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24
Smooth Wavelet Tight Frames with Zero Moments
, 2001
"... This paper considers the design of wavelet tight frames based on iterated oversampled filter banks. The greater design freedom available makes possible the construction of wavelets with a high degree of smoothness, in comparison with orthonormal wavelet bases. In particular, this paper takes up the ..."
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Cited by 35 (5 self)
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This paper considers the design of wavelet tight frames based on iterated oversampled filter banks. The greater design freedom available makes possible the construction of wavelets with a high degree of smoothness, in comparison with orthonormal wavelet bases. In particular, this paper takes up the design of systems that are analogous to Daubechies orthonormal wavelets—that is, the design of minimal length wavelet filters satisfying certain polynomial properties, but now in the oversampled case. Gröbner bases are used to obtain the solutions to the nonlinear design equations. Following the dualtree DWT of Kingsbury, one goal is to achieve near shift invariance while keeping the redundancy factor bounded by 2, instead of allowing it to grow as it does for the undecimated DWT (which is exactly shift invariant). Like the dual tree, the overcomplete DWT described in this paper is less shiftsensitive than an orthonormal wavelet basis. Like the examples of Chui and He, and Ron and Shen, the wavelets are much smoother than what is possible in the orthonormal case.
Nonideal Sampling and Regularization Theory
, 2008
"... Shannon’s sampling theory and its variants provide effective solutions to the problem of reconstructing a signal from its samples in some “shiftinvariant” space, which may or may not be bandlimited. In this paper, we present some further justification for this type of representation, while addressi ..."
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Cited by 15 (3 self)
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Shannon’s sampling theory and its variants provide effective solutions to the problem of reconstructing a signal from its samples in some “shiftinvariant” space, which may or may not be bandlimited. In this paper, we present some further justification for this type of representation, while addressing the issue of the specification of the best reconstruction space. We consider a realistic setting where a multidimensional signal is prefiltered prior to sampling, and the samples are corrupted by additive noise. We adopt a variational approach to the reconstruction problem and minimize a data fidelity term subject to a Tikhonovlike (continuous domain) 2regularization to obtain the continuousspace solution. We present theoretical justification for the minimization of this cost functional and show that the globally minimal continuousspace solution belongs to a shiftinvariant space generated by a function (generalized Bspline) that is generally not bandlimited. When the sampling is ideal, we recover some of the classical smoothing spline estimators. The optimal reconstruction space is characterized by a condition that links the generating function to the regularization operator and implies the existence of a Bsplinelike basis. To make the scheme practical, we specify the generating functions corresponding to the most popular families of regularization operators (derivatives, iterated Laplacian), as well as a new, generalized one that leads to a new brand of Matérn splines. We conclude the paper by proposing a stochastic interpretation of the reconstruction algorithm and establishing an equivalence with the minimax and minimum mean square error (MMSE/Wiener) solutions of the generalized sampling problem.
Iterated Oversampled Filter Banks and Wavelet Frames
 In Wavelet Applications VII, Proceedings of SPIE
, 2000
"... This paper takes up the design of wavelet tight frames that are analogous to Daubechies orthonormal wavelets  that is, the design of minimal length wavelet filters satisfying certain polynomial properties, but now in the oversampled case. The oversampled dyadic DWT considered in this paper is bas ..."
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Cited by 12 (2 self)
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This paper takes up the design of wavelet tight frames that are analogous to Daubechies orthonormal wavelets  that is, the design of minimal length wavelet filters satisfying certain polynomial properties, but now in the oversampled case. The oversampled dyadic DWT considered in this paper is based on a single scaling function and two distinct wavelets. Having more wavelets than necessary gives a closer spacing between adjacent wavelets within the same scale. As a result, the transform (like Kingsbury's dualtree DWT) is nearly shiftinvariant, and can be used to improve denoising. Because the associated timefrequency lattice preserves the dyadic structure of the critically sampled DWT (which the undecimated DWT does not) it can be used with treebased denoising algorithms that exploit parentchild correlation. Keywords: Wavelet, tight frame, FIR filter banks, orthonormal transform, Grobner bases. 1. INTRODUCTION This paper describes new wavelet tight frames 1 (`overcomplete b...
Mband Compactly Supported Orthogonal Symmetric Interpolating Scaling Functions
 IEEE Trans. Sig. Proc
, 2001
"... Abstract—In many applications, wavelets are usually expected to have the following properties: compact support, orthogonality, linearphase, regularity, and interpolation. To construct such wavelets, it is crucial designing scaling functions with the above properties. In two and threeband cases, e ..."
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Cited by 8 (0 self)
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Abstract—In many applications, wavelets are usually expected to have the following properties: compact support, orthogonality, linearphase, regularity, and interpolation. To construct such wavelets, it is crucial designing scaling functions with the above properties. In two and threeband cases, except for the Haar functions, there exists no scaling function with the above five properties. Inband case ( 4), more free degrees available in design enable us to construct such scaling functions. In this paper, a novel approach to designing such scaling functions is proposed. First, we extend the twoband Dubuc filters toband case. Next, theband FIR regular symmetric interpolating scaling filters are parameterized, and then,band FIR regular orthogonal symmetric interpolating scaling filters (OSISFs) are designed via optimal selection of parameters. Finally, two family of fourband and fiveband OSISFs and scaling functions are developed, and their smoothnesses are estimated. Index Terms—Cardinal interpolation, linearphase, scaling function, Sobolev exponent, wavelet sampling. I.
