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Dictionaries for Sparse Representation Modeling
"... Sparse and redundant representation modeling of data assumes an ability to describe signals as linear combinations of a few atoms from a prespecified dictionary. As such, the choice of the dictionary that sparsifies the signals is crucial for the success of this model. In general, the choice of a p ..."
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Cited by 44 (3 self)
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Sparse and redundant representation modeling of data assumes an ability to describe signals as linear combinations of a few atoms from a prespecified dictionary. As such, the choice of the dictionary that sparsifies the signals is crucial for the success of this model. In general, the choice of a proper dictionary can be done using one of two ways: (i) building a sparsifying dictionary based on a mathematical model of the data, or (ii) learning a dictionary to perform best on a training set. In this paper we describe the evolution of these two paradigms. As manifestations of the first approach, we cover topics such as wavelets, wavelet packets, contourlets, and curvelets, all aiming to exploit 1D and 2D mathematical models for constructing effective dictionaries for signals and images. Dictionary learning takes a different route, attaching the dictionary to a set of examples it is supposed to serve. From the seminal work of Field and Olshausen, through the MOD, the KSVD, the Generalized PCA and others, this paper surveys the various options such training has to offer, up to the most recent contributions and structures.
Resolution of the Wavefront Set using Continuous Shearlets
, 2008
"... Abstract. It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unabl ..."
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Cited by 37 (20 self)
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Abstract. It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unable to describe the geometry of the set of singularities of f and, in particular, identify the wavefront set of a distribution. In this paper, we employ the same framework of affine systems which is at the core of the construction of the wavelet transform to introduce the Continuous Shearlet Transform. This is defined by SHψf(a, s, t) = 〈f, ψast〉, where the analyzing elements ψast are dilated and translated copies of a single generating function ψ. The dilation matrices form a twoparameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements {ψast} form a system of smooth functions at continuous scales a> 0, locations t ∈ R 2, and oriented along lines of slope s ∈ R in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution f. 1.
A Shearlet Approach to Edge Analysis and Detection”, IEEE Trans. Image Proc. 18(5) 929–941, 2009. 7. Results of ball spiraling surface detected for 2D and 3D shearlet routines without noise added to data. Figure 8. Results of ball spiraling surface detect
"... Abstract—It is well known that the wavelet transform provides a very effective framework for analysis of multiscale edges. In this paper, we propose a novel approach based on the shearlet transform: a multiscale directional transform with a greater ability to localize distributed discontinuities suc ..."
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Cited by 20 (8 self)
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Abstract—It is well known that the wavelet transform provides a very effective framework for analysis of multiscale edges. In this paper, we propose a novel approach based on the shearlet transform: a multiscale directional transform with a greater ability to localize distributed discontinuities such as edges. Indeed, unlike traditional wavelets, shearlets are theoretically optimal in representing images with edges and, in particular, have the ability to fully capture directional and other geometrical features. Numerical examples demonstrate that the shearlet approach is highly effective at detecting both the location and orientation of edges, and outperforms methods based on wavelets as well as other standard methods. Furthermore, the shearlet approach is useful to design simple and effective algorithms for the detection of corners and junctions. Index Terms—Curvelets, edge detection, feature extraction, shearlets, singularities, wavelets. I.
Construction of Regular and Irregular Shearlet Frames
 J. Wavelet Theory and Appl
"... Abstract. In this paper, we study the construction of irregular shearlet systems, i.e., 3 − systems of the form SH(ψ, Λ) = {a 4 ψ(A−1 a S−1 s (x − t)) : (a, s, t) ∈ Λ}, where ψ ∈ L2 (R2), Λ is an arbitrary sequence in R + × R × R2, Aa is a parabolic scaling matrix and Ss a shear matrix. These syst ..."
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Cited by 9 (4 self)
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Abstract. In this paper, we study the construction of irregular shearlet systems, i.e., 3 − systems of the form SH(ψ, Λ) = {a 4 ψ(A−1 a S−1 s (x − t)) : (a, s, t) ∈ Λ}, where ψ ∈ L2 (R2), Λ is an arbitrary sequence in R + × R × R2, Aa is a parabolic scaling matrix and Ss a shear matrix. These systems are obtained by appropriately sampling the Continuous Shearlet Transform. We derive sufficient conditions for such a discrete system to form a frame for L 2 (R 2), and provide explicit estimates for the frame bounds. Among the examples of such discrete systems, one is the Parseval frame of shearlets previously introduced by the authors, which is optimal in approximating 2D smooth functions with discontinuities along C 2curves. This study provides the framework for the construction of a variety of discrete directional multiscale systems with the ability to detect orientations inherited from the Continuous Shearlet Transform. 1.
Characterization and analysis of edges using the continuous shearlet transform
"... Abstract. This paper shows that the continuous shearlet transform, a novel directional multiscale transform recently introduced by the authors and their collaborators, provides a precise geometrical characterization for the boundary curves of very general planar regions. This study is motivated by i ..."
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Cited by 6 (4 self)
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Abstract. This paper shows that the continuous shearlet transform, a novel directional multiscale transform recently introduced by the authors and their collaborators, provides a precise geometrical characterization for the boundary curves of very general planar regions. This study is motivated by imaging applications, where such boundary curves represent edges of images. The shearlet approach is able to characterize both locations and orientations of the edge points, including corner points and junctions, where the edge curves exhibit abrupt changes in tangent or curvature. Our results encompass and greatly extend previous results based on the shearlet and curvelet transforms which were limited to very special cases such as polygons and smooth boundary curves with nonvanishing curvature. Key words. Analysis of singularities, continuous wavelets, curvelets, directional wavelets, edge detection, shearlets, wavelets AMS subject classifications. 42C15, 42C40
Compactly supported shearlets
, 2010
"... Abstract Shearlet theory has become a central tool in analyzing and representing 2D data with anisotropic features. Shearlet systems are systems of functions generated by one single generator with parabolic scaling, shearing, and translation operators applied to it, in much the same way wavelet syst ..."
