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39
Construction of Multiscaling Functions with Approximation and Symmetry
, 1998
"... . This paper presents a new and e#cient way to create multiscaling functions with given approximation order, regularity, symmetry, and short support. Previous techniques were operating in time domain and required the solution of large systems of nonlinear equations. By switching to the frequency dom ..."
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Cited by 40 (10 self)
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. This paper presents a new and e#cient way to create multiscaling functions with given approximation order, regularity, symmetry, and short support. Previous techniques were operating in time domain and required the solution of large systems of nonlinear equations. By switching to the frequency domain and employing the latest results of the multiwavelet theory we are able to elaborate a simple and e#cient method of construction of multiscaling functions. Our algorithm is based on a recently found factorization of the refinement mask through the twoscale similarity transform (TST). Theoretical results and new examples are presented. Key words. approximation order, symmetry, multiscaling functions, multiwavelets AMS subject classifications. 41A25, 42A38, 39B62 PII. S0036141096297182 1. Introduction. This paper discusses the construction of multiscaling functions which generate a multiresolution analysis (MRA) and lead to multiwavelets. A standard (scalar) MRA assumes that there is ...
New Image Compression Techniques Using Multiwavelets and Multiwavelet Packets
, 2001
"... Advances in wavelet transforms and quantization methods have produced algorithms capable of surpassing the existing image compression standards like the Joint Photographic Experts Group (JPEG) algorithm. For best performance in image compression, wavelet transforms require filters that combine a num ..."
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Cited by 27 (0 self)
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Advances in wavelet transforms and quantization methods have produced algorithms capable of surpassing the existing image compression standards like the Joint Photographic Experts Group (JPEG) algorithm. For best performance in image compression, wavelet transforms require filters that combine a number of desirable properties, such as orthogonality and symmetry. However, the design possibilities for wavelets are limited because they cannot simultaneously possess all of the desirable properties. The relatively new field of multiwavelets shows promise in obviating some of the limitations of wavelets. Multiwavelets offer more design options and are able to combine several desirable transform features. The few previously published results of multiwaveletbased image compression have mostly fallen short of the performance enjoyed by the current wavelet algorithms. This paper presents new multiwavelet transform and quantization methods and introduces multiwavelet packets. Extensive experimental results demonstrate that our techniques exhibit performance equal to, or in several cases superior to, the current wavelet filters.
Theory and Algorithms for NonUniform Spline Wavelets
, 2001
"... We investigate mutually orthogonal spline wavelet spaces on nonuniform partitions of a bounded interval, addressing the existence, uniqueness and construction of bases of minimally supported spline wavelets. The relevant algorithms for decomposition and reconstruction are considered as well as some ..."
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Cited by 18 (0 self)
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We investigate mutually orthogonal spline wavelet spaces on nonuniform partitions of a bounded interval, addressing the existence, uniqueness and construction of bases of minimally supported spline wavelets. The relevant algorithms for decomposition and reconstruction are considered as well as some stabilityrelated questions. In addition, we briefly review the bivariate case for tensor products and arbitrary triangulations. We conclude the paper with a discussion of some special cases.
Multiwavelets: Regularity, Orthogonality and Symmetry via Twoscale Similarity Transform
 Stud. Appl. Math
, 1996
"... An important object in wavelet theory is the scaling function OE(t), satisfying a dilation equation OE(t) = P C k OE(2t \Gamma k). Properties of a scaling function are closely related to the properties of the symbol or mask P (!) = P C k e \Gammai!k . The approximation order provided by OE(t) ..."
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Cited by 17 (7 self)
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An important object in wavelet theory is the scaling function OE(t), satisfying a dilation equation OE(t) = P C k OE(2t \Gamma k). Properties of a scaling function are closely related to the properties of the symbol or mask P (!) = P C k e \Gammai!k . The approximation order provided by OE(t) is the number of zeros of P (!) at ! = ß, or in other words the number of factors (1+e \Gammai! ) in P (!). In the case of multiwavelets P (!) becomes a matrix trigonometric polynomial. The factors (1+e \Gammai! ) are replaced by a matrix factorization of P (!), which defines the approximation order of the multiscaling function. This matrix factorization is based on the twoscale similarity transform (TST). In this paper we study properties of the TST and show how it is connected with the theory of multiwavelets. This approach leads us to new results on regularity, symmetry and orthogonality of multiscaling functions and opens an easy way to their construction. Key words: approximation...
