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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 109 (4 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.
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 73 (10 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.
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 60 (21 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
A Shearlet Approach to Edge Analysis and Detection
"... 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 edg ..."
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Cited by 57 (27 self)
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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.
1 Compressive Video Sampling with Approximate Message Passing Decoding
"... In this paper, we apply compressed sensing to video compression. Compressed sensing (CS) techniques exploit the observation that one needs much fewer random measurements than given by the ShannonNyquist sampling theory to recover an object if this object is compressible (i.e., sparse in the spatial ..."
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Cited by 38 (2 self)
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In this paper, we apply compressed sensing to video compression. Compressed sensing (CS) techniques exploit the observation that one needs much fewer random measurements than given by the ShannonNyquist sampling theory to recover an object if this object is compressible (i.e., sparse in the spatial domain or in a transform domain). In the CS framework, we can achieve sensing, compression and denoising simultaneously. We propose a fast and simple online encoding by application of pseudorandom downsampling of the twodimensional fast Fourier transform to video frames. For offline decoding, we apply a modification of the recently proposed approximate message passing (AMP) algorithm. The AMP method has been derived using the statistical concept of ’state evolution’, and it has been shown to considerably accelerate the convergence rate in special CSdecoding applications. We shall prove that the AMP method can be rewritten as a forwardbackward splitting algorithm. This new representation enables us to give conditions that ensure convergence of the AMP method and to modify the algorithm in order to achieve higher robustness. The success of reconstruction methods for video decoding also essentially depends on the chosen transform, where sparsity of the video signals is assumed. We propose to incorporate the 3D dualtree complex wavelet transform that possesses sufficiently good properties of directional selectivity and shift invariance while being computationally less expensive and less redundant than other directional 3D wavelet transforms.
Nested iterative algorithms for convex constrained image recovery problems
 IEEE Journal of Selected Topics in Signal Processing
, 2007
"... The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions f and g, ..."
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Cited by 30 (8 self)
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The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions f and g, where f may be nonsmooth and g is differentiable with a Lipschitzcontinuous gradient. To reach this goal, we derive two types of algorithms that combine forwardbackward and DouglasRachford iterations. The weak convergence of the proposed algorithms is proved. In the case when the Lipschitzcontinuity property of the gradient of g is not satisfied, we also show that, under some assumptions, it remains possible to apply these methods to the considered optimization problem by making use of a quadratic extension technique. The effectiveness of the algorithms is demonstrated for two waveletbased image restoration problems involving a signaldependent Gaussian noise and a Poisson noise, respectively. 1
Parallel proximal algorithm for image restoration using hybrid regularization
 IEEE Transactions on Image Processing
, 2011
"... Regularization approaches have demonstrated their effectiveness for solving illposed problems. However, in the context of variational restoration methods, a challenging question remains, namely how to find a good regularizer. While total variation introduces staircase effects, wavelet domain regula ..."
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Cited by 25 (8 self)
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Regularization approaches have demonstrated their effectiveness for solving illposed problems. However, in the context of variational restoration methods, a challenging question remains, namely how to find a good regularizer. While total variation introduces staircase effects, wavelet domain regularization brings other artefacts, e.g. ringing. However, a tradeoff can be made by introducing a hybrid regularization including several terms non necessarily acting in the same domain (e.g. spatial and wavelet transform domains). While this approachwas shown to provide good results for solving deconvolution problems in the presence of additive Gaussian noise, an important issue is to efficiently deal with this hybrid regularization for more general noise models. To solve this problem, we adopt a convex optimization framework where the criterion to be minimized is split in the sum of more than two terms. For spatial domain regularization, isotropic or anisotropic total variation definitions using various gradient filters are considered. An accelerated version of the Parallel Proximal Algorithm is proposed to perform the minimization. Some difficulties in the computation of the proximity operators involved in this algorithm are also addressed in this paper. Numerical experiments performed in the context of Poisson data recovery, show the good behaviour of the algorithm as well as promising results concerning the use of hybrid regularization techniques.
A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity
, 2011
"... The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smoot ..."
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Cited by 21 (8 self)
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The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multiorientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to nonEuclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping “pictures”.