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204
Framelets: MRABased Constructions of Wavelet Frames
, 2001
"... We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spl ..."
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Cited by 130 (54 self)
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We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spline, pseudospline tight frames and symmetric biframes with short supports and high approximation orders. Several explicit examples are discussed. The connection of these frames with multiresolution analysis guarantees the existence of fast implementation algorithms, which we discuss briefly as well.
Diffusion Wavelets
, 2004
"... We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their ..."
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Cited by 75 (12 self)
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We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. In this case our construction can be viewed as a farreaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the nonstandard wavelet representation of CalderónZygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical LittlewoodPaley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.
Optimally sparse multidimensional representations using shearlets, preprint
, 2006
"... Abstract. Recent advances in applied mathematics and signal processing have shown that, in order to obtain sparse representations of multidimensional functions and signals, one has to use representation elements distributed not only at various scales and locations – as in classical wavelet theory – ..."
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Cited by 61 (29 self)
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Abstract. Recent advances in applied mathematics and signal processing have shown that, in order to obtain sparse representations of multidimensional functions and signals, one has to use representation elements distributed not only at various scales and locations – as in classical wavelet theory – but also at various directions. In this paper, we show that we obtain a construction having exactly these properties by using the framework of affine systems. The representation elements that we obtain are generated by translations, dilations, and shear transformations of a single mother function, and are called shearlets. The shearlets provide optimally sparse representations for 2D functions that are smooth away from discontinuities along curves. Another benefit of this approach is that, thanks to their mathematical structure, these systems provide a Multiresolution analysis similar to the one associated with classical wavelets, which is very useful for the development of fast algorithmic implementations.
Sparse Directional Image Representations using the Discrete Shearlet Transform
 Appl. Comput. Harmon. Anal
"... It is now widely acknowledged that traditional wavelets are not very effective in dealing with multidimensional signals containing distributed discontinuities. To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. This paper introduces a n ..."
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Cited by 48 (28 self)
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It is now widely acknowledged that traditional wavelets are not very effective in dealing with multidimensional signals containing distributed discontinuities. To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. This paper introduces a new discrete multiscale directional representation called the Discrete Shearlet Transform. This approach, which is based on the shearlet transform, combines the power of multiscale methods with a unique ability to capture the geometry of multidimensional data and is optimally efficient in representing images containing edges. We describe two different methods of implementing the shearlet transform. The numerical experiments presented in this paper demonstrate that the Discrete Shearlet Transform is very competitive in denoising applications both in terms of performance and computational efficiency.
OPTIMALLY SPARSE MULTIDIMENSIONAL REPRESENTATION USING SHEARLETS
"... Abstract. In this paper we show that the shearlets, an affinelike system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2–dimensional functions f that are C2 except for discontinuities along C2 curves. More specifically, if f S N is ..."
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Cited by 37 (16 self)
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Abstract. In this paper we show that the shearlets, an affinelike system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2–dimensional functions f that are C2 except for discontinuities along C2 curves. More specifically, if f S N is the N–term reconstruction of f obtained by using the N largest coefficients in the shearlet representation, then the asymptotic approximation error decays as ‖f − f S N ‖2 2 ≃ N −2 (log N) 3, N → ∞, which is essentially optimal, and greatly outperforms the corresponding asymptotic approximation rate N −1 associated with wavelet approximations. Unlike the curvelets, that have similar sparsity properties, the shearlets form an affinelike system and have a simpler mathematical structure. In fact, the elements of this system form a Parseval frame and are generated by applying dilations, shear transformations and translations to a single welllocalized window function.
Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix
 J. Comput. Appl. Math
"... Tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas. In this paper, for any d × d dilation matrix M, we demonstrate in a constructive way that we can construct compactly supported tight Mwavelet frames and orthonormal Mwavelet bases in ..."