Multiwavelet Construction via the Lifting Scheme
 in Wavelet Analysis and Multiresolution Methods, T.X. He, ed., Lecture Notes in Pure and Appl. Math
, 1999
"... Lifting provides a simple method for constructing biorthogonal wavelet bases. We generalize lifting to the case of multiwavelets, and in so doing provide useful intuition about the additional degrees of freedom made available in the construction of multiwavelets. We show that any compactly suppo ..."
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Cited by 9 (0 self)
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Lifting provides a simple method for constructing biorthogonal wavelet bases. We generalize lifting to the case of multiwavelets, and in so doing provide useful intuition about the additional degrees of freedom made available in the construction of multiwavelets. We show that any compactly supported multiwavelet transform can be decomposed into a sequence of lifting steps. Finally, we compare lifting to the twoscale similarity transform construction method. 1 Introduction The recent work of Geronimo et al [5] has generated considerable interest in multiwavelet constructions. In contrast to the scalar wavelet case, in which all basis functions are generated from translations and dilations of a single wavelet and scaling function, multiwavelet bases are constructed from translates and dilations of a vector of wavelets and scaling functions. Allowing multiple prototypes for the basis elements provides additional degrees of freedom that can be used to construct basis functions wit...
Lifting scheme for biorthogonal multiwavelets originated from Hermite spline
 IEEE Trans. Signal Proc
"... Abstract—We present new multiwavelet transforms of multiplicity 2 for manipulation of discretetime signals. The transforms are implemented in two phases: 1) Pre (post)processing, which transforms the scalar signal into a vector signal (and back) and 2) wavelet transforms of the vector signal. Both ..."
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Cited by 6 (2 self)
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Abstract—We present new multiwavelet transforms of multiplicity 2 for manipulation of discretetime signals. The transforms are implemented in two phases: 1) Pre (post)processing, which transforms the scalar signal into a vector signal (and back) and 2) wavelet transforms of the vector signal. Both phases are performed in a lifting manner. We use the cubic interpolatory Hermite splines as a predicting aggregate in the vector wavelet transform. We present new pre(post)processing algorithms that do not degrade the approximation accuracy of the vector wavelet transforms. We describe two types of vector wavelet transforms that are dual to each other but have similar properties and three pre(post)processing algorithms. As a result, we get fast biorthogonal algorithms to transform discretetime signals that are exact on sampled cubic polynomials. The bases for the transform are symmetric and have short support. Index Terms—Hermite spline, lifting scheme, multifilter, multiwavelet transform. I.
Applications of Multiwavelets to Image Compression
, 1999
"... Methods for digital image compression have been the subject of much study over the past decade. Advances in wavelet transforms and quantization methods have produced algorithms capable of surpassing the existing image compression standards like the Joint Photographic Experts Group (JPEG) algorithm. ..."
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Cited by 5 (1 self)
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Methods for digital image compression have been the subject of much study over the past decade. Advances in wavelet transforms and quantization methods have produced algorithms capable of surpassing the existing image compression standards like the Joint Photographic Experts Group (JPEG) algorithm. For best performance in image compression, wavelet transforms require filters that combine a number of desirable properties, such as orthogonality and symmetry. However, the design possibilities for wavelets are limited because they cannot simultaneously possess all of these desirable properties. The relatively new field of multiwavelets shows promise in removing some of the limitations of wavelets. Multiwavelets o#er more design options and hence can combine all desirable transform features. The few previously published results of multiwaveletbased image compression have mostly fallen short of the performance enjoyed by the current wavelet algorithms. This thesis presents new multiwavelet transform methods and measurements that verify the potential benefits of multiwavelets. Using a zerotree quantization scheme modified to better match the unique decomposition properties of multiwavelets, it is shown that the latest multiwavelet filters can give performance equal to, or in many cases superior to, the current wavelet filters. The performance of multiwavelet packets is also explored for the first time and is shown to be competitive to that of wavelet packets in some cases. The wavelet and multiwavelet filter banks are tested on a much wider range of images than in the usual literature, providing a better analysis of the benefits and drawbacks of each.
