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39
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 443 (7 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 385 (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...
On compactly supported splinewavelets and a duality principle
 Trans. Amer. Soc
, 1992
"... Abstract. Let • • • C K _ ] c Vq c Vx c • • • be a multiresolution analysis of L2 generated by the mth order 5spline Nm{x). In this paper, we exhibit a compactly supported basic wavelet i//m(x) that generates the corresponding orthogonal complementary wavelet subspaces..., W _ \, Wo, Wx,.... Co ..."
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Cited by 108 (6 self)
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Abstract. Let • • • C K _ ] c Vq c Vx c • • • be a multiresolution analysis of L2 generated by the mth order 5spline Nm{x). In this paper, we exhibit a compactly supported basic wavelet i//m(x) that generates the corresponding orthogonal complementary wavelet subspaces..., W _ \, Wo, Wx,.... Consequently, the two finite sequences that describe the twoscale relations of Nm(x) and i//m(x) in terms of Nm(2x j),;6Z, yield an efficient reconstruction algorithm. To give an efficient wavelet decomposition algorithm based on these two finite sequences, we derive a duality principle, which also happens to yield the dual bases {Nm(x j)} and {y/m(x j)} , relative to {Nm(x j)} and {y/m(x j)}, respectively. 1.
Wavelets of multiplicity r
 Trans. Amer. Math. Soc
, 1994
"... Abstract. A multiresolution approximation (Km)m€Z of L2(R) is of multiplicity r> 0 if there are r functions <¡>x,..., <j>r whose translates form a Riesz basis for V $. In the general theory we derive necessary and sufficient conditions for the translates of <j>x,..., <j>r, ..."
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Cited by 77 (7 self)
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Abstract. A multiresolution approximation (Km)m€Z of L2(R) is of multiplicity r> 0 if there are r functions <¡>x,..., <j>r whose translates form a Riesz basis for V $. In the general theory we derive necessary and sufficient conditions for the translates of <j>x,..., <j>r, y/x,..., y/r to form a Riesz basis for V \. The resulting reconstruction and decomposition sequences lead to the construction of dual bases for V0 and its orthogonal complement W0 in Vx. The general theory is applied in the construction of spline wavelets with multiple knots. Algorithms for the construction of these wavelets for some special cases are given. Let r be a positive integer and 1.
Compression Domain Volume Rendering for Distributed Environments
, 1998
"... This paper describes a method for volume data compression and rendering which bases on wavelet splats. The underlying concept is especially designed for distributed and networked applications, where we assume a remote server to maintain large scale volume data sets, being inspected, browsed throug ..."
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Cited by 43 (4 self)
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This paper describes a method for volume data compression and rendering which bases on wavelet splats. The underlying concept is especially designed for distributed and networked applications, where we assume a remote server to maintain large scale volume data sets, being inspected, browsed through and rendered interactively by a local client. Therefore, we encode the server`s volume data using a newly designed wavelet based volume compression method. A local client can render the volumes immediately from the compression domain by using wavelet footprints, a method proposed earlier. In addition, our setup features full progression, where the rendered image is refined progressively as data comes in. Furthermore, framerate constraints are considered by controlling the quality of the image both locally and globally depending on the current network bandwidth or computational capabilities of the client. As a very important aspect of our setup, the client does not need to provide st...
Intertwining Multiresolution Analyses and the Construction of Piecewise Polynomial Wavelets
, 1994
"... Let (Vp ) be a local multiresolution analysis (MRA) of L 2 (R) of multiplicity r 1, i.e., V0 is generated by r compactly supported scaling functions. If the scaling functions generate an orthogonal basis of V0 then (Vp) is called an orthogonal MRA. We prove that there exists an orthogonal local M ..."
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Cited by 26 (6 self)
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Let (Vp ) be a local multiresolution analysis (MRA) of L 2 (R) of multiplicity r 1, i.e., V0 is generated by r compactly supported scaling functions. If the scaling functions generate an orthogonal basis of V0 then (Vp) is called an orthogonal MRA. We prove that there exists an orthogonal local MRA (V 0 p ) of multiplicity r 0 such that Vq ae V 0 0 ae Vq+n for some integers q 0, n 1 and r 0 ? 1. In particular, this shows that compactly supported orthogonal polynomial spline wavelets and scaling functions (of mulitplicity r 0 ? 1) of arbitrary regularity exist and we give several such examples. 1 Introduction The starting point for most wavelet constructions is a single function OE 2 L 2 (R) called a scaling function whose integer translates form a Riesz basis for a closed linear subspace V 0 ae L 2 (R). If the scaling function is compactly supported and generates an orthogonal basis of V 0 , then the associated wavelet will also be compactly supported and generate...
Asymptotic error expansion of wavelet approximations of smooth functions II
, 1994
"... ongs to the space L p (R), 1 p ! 1, if kf(x)k p = `Z +1 \Gamma1 jf(x)j p dx '1=p ! 1 ; and to L 1 (R) if kf(x)k1 = sup x2R jf(x)j ! 1 : The inner product of two functions f(x) and g(x) of the Hilbert space L 2 (R) is defined as h f; g i = Z +1 \Gamma1 f(x) g(x) dx : Corre ..."
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Cited by 15 (1 self)
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ongs to the space L p (R), 1 p ! 1, if kf(x)k p = `Z +1 \Gamma1 jf(x)j p dx '1=p ! 1 ; and to L 1 (R) if kf(x)k1 = sup x2R jf(x)j ! 1 : The inner product of two functions f(x) and g(x) of the Hilbert space L 2 (R) is defined as h f; g i = Z +1 \Gamma1 f(x) g(x) dx : Corresponden
Complex Wavelet Bases, Steerability, and the MarrLike Pyramid
, 2008
"... Our aim in this paper is to tighten the link between wavelets, some classical imageprocessing operators, and David Marr’s theory of early vision. The cornerstone of our approach is a new complex wavelet basis that behaves like a smoothed version of the GradientLaplace operator. Starting from firs ..."
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Cited by 13 (7 self)
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Our aim in this paper is to tighten the link between wavelets, some classical imageprocessing operators, and David Marr’s theory of early vision. The cornerstone of our approach is a new complex wavelet basis that behaves like a smoothed version of the GradientLaplace operator. Starting from first principles, we show that a singlegenerator wavelet can be defined analytically and that it yields a semiorthogonal complex basis of, irrespective of the dilation matrix used. We also provide an efficient FFTbased filterbank implementation. We then propose a slightly redundant version of the transform that is nearly translationinvariant and that is optimized for better steerability (Gaussianlike smoothing kernel). We call it the Marrlike wavelet pyramid because it essentially replicates the processing steps in Marr’s theory of early vision. We use it to derive a primal wavelet sketch which is a compact description of the image by a multiscale, subsampled edge map. Finally, we provide an efficient iterative