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Cited by 26 (21 self)
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Tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas. In this paper, for any d × d dilation matrix M, we demonstrate in a constructive way that we can construct compactly supported tight Mwavelet frames and orthonormal Mwavelet bases in L2(R d) of exponential decay, which are derived from compactly supported Mrefinable functions, such that they can have both arbitrarily high smoothness and any preassigned order of vanishing moments. This paper improves several
Interference Avoidance for Wireless Systems
 In Vehicular Technology Conference
, 1999
"... Motivated by the emergence of programmable radios, we seek to understand a new class of communication system where pairs of transmitters and receivers can adapt their modulation/demodulation method in the presence of interference to achieve better performance. Using signal to interference ratio as a ..."
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Cited by 16 (9 self)
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Motivated by the emergence of programmable radios, we seek to understand a new class of communication system where pairs of transmitters and receivers can adapt their modulation/demodulation method in the presence of interference to achieve better performance. Using signal to interference ratio as a metric and a general signal space approach, we present a class of iterative distributed algorithms for synchronous systems which results in an ensemble of optimal waveforms for multiple users connected to a common receiver (or colocated independent receivers). That is, the waveform ensemble meets the Welch Bound with equality and therefore achieves minimum average interference over the ensemble of signature waveforms. We describe fixed points for a number of scenarios. 1 Introduction Wireless system designers have always had to contend with interference from both natural sources and other users of the medium. Thus, the classical wireless communications design cycle has consisted of measu...
A Bayesian Decision Theoretic Approach To Wavelet Thresholding
 J. AMER. STATIST. ASSOC
, 1995
"... ... In this paper we find hard thresholding decision rules that minimize Bayes risk for broad classes of underlying models. Standard DonohoJohnstone test signals are used to evaluate performance of such rules. We show that a decision theoretic hard thresholding rule can achieve smaller mean squared ..."
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Cited by 16 (1 self)
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... In this paper we find hard thresholding decision rules that minimize Bayes risk for broad classes of underlying models. Standard DonohoJohnstone test signals are used to evaluate performance of such rules. We show that a decision theoretic hard thresholding rule can achieve smaller mean squared error than some standard wavelet thresholding methods, if the prior information on the level noise is precise.
The Construction Of Single Wavelets In DDimensions
 J. GEOM. ANAL
, 1999
"... Sets K in ddimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic function 1K is a single dyadic orthonormal wavelet. The construction is characterized by its generality in terms of a quantitative iterative procedure, by its computat ..."
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Cited by 16 (2 self)
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Sets K in ddimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic function 1K is a single dyadic orthonormal wavelet. The construction is characterized by its generality in terms of a quantitative iterative procedure, by its computational implementation, and by its simplicity. The general case in which the inverse Fourier transforms of the characteristic functions 1 K 1 ; : : : ; 1 K L are a family of orthonormal wavelets is treated in [Leo99].
Construction of Multivariate Biorthogonal Wavelets by CBC Algorithm
 Adv. Comput. Math
, 1998
"... In applications, it is well known that short support, high vanishing moments and reasonable smoothness are the three most important properties of a biorthogonal wavelet. Based on our previous work on analysis and construction of optimal fundamental refinable functions and optimal biorthogonal wavele ..."
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Cited by 15 (11 self)
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In applications, it is well known that short support, high vanishing moments and reasonable smoothness are the three most important properties of a biorthogonal wavelet. Based on our previous work on analysis and construction of optimal fundamental refinable functions and optimal biorthogonal wavelets, in this paper, we shall discuss the mutual relations among these three properties. For example, we shall see that any orthogonal scaling function, which is supported on [0; 2r \Gamma 1] s for some positive integer r and has accuracy order r, has Lp (1 p 1) smoothness not exceeding that of the univariate Daubechies orthogonal scaling function which is supported on [0; 2r \Gamma 1]. Similar results hold true for fundamental refinable functions and biorthogonal wavelets. Then, we shall discuss the relation between symmetry and the smoothness of a refinable function. Next, we discuss the coset by coset (CBC) algorithm reported in Han [29] to construct biorthogonal wavelets with arbitrar...