Raising Multiwavelet Approximation Order through Lifting
 SIAM J. Math. Anal. EXTENSION OF TWOSCALE SIMILARITY TRANSFORMS
, 1999
"... . Given a pair of biorthogonal, compactly supported multiwavelets, we present an algorithm for raising their approximation orders to any desired level, using one lifting step and one dual lifting step. Free parameters in the algorithm are explicitly identified, and can be used to optimize the result ..."
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Cited by 5 (1 self)
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. Given a pair of biorthogonal, compactly supported multiwavelets, we present an algorithm for raising their approximation orders to any desired level, using one lifting step and one dual lifting step. Free parameters in the algorithm are explicitly identified, and can be used to optimize the result with respect to other criteria. Key words. wavelets, multiwavelets, lifting, approximation order AMS subject classification. 42C15 1. Introduction. A refinable function vector of multiplicity r and dilation factor m is a vector OE (0) of r realvalued functions OE (0) (x) = 0 B B @ OE (0) 1 (x) . . . OE (0) r (x) 1 C C A ; x 2 R; (1.1) which satisfies a matrix refinement equation OE (0) (x) = p m X k2Z h (0) k OE (0) (mx \Gamma k): (1.2) The sequence H (0) = fh (0) k g k2Z of coefficient matrices is called the mask of the function. We assume that only finitely many h (0) k are nonzero, and that all OE (0) j have compact support. We call OE (0) a multis...
A Multivariate Thresholding Technique for Image Denoising Using Multiwavelets
 EURASIP JOURNAL ON APPLIED SIGNAL PROCESSING 2005:8, 1205–1211
, 2005
"... Multiwavelets, wavelets with several scaling functions, offer simultaneous orthogonality, symmetry, and short support, which is not possible with ordinary (scalar) wavelets. These properties make multiwavelets promising for signal processing applications, such as image denoising. The common approach ..."
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Cited by 4 (1 self)
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Multiwavelets, wavelets with several scaling functions, offer simultaneous orthogonality, symmetry, and short support, which is not possible with ordinary (scalar) wavelets. These properties make multiwavelets promising for signal processing applications, such as image denoising. The common approach for image denoising is to get the multiwavelet decomposition of a noisy image and apply a common threshold to each coefficient separately. This approach does not generally give sufficient performance. In this paper, we propose a multivariate thresholding technique for image denoising with multiwavelets. The proposed technique is based on the idea of restoring the spatial dependence of the pixels of the noisy image that has undergone a multiwavelet decomposition. Coefficients with high correlation are regarded as elements of a vector and are subject to a common thresholding operation. Simulations with several multiwavelets illustrate that the proposed technique results in a better performance.
Pseudobiorthogonal Multiwavelets and Finite Elements
, 1997
"... This paper has two objectives. One is to propose a way to build perfect reconstruction multifilters. This requires four r \Theta r matrix polynomials H 0 , H 1 , F 0 , F 1 such that H 0 (z) H 0 (\Gammaz) H 1 (z) H 1 (\Gammaz) F 0 (z) F 1 (z) F 0 (\Gammaz) F 1 (\Gammaz) = cI: ..."
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Cited by 4 (1 self)
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This paper has two objectives. One is to propose a way to build perfect reconstruction multifilters. This requires four r \Theta r matrix polynomials H 0 , H 1 , F 0 , F 1 such that H 0 (z) H 0 (\Gammaz) H 1 (z) H 1 (\Gammaz) F 0 (z) F 1 (z) F 0 (\Gammaz) F 1 (\Gammaz) = cI: In the scalar case (r = 1) there are standard constructions of H 1 , F 0 , F 1 from a suitable H 0 . A new procedure is needed for multifilters, because matrices do not commute. Our second purpose is to produce two specific pseudobiorthogonal wavelet bases for L 2 . We start with the piecewise cubic Hermite functions OE 0 (t) and OE 1 (t). These are "finite elements" supported on [0; 2]. They are the scaling functions for a particular multifilter H 0 with r = 2. The connection between the filter coefficients h 0 (k) and the Hermite cubics is the matrix dilation equation OE(t) = X h 0 (k)OE(2t \Gamma k); OE(t) = [OE 0 (t) OE 1 (t)] T The basis fOE 0 (t \Gamma k); OE 1 (t \Gamma k)g...