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Cited by 5 (0 self)
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Abstract Shearlet theory has become a central tool in analyzing and representing 2D data with anisotropic features. Shearlet systems are systems of functions generated by one single generator with parabolic scaling, shearing, and translation operators applied to it, in much the same way wavelet systems are dyadic scalings and translations of a single function, but including a precise control of directionality. Of the many directional representation systems proposed in the last decade, shearlets are among the most versatile and successful systems. The reason for this being an extensive list of desirable properties: shearlet systems can be generated by one function, they provide precise resolution of wavefront sets, they allow compactly supported analyzing elements, they are associated with fast decomposition algorithms, and they provide a unified treatment of the continuum and the digital realm. The aim of this paper is to introduce some key concepts in directional representation systems and to shed some light on the success of shearlet systems as directional representation systems. In particular, we will give an overview of the different paths taken in shearlet theory with focus on separable and compactly supported shearlets in 2D and 3D. We will present constructions of compactly supported shearlet frames in those dimensions as well as discuss recent results on the ability of compactly supported shearlet frames satisfying weak decay, smoothness, and directional moment conditions to provide optimally sparse approximations of cartoonlike images in 2D as well as in 3D. Finally, we will show that these compactly supported shearlet systems provide optimally sparse approximations of an even generalized model of
Radon Transform Inversion using the Shearlet Representation
, 2010
"... The inversion of the Radon transform is a classical illposed inverse problem where some method of regularization must be applied in order to accurately recover the objects of interest from the observable data. A wellknown consequence of the traditional regularization methods is that some important ..."
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Cited by 4 (4 self)
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The inversion of the Radon transform is a classical illposed inverse problem where some method of regularization must be applied in order to accurately recover the objects of interest from the observable data. A wellknown consequence of the traditional regularization methods is that some important features to be recovered are lost, as evident in imaging applications where the regularized reconstructions are blurred versions of the original. In this paper, we show that the affinelike system of functions known as the shearlet system can be applied to obtain a highly effective reconstruction algorithm which provides nearoptimal rate of convergence in estimating a large class of images from noisy Radon data. This is achieved by introducing a shearletbased decomposition of the Radon operator and applying a thresholding scheme on the noisy shearlet transform coefficients. For a given noise level ɛ, the proposed shearlet shrinkage method can be tuned so that the estimator will attain the essentially optimal mean square error O(log(ɛ −1)ɛ 4/5), as ɛ → 0. Several numerical demonstrations show that its performance improves upon similar competitive strategies based on wavelets and curvelets. Key words: directional wavelets; inverse problems, Radon transform shearlets; wavelets
Shearletbased deconvolution
 IEEE Trans. Image Process
, 2009
"... Abstract—In this paper, a new type of deconvolution algorithm is proposed that is based on estimating the image from a shearlet decomposition. Shearlets provide a multidirectional and multiscale decomposition that has been mathematically shown to represent distributed discontinuities such as edges b ..."
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Cited by 4 (4 self)
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Abstract—In this paper, a new type of deconvolution algorithm is proposed that is based on estimating the image from a shearlet decomposition. Shearlets provide a multidirectional and multiscale decomposition that has been mathematically shown to represent distributed discontinuities such as edges better than traditional wavelets. Constructions such as curvelets and contourlets share similar properties, yet their implementations are significantly different from that of shearlets. Taking advantage of unique properties of a new Mchannel implementation of the shearlet transform, we develop an algorithm that allows for the approximation inversion operator to be controlled on a multiscale and multidirectional basis. A key improvement over closely related approaches such as ForWaRD is the automatic determination of the threshold values for the noise shrinkage for each scale and direction without explicit knowledge of the noise variance using a generalized cross validation (GCV). Various tests show that this method can perform significantly better than many competitive deconvolution algorithms. Index Terms—Deconvolution, generalized cross validation, shearlets, wavelets. I.
Irregular shearlet frames: Geometry and approximation properties
 J. Fourier Anal. Appl
, 2010
"... Abstract. Recently, shearlet systems were introduced as a means to derive efficient encoding methodologies for anisotropic features in 2dimensional data with a unified treatment of the continuum and digital setting. However, only very few construction strategies for discrete shearlet systems are kn ..."
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Cited by 3 (1 self)
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Abstract. Recently, shearlet systems were introduced as a means to derive efficient encoding methodologies for anisotropic features in 2dimensional data with a unified treatment of the continuum and digital setting. However, only very few construction strategies for discrete shearlet systems are known so far. In this paper, we take a geometric approach to this problem. Utilizing the close connection with group representations, we first introduce and analyze an upper and lower weighted shearlet density based on the shearlet group. We then apply this geometric measure to provide necessary conditions on the geometry of the sets of parameters for the associated shearlet systems to form a frame for L 2 (R 2), either when using all possible generators or a large class exhibiting some decay conditions. While introducing such a feasible class of shearlet generators, we analyze approximation properties of the associated shearlet systems, which themselves lead to interesting insights into homogeneous approximation abilities of shearlet frames. We also present examples, such a oversampled shearlet systems and coshearlet systems, to illustrate the usefulness of our geometric approach to the construction of shearlet frames. 1.