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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 101 (46 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.
Wavelet transforms versus Fourier transforms
 Department of Mathematics, MIT, Cambridge MA
, 213
"... Abstract. This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The "wavelet transform " maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and t ..."
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Cited by 82 (2 self)
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Abstract. This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The "wavelet transform " maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and the transform is fast (faster than the Fast Fourier Transform, which we briefly explain), but approximation by piecewise constants is poor. To improve this first wavelet, we are led to dilation equations and their unusual solutions. Higherorder wavelets are constructed, and it is surprisingly quick to compute with them — always indirectly and recursively. We comment informally on the contest between these transforms in signal processing, especially for video and image compression (including highdefinition television). So far the Fourier Transform — or its 8 by 8 windowed version, the Discrete Cosine Transform — is often chosen. But wavelets are already competitive, and they are ahead for fingerprints. We present a sample of this developing theory. 1. The Haar wavelet To explain wavelets we start with an example. It has every property we hope for, except one. If that one defect is accepted, the construction is simple and the computations are fast. By trying to remove the defect, we are led to dilation equations and recursively defined functions and a small world of fascinating new problems — many still unsolved. A sensible person would stop after the first wavelet, but fortunately mathematics goes on. The basic example is easier to draw than to describe: W(x)
Recovering Edges in IllPosed Inverse Problems: Optimality of Curvelet Frames
, 2000
"... We consider a model problem of recovering a function f(x1,x2) from noisy Radon data. The function f to be recovered is assumed smooth apart from a discontinuity along a C2 curve – i.e. an edge. We use the continuum white noise model, with noise level ɛ. Traditional linear methods for solving such in ..."
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Cited by 80 (14 self)
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We consider a model problem of recovering a function f(x1,x2) from noisy Radon data. The function f to be recovered is assumed smooth apart from a discontinuity along a C2 curve – i.e. an edge. We use the continuum white noise model, with noise level ɛ. Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model Mean Squared Errors (MSEs) that tend to zero with noise level ɛ only as O(ɛ1/2)asɛ → 0. A recent innovation – nonlinear shrinkage in the wavelet domain – visually improves edge sharpness and improves MSE convergence to O(ɛ2/3). However, as we show here, this rate is not optimal. In fact, essentially optimal performance is obtained by deploying the recentlyintroduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curveletbased biorthogonal decomposition
On Dual Wavelet Tight Frames
 Appl. Comput. Harmon. Anal
"... . A characterization of multivariate dual wavelet tight frames for any general dilation matrix is presented in this paper. As an application, Lawton's result on wavelet tight frames in L 2 (IR) is generalized to the ndimensional case. Two ways of constructing certain dual wavelet tight frame ..."
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Cited by 71 (35 self)
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. A characterization of multivariate dual wavelet tight frames for any general dilation matrix is presented in this paper. As an application, Lawton's result on wavelet tight frames in L 2 (IR) is generalized to the ndimensional case. Two ways of constructing certain dual wavelet tight frames in L 2 (IR n ) are suggested. Finally examples of smooth wavelet tight frames in L 2 (IR) and H 2 (IR) are provided. In particular, an example is given to demonstrate that there is a function /, whose Fourier transform is positive, compactly supported and infinitely differentiable, which generates a nonMRA wavelet tight frame in H 2 (IR). Keywords: Bessel sequence, dual wavelet tight frame, multiresolution analysis, wavelet basis. AMS Subject Classification: 41A30, 41A63, 42C10. x1. Introduction A dual wavelet tight frame (DWTF) differs from a biorthogonal wavelet basis in that it may be linearly dependent. In many applications such as signal analysis, this freedom of redundancy is...
On the Importance of Combining WaveletBased NonLinear Approximation With Coding Strategies
, 2000
"... ..."
A NonLinear Variational Problem For Image Matching
 SIAM J. Sci. Comput
, 1994
"... Minimizing a nonlinear functional is presented as a way of obtaining a planar mapping which matches two similar images. A smoothing term is added to the nonlinear functional in order to penalize discontinuous and irregular solutions. One option for the smoothing term is a quadratic form generated ..."
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Cited by 71 (3 self)
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Minimizing a nonlinear functional is presented as a way of obtaining a planar mapping which matches two similar images. A smoothing term is added to the nonlinear functional in order to penalize discontinuous and irregular solutions. One option for the smoothing term is a quadratic form generated by a linear differential operator. The functional is then minimized using the Fourier representation of the planar mapping. With this representation the quadratic form is diagonalized. Another option is a quadratic form generated via a basis of compactly supported wavelets. In both cases a natural approximation scheme is described. Both quadratic forms are shown to impose the same smoothing. However in terms of the finite dimensional approximations it turns out that it is easier to accommodate local deformations using the wavelet basis.
Multiresolution and wavelets
 Proc. Edinburgh Math. Soc
, 1994
"... Multiresolution is investigated on the basis of shiftinvariant spaces. Given a finitely generated shiftinvariant subspace S of L2(IR d), let Sk be the 2 kdilate of S (k ∈ Z). A necessary and sufficient condition is given for the sequence {Sk}k ∈ Z to form a multiresolution of L2(IR d). A general ..."
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Cited by 68 (32 self)
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Multiresolution is investigated on the basis of shiftinvariant spaces. Given a finitely generated shiftinvariant subspace S of L2(IR d), let Sk be the 2 kdilate of S (k ∈ Z). A necessary and sufficient condition is given for the sequence {Sk}k ∈ Z to form a multiresolution of L2(IR d). A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties. With the aid of the general theory of vector fields on spheres, it is demonstrated that the intrinsic properties of the scaling function must be used in constructing orthogonal wavelets with a certain decay rate. When the scaling function is skewsymmetric about some point, orthogonal wavelets and prewavelets are constructed in such a way that they possess certain attractive properties. Several examples are provided to illustrate the general theory.
TIMEFREQUENCY ANALYSIS OF SJÖSTRAND’S CLASS
, 2004
"... We investigate the properties an exotic symbol class of pseudodifferential operators, Sjöstrand’s class, with methods of timefrequency analysis (phase space analysis). Compared to the classical treatment, the timefrequency approach leads to striklingly simple proofs of Sjöstrand’s fundamental resu ..."
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Cited by 61 (14 self)
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We investigate the properties an exotic symbol class of pseudodifferential operators, Sjöstrand’s class, with methods of timefrequency analysis (phase space analysis). Compared to the classical treatment, the timefrequency approach leads to striklingly simple proofs of Sjöstrand’s fundamental results and to farreaching generalizations.
Wavelets: The Mathematical Background
 Proc. IEEE
, 1996
"... We present here the mathematical foundations of the wavelet transform, multiresolution analysis and discretetime transforms and algorithms. This article serves as background material for the rest of the special issue. 1 Introduction When we deal with a given physical object, we encounter many of i ..."
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Cited by 61 (0 self)
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We present here the mathematical foundations of the wavelet transform, multiresolution analysis and discretetime transforms and algorithms. This article serves as background material for the rest of the special issue. 1 Introduction When we deal with a given physical object, we encounter many of its different faces, or, representations. For example, we can represent numbers in various systems depending on the application; in everyday life, we use the decimal system, while for use in computers we employ the binary representation. Consequently, in many fields, such as numerical analysis or signal processing, a preliminary task is to find an adapted representation of the signal that may be particularly suitable for a problem at hand. For example, in images, one of the common tasks is to attempt a representation that will facilitate extraction of features. A way to obtain a specific representation is to decompose a signal x into elementary building blocks x i , of some importance, as fo...