Results 1  10
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258
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
, 2007
"... A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinato ..."
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Cited by 202 (31 self)
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A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easilyverifiable conditions under which optimallysparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several wellknown signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical
Optimal spatial adaptation for patchbased image denoising
 IEEE Trans. Image Process
, 2006
"... Abstract—A novel adaptive and patchbased approach is proposed for image denoising and representation. The method is based on a pointwise selection of small image patches of fixed size in the variable neighborhood of each pixel. Our contribution is to associate with each pixel the weighted sum of da ..."
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Cited by 68 (10 self)
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Abstract—A novel adaptive and patchbased approach is proposed for image denoising and representation. The method is based on a pointwise selection of small image patches of fixed size in the variable neighborhood of each pixel. Our contribution is to associate with each pixel the weighted sum of data points within an adaptive neighborhood, in a manner that it balances the accuracy of approximation and the stochastic error, at each spatial position. This method is general and can be applied under the assumption that there exists repetitive patterns in a local neighborhood of a point. By introducing spatial adaptivity, we extend the work earlier described by Buades et al. which can be considered as an extension of bilateral filtering to image patches. Finally, we propose a nearly parameterfree algorithm for image denoising. The method is applied to both artificially corrupted (white Gaussian noise) and real images and the performance is very close to, and in some cases even surpasses, that of the already published denoising methods. I.
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 56 (24 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.
NONSUBSAMPLED CONTOURLET TRANSFORM: FILTER DESIGN AND APPLICATIONS IN DENOISING
"... In this paper we study the nonsubsampled contourlet transform. We address the corresponding filter design problem using the McClellan transformation. We show how zeroes can be imposed in the filters so that the iterated structure produces regular basis functions. The proposed design framework yields ..."
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Cited by 53 (4 self)
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In this paper we study the nonsubsampled contourlet transform. We address the corresponding filter design problem using the McClellan transformation. We show how zeroes can be imposed in the filters so that the iterated structure produces regular basis functions. The proposed design framework yields filters that can be implemented efficiently through a lifting factorization. We apply the constructed transform in image noise removal where the results obtained are comparable to the stateofthe art, being superior in some cases.
A douglasRachford splitting approach to nonsmooth convex variational signal recovery
 IEEE Journal of Selected Topics in Signal Processing
, 2007
"... Abstract — Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is propo ..."
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Cited by 49 (15 self)
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Abstract — Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is proposed to solve it. The convergence of the method, which is based on the DouglasRachford algorithm for monotone operatorsplitting, is obtained under general conditions. Applications to nonGaussian image denoising in a tight frame are also demonstrated. Index Terms — Convex optimization, denoising, DouglasRachford, frame, nondifferentiable optimization, Poisson noise,
Wave atoms and sparsity of oscillatory patterns
 Appl. Comput. Harmon. Anal
, 2006
"... We introduce “wave atoms ” as a variant of 2D wavelet packets obeying the parabolic scaling wavelength ∼ (diameter) 2. We prove that warped oscillatory functions, a toy model for texture, have a significantly sparser expansion in wave atoms than in other fixed standard representations like wavelets, ..."
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Cited by 45 (5 self)
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We introduce “wave atoms ” as a variant of 2D wavelet packets obeying the parabolic scaling wavelength ∼ (diameter) 2. We prove that warped oscillatory functions, a toy model for texture, have a significantly sparser expansion in wave atoms than in other fixed standard representations like wavelets, Gabor atoms, or curvelets. We propose a novel algorithm for a tight frame of wave atoms with redundancy two, directly in the frequency plane, by the “wrapping ” technique. We also propose variants of the basic transform for applications in image processing, including an orthonormal basis, and a shiftinvariant tight frame with redundancy four. Sparsity and denoising experiments on both seismic and fingerprint images demonstrate the potential of the tool introduced.
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.
A variational formulation for framebased inverse problems
 Inverse Problems
, 2007
"... A convex variational framework is proposed for solving inverse problems in Hilbert spaces with a priori information on the representation of the target solution in a frame. The objective function to be minimized consists of a separable term penalizing each frame coefficient individually and of a smo ..."
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Cited by 42 (19 self)
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A convex variational framework is proposed for solving inverse problems in Hilbert spaces with a priori information on the representation of the target solution in a frame. The objective function to be minimized consists of a separable term penalizing each frame coefficient individually and of a smooth term modeling the data formation model as well as other constraints. Sparsityconstrained and Bayesian formulations are examined as special cases. A splitting algorithm is presented to solve this problem and its convergence is established in infinitedimensional spaces under mild conditions on the penalization functions, which need not be differentiable. Numerical simulations demonstrate applications to framebased image restoration. 1
Directionlets: Anisotropic MultiDirectional Representation With Separable Filtering
 Ph.D. dissertation, School Comput. Commun. Sci., Swiss Federal Inst. Technol. Lausanne (EPFL
, 2005
"... Abstract—In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. Onedimensional (1D) discontinuities in images ..."
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Cited by 41 (7 self)
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Abstract—In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. Onedimensional (1D) discontinuities in images (edges and contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a nonsparse representation. To efficiently capture these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, a more complex multidirectional (MDIR) and anisotropic transform is required. We present a new latticebased perfect reconstruction and critically sampled anisotropic MDIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard twodimensional WT, unlike in the case of some other directional transform constructions (e.g., curvelets, contourlets, or edgelets). The corresponding anisotropic basis unctions (directionlets) have directional vanishing moments along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for nonlinear approximation of images, achieving the approximation power ( 1 55), which, while slower than the optimal rate ( 2), is much better than ( 1) achieved with wavelets, but at similar complexity. Index Terms—Directional vanishing moments, directionlets, filter banks, geometry, multidirection, multiresolution, separable filtering, sparse image representation, wavelets. I.