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29
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 434 (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.
Wavelet Families Of Increasing Order In Arbitrary Dimensions
, 1997
"... . We build compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The resulting scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its ..."
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Cited by 46 (0 self)
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. We build compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The resulting scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its advantages: fast transform, inplace calculation, and integerto integer transforms. We show that two lifting steps suffice: predict and update. The predict step can be built using multivariate polynomial interpolation, while update is a multiple of the adjoint of predict. Submitted to IEEE Transactions on Image Processing Over the last decade several constructions of compactly supported wavelets have originated both from signal processing and mathematical analysis. In signal processing, critically sampled wavelet transforms are known as filter banks or subband transforms [32, 43, 54, 56]. In mathematical analysis, wavelets are defined as translates and dilates of one fixed function and ar...
Multiresolution signal decomposition schemes. Part 1: Linear and morphological pyramids
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2000
"... ..."
Adaptive polyphase subband decomposition structures for image compression
 IEEE Transactions on Image Processing
, 2000
"... Abstract—Subband decomposition techniques have been extensively used for data coding and analysis. In most filter banks, the goal is to obtain subsampled signals corresponding to different spectral regions of the original data. However, this approach leads to various artifacts in images having spati ..."
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Cited by 40 (11 self)
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Abstract—Subband decomposition techniques have been extensively used for data coding and analysis. In most filter banks, the goal is to obtain subsampled signals corresponding to different spectral regions of the original data. However, this approach leads to various artifacts in images having spatially varying characteristics, such as images containing text, subtitles, or sharp edges. In this paper, adaptive filter banks with perfect reconstruction property are presented for such images. The filters of the decomposition structure which can be either linear or nonlinear vary according to the nature of the signal. This leads to improved image compression ratios. Simulation examples are presented. Index Terms—Adaptive polyphase structures, adaptive subband decomposition, image coding, lifting structures. I.
Matrix Factorizations for Reversible Integer Mapping
, 2001
"... Reversible integer mapping is essential for lossless source coding by transformation. A general matrix factorization theory for reversible integer mapping of invertible linear transforms is developed in this paper. Concepts of the integer factor and the elementary reversible matrix (ERM) for integer ..."
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Cited by 22 (8 self)
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Reversible integer mapping is essential for lossless source coding by transformation. A general matrix factorization theory for reversible integer mapping of invertible linear transforms is developed in this paper. Concepts of the integer factor and the elementary reversible matrix (ERM) for integer mapping are introduced, and two forms of ERMtriangular ERM (TERM) and singlerow ERM (SERM)are studied. We prove that there exist some approaches to factorize a matrix into TERMs or SERMs if the transform is invertible and in a finitedimensional space. The advantages of the integer implementations of an invertible linear transform are i) mapping integers to integers, ii) perfect reconstruction, and iii) inplace calculation. We find that besides a possible permutation matrix, the TERM factorization of anbynonsingular matrix has at most three TERMs, and its SERM factorization has at most +1SERMs. The elementary structure of ERM transforms is the ladder structure. An executable factorization algorithm is also presented. Then, the computational complexity is compared, and some optimization approaches are proposed. The error bounds of the integer implementations are estimated as well. Finally, three ERM factorization examples of DFT, DCT, and DWT are given.
On Ladder Structures and Linear Phase Conditions for Multidimensional BiOrthogonal Filter Banks
 In Proceedings of ICASSP94
, 1993
"... In this paper some new results for multidimensional mband biorthogonal filter banks are presented. In the first part of the paper we introduce the ladder structure as a method for the design and implementation of aforementioned filter banks. The theory of ladders and the relation to filter banks i ..."
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Cited by 17 (1 self)
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In this paper some new results for multidimensional mband biorthogonal filter banks are presented. In the first part of the paper we introduce the ladder structure as a method for the design and implementation of aforementioned filter banks. The theory of ladders and the relation to filter banks is presented. The second part of the paper focuses upon linear phase conditions for biorthogonal filter banks. Finally it is shown how the linear phase requirements and the ladder structure can be merged into a single framework. Keywords Filter banks, biorthogonal, multirate systems, linear phase filters, cascade, ladder structure. EDICS NrFilter Bank Theory 2.4.2 Dr. Ton A.C.M. Kalker is with Philips Research Laboratories, P.O. Box 80000, 5600 JA, Eindhoven, The Netherlands, Tel. 3140744174, Fax 3140744675, email kalker@prl.philips.nl. M. Sc. Imran A. Shah is with Philips Research Laboratories, 345 Scarborough Road, Briarcliff Manor, USA. I . Introduction Multirate perfect reco...
37 other authors
 Science
, 1995
"... Symmetric extension is explored as a means for constructing nonexpansive reversible integertointeger (ITI) wavelet transforms for finitelength signals. Two families of reversible ITI wavelet transforms are introduced, and their constituent transforms are shown to be compatible with symmetric exte ..."
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Cited by 16 (1 self)
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Symmetric extension is explored as a means for constructing nonexpansive reversible integertointeger (ITI) wavelet transforms for finitelength signals. Two families of reversible ITI wavelet transforms are introduced, and their constituent transforms are shown to be compatible with symmetric extension. One of these families is then studied in detail, and several interesting results concerning its member transforms are presented. In addition, some new reversible ITI structures are derived that are useful in conjunction with techniques like symmetric extension. Lastly, the relationship between symmetric extension and perliftingstep extension is explored, and some new results are obtained in this regard.
Ladder structures for Multidimensional Linear Phase Perfect Reconstruction filter banks and Wavelets
, 1992
"... The design of multidimensional filter banks and wavelets have been areas of active research for use in video and image communication systems. At the same time efficient structures for the implementation of such filters are of importance. In 1D, the well known lattice structure and the recently intr ..."
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Cited by 13 (2 self)
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The design of multidimensional filter banks and wavelets have been areas of active research for use in video and image communication systems. At the same time efficient structures for the implementation of such filters are of importance. In 1D, the well known lattice structure and the recently introduced ladder structure [1] are attractive. However, their extensions to higher dimensions (mD) have been limited. In this paper we reintroduce the ladder structure, with the purpose of transforming the structure into mD using the McClellan transform. 1 Introduction Recently the ladder structure was introduced for the design and efficient implementation of 1D perfect reconstructing filter banks (PRFB) [1]. These structures have the advantage of not only being insensitive to coefficient quantization but, under fairly general conditions, also to quantization of intermediate results. In [6] it is shown that the class of 1D 2channel PRFB's which can be realized by a ladder structure coinci...
Multiplierless Approximation of Transforms with Adder Constraint
 IEEE SIGNAL PROCESSING LETTERS
, 2002
"... This letter describes an algorithm for systematically finding a multiplierless approximation of transforms by replacing floatingpoint multipliers with VLSIfriendly binary coefficients of the form k/2^n. Assuming the cost of hardware binary shifters is negligible, the total number of binary adders ..."
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Cited by 13 (3 self)
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This letter describes an algorithm for systematically finding a multiplierless approximation of transforms by replacing floatingpoint multipliers with VLSIfriendly binary coefficients of the form k/2^n. Assuming the cost of hardware binary shifters is negligible, the total number of binary adders employed to approximate the transform can be regarded as an index of complexity. Because the new algorithm is more systematic and faster than trialanderror binary approximations with adder constraint, it is a much more efficient design tool. Furthermore, the algorithm is not limited to a specific transform; various approximations of the discrete cosine transform are presented as examples of its versatility.