 2 V j , v(2x) 2 V j+1 , 3. v(x) 2 V 0 , v(x + 1) 2 V 0 , 4. +1 [ j=\Gamma1 V j is dense in L 2 (IR), 5. +1 " j=\Gamma1 V j = f0g, To appear in the proceedings of the Joint Statistical Meetings, San Francisco, August 1993. y Research Assistant of the National Fund of Scientific Research Belgium and partially supported by ONR Grant N0001490J1343. 6. A scaling function '(x) 2 V 0 exists such that the set f'(x \Gamma l) j l 2 ZZg is a Riesz b...
Wavelet representation of loweratmospheric long nonlinear wave dynamics, governed by the BenjaminDavisOnoBurgers equation
, 1995
"... A modified technique is presented for projecting a large class of nonlinear partial differential equations with respect to (x; t) onto a finite number of ordinary differential equations with respect to t. Improved description compared to standard finitedifference or Fourier spectral methods involve ..."
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Cited by 4 (2 self)
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A modified technique is presented for projecting a large class of nonlinear partial differential equations with respect to (x; t) onto a finite number of ordinary differential equations with respect to t. Improved description compared to standard finitedifference or Fourier spectral methods involves using an orthonormal basis of wavelet functions / ;n (x). Whereas Fourier projection represents the interaction between spatial scales throughout the xdomain, wavelet representation does the same locally. This technique is applied to solving the BDOBurgers equation, extending previous results 1,2,5,6 for the Burgers equation. Keywords: wavelets, wavelet transform of PDE to ODE, Burgers equation, BenjaminDavisOno equation, solitons, Galerkin method, Hilbert transform, dispersive waves, atmospheric dynamics, Lyapunov exponents 1 INTRODUCTION A stably stratified lower atmosphere can act as a waveguide for long, finiteamplitude internal gravity waves. Generally, these waves transport...
AverageInterpolating Wavelet Bases on Irregular Meshes on the Interval
, 2000
"... this paper and to avoid tenorrealization of basis functions, we restate the basics of nonstationary subdivision in 2 and L2 ..."
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Cited by 2 (0 self)
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this paper and to avoid tenorrealization of basis functions, we restate the basics of nonstationary subdivision in 2 and L2