Symbolic computation and signal processing
 INRIA of the
"... Many problems in digital signal processing can be converted to algebraic problems over polynomial and Laurent polynomial rings, and can be solved using the existing methods of algebraic and symbolic computation. This paper aims to establish this connection in a systematic manner, and demonstrate how ..."
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Cited by 8 (0 self)
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Many problems in digital signal processing can be converted to algebraic problems over polynomial and Laurent polynomial rings, and can be solved using the existing methods of algebraic and symbolic computation. This paper aims to establish this connection in a systematic manner, and demonstrate how it can be used to solve various problems arising from multidimensional signal processing. The method of Gröbner bases is used as a main computational tool. © 2003 Elsevier Ltd. All rights reserved.
Mimo Biorthogonal Partners And Applications
 IEEE Trans. Signal Processing
, 2001
"... Multiple Input MultipleOS put (MIMO biorthogonal partners arise in many different contexts, one of them being multiwavelet theory. They also play a central role in the theory of MIMO channel equalization, especially with fractionally spaced equalizers. In this paper we first derive some theoretical ..."
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Cited by 7 (3 self)
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Multiple Input MultipleOS put (MIMO biorthogonal partners arise in many different contexts, one of them being multiwavelet theory. They also play a central role in the theory of MIMO channel equalization, especially with fractionally spaced equalizers. In this paper we first derive some theoretical properties of MIMO biorthogonal partners. We develop conditions for the existence of MIMO biorthogonal partners and conditions under which FIR solutions are possible. In the process of constructing FIR MIMO biorthogonal partners we exploit the nonuniqueness of the solution. This will lead to the design of flexible fractionally spaced zeroforcing equalizers. The additional flexibility in design makes these equalizers more robust to channel noise. Finally, other situations where MIMO biorthogonal partners occur will also be considered, such as prefiltering in multiwavelet theory and deriving the vector version of the least squares signal projection problem.
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 5 (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.
ANALYSIS AND CONSTRUCTION OF MULTIVARIATE INTERPOLATING REFINABLE FUNCTION VECTORS
"... Abstract. In this paper, we shall introduce and study a family of multivariate interpolating refinable function vectors with some prescribed interpolation property. Such interpolating refinable function vectors are of interest in approximation theory, sampling theorems, and wavelet analysis. In this ..."
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Cited by 3 (2 self)
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Abstract. In this paper, we shall introduce and study a family of multivariate interpolating refinable function vectors with some prescribed interpolation property. Such interpolating refinable function vectors are of interest in approximation theory, sampling theorems, and wavelet analysis. In this paper, we characterize a multivariate interpolating refinable function vector in terms of its mask and analyze the underlying sum rule structure of its generalized interpolatory matrix mask. We also discuss the symmetry property of multivariate interpolating refinable function vectors. Based on these results, we construct a family of univariate generalized interpolatory matrix masks with increasing orders of sum rules and with symmetry for interpolating refinable function vectors. Such a family includes several known important families of univariate refinable function vectors as special cases. Several examples of bivariate interpolating refinable function vectors with symmetry will also be presented. 1.
Applications of multiwavelet techniques to image denoising
 Proceedings of International Conference on Image Processing
"... The developments in wavelet theory have given rise to the wavelet thresholding method, for extracting a signal from noisy data [1,2]. Multiwavelets, wavelets with several scaling functions, have recently been introduced and they offer simultaneous orthogonality, symmetry and short support; which is ..."
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Cited by 2 (0 self)
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The developments in wavelet theory have given rise to the wavelet thresholding method, for extracting a signal from noisy data [1,2]. Multiwavelets, wavelets with several scaling functions, have recently been introduced and they offer simultaneous orthogonality, symmetry and short support; which is not possible with ordinary wavelets, also called scalar wavelets [3]. This property makes multiwavelets more suitable for various signal processing applications, especially compression and denoising. Like scalar wavelets, multiwavelets can be realized as filterbanks, however the filterbanks are now matrixvalued; requiring two or more input streams, which can be accomplished by prefiltering. In this paper, several thresholding methods to be used with different multiwavelets for image denoising are presented. The performances of multiwavelets are compared with those of scalar wavelets. Simulations reveal that multiwavelet based image denoising schemes outperform wavelet based methods both subjectively and objectively.
Nonseparable orthonormal interpolating scaling vectors
 Appl. Comput. Harmon. Anal
"... In this paper we introduce an algorithm for the construction of interpolating scaling vectors on Rd with compact support and orthonormal integer translates. Our method is substantiated by constructing several examples of bivariate scaling vectors for quincunx and box– spline dilation matrices. As ..."
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Cited by 2 (1 self)
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In this paper we introduce an algorithm for the construction of interpolating scaling vectors on Rd with compact support and orthonormal integer translates. Our method is substantiated by constructing several examples of bivariate scaling vectors for quincunx and box– spline dilation matrices. As the main ingredients of our recipe we derive some implementable conditions for accuracy and orthonormality of an interpolating scaling vector in terms of its mask.