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46
Factoring wavelet transforms into lifting steps
 J. Fourier Anal. Appl
, 1998
"... ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This dec ..."
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Cited by 547 (8 self)
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ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is wellknown to algebraists (and expressed by the formula); it is also used in linear systems theory in the electrical engineering community. We present here a selfcontained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e, nonunitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a waveletlike transform that maps integers to integers. 1.
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 ..."
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Cited by 513 (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...
The easy path wavelet transform: A new adaptive wavelet transform for sparse representation of twodimensional data
 Multiscale Model. Simul
"... Dedicated to Manfred Tasche on the occasion of his 65th birthday We introduce a new locally adaptive wavelet transform, called Easy Path Wavelet Transform (EPWT), that works along pathways through the array of function values and exploits the local correlations of the data in a simple appropriate ma ..."
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Cited by 136 (9 self)
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Dedicated to Manfred Tasche on the occasion of his 65th birthday We introduce a new locally adaptive wavelet transform, called Easy Path Wavelet Transform (EPWT), that works along pathways through the array of function values and exploits the local correlations of the data in a simple appropriate manner. The usual discrete orthogonal and biorthogonal wavelet transform can be formulated in this approach. The EPWT can be incorporated into a multiresolution analysis structure and generates data dependent scaling spaces and wavelet spaces. Numerical results show the enormous efficiency of the EPWT for representation of twodimensional data. Key words. wavelet transform along pathways, data compression, adaptive wavelet bases, directed wavelets AMS Subject classifications. 65T60, 42C40, 68U10, 94A08 1
Optimally Sparse Image Representation by the Easy Path Wavelet Transform
"... The Easy Path Wavelet Transform (EPWT) [19] has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of f ..."
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Cited by 114 (8 self)
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The Easy Path Wavelet Transform (EPWT) [19] has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and it exploits the local correlations of the given data in a simple appropriate manner. In this paper, we show that the EPWT leads, for a suitable choice of the pathways, to optimal Nterm approximations for piecewise Hölder continuous functions with singularities along curves.
Optimal Representation of Piecewise Hölder Smooth Bivariate Functions by the Easy Path Wavelet Transform
"... The Easy Path Wavelet Transform (EPWT) [22] has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of f ..."
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Cited by 67 (3 self)
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The Easy Path Wavelet Transform (EPWT) [22] has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and it exploits the local correlations of the given data in a simple appropriate manner. Using polyharmonic spline interpolation, we show in this paper that the EPWT leads, for a suitable choice of the pathways, to optimal Nterm approximations for piecewise Hölder smooth functions with singularities along curves. Key words. sparse data representation, wavelet transform along pathways, Nterm approximation AMS Subject classifications. 41A25, 42C40, 68U10, 94A08 1
Edgeavoiding wavelets and their applications
 In Proc. ACM SIGGRAPH
"... Figure 1: Two views of the graph of the same edgeavoiding wavelet centered at the shoulder of the Cameraman. The support of the wavelet is confined within the limits set by the strong edges around the upper body. We propose a new family of secondgeneration wavelets constructed using a robust data ..."
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Cited by 44 (2 self)
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Figure 1: Two views of the graph of the same edgeavoiding wavelet centered at the shoulder of the Cameraman. The support of the wavelet is confined within the limits set by the strong edges around the upper body. We propose a new family of secondgeneration wavelets constructed using a robust dataprediction lifting scheme. The support of these new wavelets is constructed based on the edge content of the image and avoids having pixels from both sides of an edge. Multiresolution analysis, based on these new edgeavoiding wavelets, shows a better decorrelation of the data compared to common linear translationinvariant multiresolution analyses. The reduced interscale correlation allows us to avoid halo artifacts in bandindependent multiscale processing without taking any special precautions. We thus achieve nonlinear datadependent multiscale edgepreserving image filtering and processing at computation times which are linear in the number of image pixels. The new wavelets encode, in their shape, the smoothness information of the image at every scale. We use this to derive a new edgeaware interpolation scheme that achieves results, previously computed by solving an inhomogeneous Laplace equation, through an explicit computation. We thus avoid the difficulties in solving large and poorlyconditioned systems of equations. We demonstrate the effectiveness of the new wavelet basis for various computational photography applications such as multiscale dynamicrange compression, edgepreserving smoothing and detail enhancement, and image colorization.
Multiresolution Based On Weighted Averages Of The Hat Function I: Linear Reconstruction Techniques
, 1993
"... . We study the properties of the multiresolution analysis corresponding to discretization by local averages with respect to the hat function. We consider a class of reconstruction procedures which are appropriate for this multiresolution setting and describe the associated prediction operators that ..."
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Cited by 19 (4 self)
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. We study the properties of the multiresolution analysis corresponding to discretization by local averages with respect to the hat function. We consider a class of reconstruction procedures which are appropriate for this multiresolution setting and describe the associated prediction operators that allow us to climb up the ladder from coarse to finer levels of resolution. Only dataindependent (i.e. linear) reconstruction operators are considered in Part I. Linear reconstruction techniques allow us, under certain circumstances, to construct a basis of generalized wavelets for the multiresolution representation of the original data. The stability of the associated multiresolution schemes is analyzed using the general framework developed by A. Harten in [18] and the connection with the theory of recursive subdivision. Key Words. Multiscale decomposition, discretization, reconstruction. AMS(MOS) subject classifications. 41A05, 41A15, 65015 Departament de Matem`atica Aplicada. Universi...
Interpolatory wavelets for manifoldvalued data
, 2009
"... Geometric waveletlike transforms for univariate and multivariate manifoldvalued data can be constructed by means of nonlinear stationary subdivision rules which are intrinsic to the geometry under consideration. We show that in an appropriate vector bundle setting for a general class of interpolat ..."
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Cited by 14 (6 self)
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Geometric waveletlike transforms for univariate and multivariate manifoldvalued data can be constructed by means of nonlinear stationary subdivision rules which are intrinsic to the geometry under consideration. We show that in an appropriate vector bundle setting for a general class of interpolatory wavelet transforms, which applies to Riemannian geometry, Lie groups and other geometries, Hölder smoothness of functions is characterized by decay rates of their wavelet coefficients.
Adaptive Sparse Grids for Hyperbolic Conservation Laws
, 1999
"... . We report on numerical experiments using adaptive sparse grid discretization techniques for the numerical solution of scalar hyperbolic conservation laws. Sparse grids are an efficient approximation method for functions. Compared to regular, uniform grids of a mesh parameter h contain h \Gammad ..."
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Cited by 12 (4 self)
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. We report on numerical experiments using adaptive sparse grid discretization techniques for the numerical solution of scalar hyperbolic conservation laws. Sparse grids are an efficient approximation method for functions. Compared to regular, uniform grids of a mesh parameter h contain h \Gammad points in d dimensions, sparse grids require only h \Gamma1 jloghj d\Gamma1 points due to a truncated, tensorproduct multiscale basis representation. For the treatment of conservation laws two different approaches are taken: First an explicit timestepping scheme based on central differences is introduced. Sparse grids provide the representation of the solution at each time step and reduce the number of unknowns. Further reductions can be achieved with adaptive grid refinement and coarsening in space. Second, an upwind type sparse grid discretization in d + 1 dimensional spacetime is constructed. The problem is discretized both in space and in time, storing the solution at all time st...