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139
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 480 (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 427 (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 Lifting Scheme: A New Philosophy in Biorthogonal Wavelet Constructions
 in Wavelet Applications in Signal and Image Processing III
, 1995
"... In this paper we present the basic idea behind the lifting scheme, a new construction of biorthogonal wavelets which does not use the Fourier transform. In contrast with earlier papers we introduce lifting purely from a wavelet transform point of view and only consider the wavelet basis functions in ..."
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Cited by 167 (0 self)
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In this paper we present the basic idea behind the lifting scheme, a new construction of biorthogonal wavelets which does not use the Fourier transform. In contrast with earlier papers we introduce lifting purely from a wavelet transform point of view and only consider the wavelet basis functions in a later stage. We show how lifting leads to a faster, fully inplace implementation of the wavelet transform. Moreover, it can be used in the construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one function. A typical example of the latter are wavelets on the sphere. Keywords: wavelet, biorthogonal, inplace calculation, lifting 1 Introduction At the present day it has become virtually impossible to give the definition of a "wavelet". The research field is growing so fast and novel contributions are made at such a rate that even if one manages to give a definition today, it might be obsolete tomorrow. One, very vague, way of thinking about...
Nonlinear wavelet transforms for image coding via lifting
, 2003
"... We investigate central issues such as invertibility, stability, synchronization, and frequency characteristics for nonlinear wavelet transforms built using the lifting framework. The nonlinearity comes from adaptively choosing between a class of linear predictors within the lifting framework. We al ..."
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Cited by 94 (3 self)
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We investigate central issues such as invertibility, stability, synchronization, and frequency characteristics for nonlinear wavelet transforms built using the lifting framework. The nonlinearity comes from adaptively choosing between a class of linear predictors within the lifting framework. We also describe how earlier families of nonlinear filter banks can be extended through the use of prediction functions operating on a causal neighborhood of pixels. Preliminary compression results for model and realworld images demonstrate the promise of our techniques.
ThreeDimensional Embedded Subband Coding with Optimized Truncation (3D ESCOT)
 3D ESCOT)”, Applied and Computational Harmonic Analysis10
, 2001
"... This paper presents an efficient video coding algorithm: Threedimensional embedded subband coding with optimized truncation (3D ESCOT), in which coefficients in different subbands are independently coded using fractional bitplane coding and candidate truncation points are formed at the end of ..."
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Cited by 53 (20 self)
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This paper presents an efficient video coding algorithm: Threedimensional embedded subband coding with optimized truncation (3D ESCOT), in which coefficients in different subbands are independently coded using fractional bitplane coding and candidate truncation points are formed at the end of each fractional bitplane. A ratedistortion optimized truncation scheme is used to multiplex all subband bitstreams together into a layered one. A novel motion threading technique is proposed to form threads along the motion trajectories in a scene. For efficient coding of motion threads, memoryconstrained temporal wavelet transforms are applied along entire motion threads. Blockbased motion threading is implemented in conjunction with 3D ESCOT in a real video coder. Extension of 3D ESCOT to objectbased coding is also addressed. Experiments demonstrate that 3D ESCOT outperforms MPEG4 for most test sequences at the same bit rate. # 2001 Academic Press 1.
A Multiresolution Framework for Variational Subdivision
, 1998
"... Subdivision is a powerful paradigm for the generation of curves and surfaces. It is easy to implement, computationally efficient, and useful in a variety of applications because of its intimate connection with multiresolution analysis. An important task in computer graphics and geometric modeling is ..."
