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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...
Wavelet and Multiscale Methods for Operator Equations
 Acta Numerica
, 1997
"... this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of th ..."
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Cited by 172 (40 self)
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this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of these requirements as well as for their realization. This is also particularly important for understanding the severe obstructions, that keep us at present from readily materializing all the principally promising perspectives.
Wavelet Radiosity
, 1993
"... Radiosity methods have been shown to be an effective means to solve the global illumination problem in Lambertian diffuse environments. These methods approximate the radiosity integral equation by projecting the unknown radiosity function into a set of basis functions with limited support resulting ..."
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Cited by 149 (10 self)
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Radiosity methods have been shown to be an effective means to solve the global illumination problem in Lambertian diffuse environments. These methods approximate the radiosity integral equation by projecting the unknown radiosity function into a set of basis functions with limited support resulting in a set of n linear equations where n is the number of discrete elements in the scene. Classical radiosity methods required the evaluation of n 2 interaction coefficients. Efforts to reduce the number of required coefficients without compromising error bounds have focused on raising the order of the basis functions, meshing, accounting for discontinuities, and on developing hierarchical approaches, which have been shown to reduce the required interactions to O(n). In this paper we show that the hierarchical radiosity formulation is an instance of a more general set of methods based on wavelet theory. This general framework offers a unified view of both higher order element approaches to...
CHARMS: A Simple Framework for Adaptive Simulation
 ACM Transactions on Graphics
, 2002
"... Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, bui ..."
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Cited by 123 (9 self)
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Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, building adaptive solvers can be a daunting task especially for 3D finite elements. In this paper we are introducing a new approach to produce conforming, hierarchical, adaptive refinement methods (CHARMS). The basic principle of our approach is to refine basis functions, not elements. This removes a number of implementation headaches associated with other approaches and is a general technique independent of domain dimension (here 2D and 3D), element type (e.g., triangle, quad, tetrahedron, hexahedron), and basis function order (piecewise linear, higher order Bsplines, Loop subdivision, etc.). The (un)refinement algorithms are simple and require little in terms of data structure support. We demonstrate the versatility of our new approach through 2D and 3D examples, including medical applications and thinshell animations.
The application of multiwavelet filter banks to image processing
 IEEE Trans. Image Process
, 1999
"... Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrixvalued filter banks leading to wavelet bases, multiwavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar 2channel wavelet systems. After reviewing this recently d ..."
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Cited by 59 (5 self)
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Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrixvalued filter banks leading to wavelet bases, multiwavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar 2channel wavelet systems. After reviewing this recently developed theory, we examine the use of multiwavelets in a filter bank setting for discretetime signal and image processing. Multiwavelets differ from scalar wavelet systems in requiring two or more input streams to the multiwavelet filter bank. We describe two methods (repeated row and approximation/deapproximation) for obtaining such a vector input stream from a onedimensional signal. Algorithms for symmetric extension of signals at boundaries are then developed, and naturally integrated with approximationbased preprocessing. We describe an additional algorithm for multiwavelet processing of twodimensional signals, two rows at a time, and develop a new family of multiwavelets (the constrained pairs) that is wellsuited to this approach. This suite of novel techniques is then applied to two basic signal processing problems, denoising via waveletshrinkage, and data compression. After developing the approach via model problems in one dimension, we applied multiwavelet processing to images, frequently obtaining performance superior to the comparable scalar wavelet transform.
Adaptive Solution Of Partial Differential Equations In Multiwavelet Bases
 J. Comput. Phys
, 1999
"... . Representations of derivative and exponential operators, with linear boundary conditions, are constructed in multiwavelet bases, leading to a simple adaptive scheme for the solution of nonlinear timedependent partial differential equations. The emphasis on hierarchical representations of function ..."
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Cited by 19 (3 self)
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. Representations of derivative and exponential operators, with linear boundary conditions, are constructed in multiwavelet bases, leading to a simple adaptive scheme for the solution of nonlinear timedependent partial differential equations. The emphasis on hierarchical representations of functions on intervals helps to address issues both of highorder approximation and of efficient application of integral operators, and the lack of regularity of multiwavelets does not preclude their use in representing differential operators. Comparisons with finite difference, finite element, and spectral element methods are presented, as are numerical examples with the heat equation and Burgers' equation. Key words. adaptive techniques, Burgers' equation, exact linear part, highorder approximation, integrodifferential operators, Legendre polynomials, Runge phenomenon AMS subject classifications. 65D15, 65M60, 65M70, 65N30, 65N35 1. Introduction. In this paper we construct representations of oper...
Orthogonal bandlet bases for geometric images approximation
 Com. Pure and Applied Mathematics
, 2006
"... This paper introduces orthogonal bandlet bases to approximate images having some geometrical regularity. These bandlet bases are computed by applying parameterized Alpert transform operators over an orthogonal wavelet basis. These bandletization operators depend upon a multiscale geometric flow that ..."
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Cited by 18 (8 self)
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This paper introduces orthogonal bandlet bases to approximate images having some geometrical regularity. These bandlet bases are computed by applying parameterized Alpert transform operators over an orthogonal wavelet basis. These bandletization operators depend upon a multiscale geometric flow that is adapted to the image, at each wavelet scale. This bandlet construction has a hierarchical structure over wavelet coefficients taking advantage of existing regularity among these coefficients. It is proved that C α images having singularities along C α curves are approximated in a best orthogonal bandlet basis with an optimal asymptotic error decay. Fast algorithms and compression applications are described. c ○ 2000 Wiley Periodicals, Inc. I