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Cited by 43 (0 self)
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Subdivision is a powerful paradigm for the generation of curves and surfaces. It is easy to implement, computationally efficient, and useful in a variety of applications because of its intimate connection with multiresolution analysis. An important task in computer graphics and geometric modeling is the construction of curves that interpolate a given set of points and minimize a fairness functional (variational design). In the context of subdivision, fairing leads to special schemes requiring the solution of a banded linear system at every subdivision step. We present several examples of such schemes including one that reproduces nonuniform interpolating cubic splines. Expressing the construction in terms of certain elementary operations we are able to embed variational subdivision in the lifting framework, a powerful technique to construct wavelet filter banks given a subdivision scheme. This allows us to extend the traditional lifting scheme for FIR filters to a certain class of IIR filters. Consequently we show how to build variationally optimal curves and associated, stable wavelets in a straightforward fashion. The algorithms to perform the corresponding decomposition and reconstruction transformations are easy to implement and efficient enough for interactive applications.
Wavelets on Irregular Point Sets
 Phil. Trans. R. Soc. Lond. A
, 1999
"... this article we review techniques for building and analyzing wavelets on irregular point sets in one and two dimensions. We discuss current results both on the practical and theoretical side. In particular we focus on subdivision schemes and commutation rules. Several examples are included. ..."
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Cited by 39 (0 self)
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this article we review techniques for building and analyzing wavelets on irregular point sets in one and two dimensions. We discuss current results both on the practical and theoretical side. In particular we focus on subdivision schemes and commutation rules. Several examples are included.
FeedbackBypass: A New Approach to Interactive Similarity Query Processing
, 2001
"... In recent years, several methods have been proposed for implementing interactive similarity queries on multimedia databases. Common to all these methods is the idea to exploit user feedback in order to progressively adjust the query parameters and to eventually converge to an "optimal" ..."
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Cited by 39 (16 self)
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In recent years, several methods have been proposed for implementing interactive similarity queries on multimedia databases. Common to all these methods is the idea to exploit user feedback in order to progressively adjust the query parameters and to eventually converge to an "optimal" parameter setting. However, all these methods also share the drawback to "forget" user preferences across multiple query sessions, thus requiring the feedback loop to be restarted for every new query, i.e. using default parameter values. Not only is this proceeding frustrating from the user's point of view but it also constitutes a significant waste of system resources. In this paper we present FeedbackBypass, a new approach to interactive similarity query processing. It complements the role of relevance feedback engines by storing and maintaining the query parameters determined with feedback loops over time, using a waveletbased data structure (the Simplex Tree). For each query, a favorable set of query parameters can be determined and used to either "bypass" the feedback loop completely for alreadyseen queries, or to start the search process from a nearoptimal configuration. FeedbackBypass can be combined well with all stateoftheart relevance feedback techniques working in highdimensional vector spaces. Its storage requirements scale linearly with the dimensionality of the query space, thus making even sophisticated query spaces amenable. Experimen Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large ...
Waveletbased image coding: An overview
 Applied and Computational Control, Signals, and Circuits
, 1998
"... ABSTRACT This paper presents an overview of waveletbased image coding. We develop the basics of image coding with a discussion of vector quantization. We motivate the use of transform coding in practical settings,and describe the properties of various decorrelating transforms. We motivate the use o ..."
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Cited by 38 (3 self)
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ABSTRACT This paper presents an overview of waveletbased image coding. We develop the basics of image coding with a discussion of vector quantization. We motivate the use of transform coding in practical settings,and describe the properties of various decorrelating transforms. We motivate the use of the wavelet transform in coding using ratedistortion considerations as well as approximationtheoretic considerations. Finally,we give an overview of current coders in the literature. 1
Multilevel Solvers For Unstructured Surface Meshes
 SIAM J. Sci. Comput
"... Parameterization of unstructured surface meshes is of fundamental importance in many applications of Digital Geometry Processing. Such parameterization approaches give rise to large and exceedingly illconditioned systems which are difficult or impossible to solve without the use of sophisticated mu ..."
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Cited by 36 (2 self)
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Parameterization of unstructured surface meshes is of fundamental importance in many applications of Digital Geometry Processing. Such parameterization approaches give rise to large and exceedingly illconditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner.