Results 1  10
of
36
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 combin ..."
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Cited by 423 (37 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
Optimally Sparse Image Representation by the Easy Path Wavelet Transform
"... The Easy Path Wavelet Transform (EPWT) [19] has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of f ..."
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Cited by 115 (8 self)
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The Easy Path Wavelet Transform (EPWT) [19] has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and it exploits the local correlations of the given data in a simple appropriate manner. In this paper, we show that the EPWT leads, for a suitable choice of the pathways, to optimal Nterm approximations for piecewise Hölder continuous functions with singularities along curves.
On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and SIDEs
 SIAM J. NUMER. ANAL
, 2004
"... Soft wavelet shrinkage, total variation (TV) diffusion, TV regularization, and a dynamical system called SIDEs are four useful techniques for discontinuity preserving denoising of signals and images. In this paper we investigate under which circumstances these methods are equivalent in the onedimen ..."
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Cited by 91 (19 self)
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Soft wavelet shrinkage, total variation (TV) diffusion, TV regularization, and a dynamical system called SIDEs are four useful techniques for discontinuity preserving denoising of signals and images. In this paper we investigate under which circumstances these methods are equivalent in the onedimensional case. First, we prove that Haar wavelet shrinkage on a single scale is equivalent to a single step of spacediscrete TV diffusion or regularization of twopixel pairs. In the translationally invariant case we show that applying cycle spinning to Haar wavelet shrinkage on a single scale can be regarded as an absolutely stable explicit discretization of TV diffusion. We prove that spacediscrete TV diffusion and TV regularization are identical and that they are also equivalent to the SIDEs system when a specific force function is chosen. Afterwards, we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularization of the Laplacian pyramid of the signal. We analyze possibilities to avoid Gibbslike artifacts for multiscale Haar wavelet shrinkage by scaling the thresholds. Finally, we present experiments where hybrid methods are designed that combine the advantages of wavelets and PDE/variational approaches. These methods are based on iterated shiftinvariant wavelet shrinkage at multiple scales with scaled thresholds.
A MULTISCALE IMAGE REPRESENTATION USING HIERARCHICAL (BV, L²) DECOMPOSITIONS
 MULTISCALE MODEL. SIMUL.
, 2004
"... We propose a new multiscale image decomposition which offers a hierarchical, adaptive representation for the different features in general images. The starting point is a variational decomposition of an image, f = u0 { + v0, where [u0,v0] is the minimizer of a Jfunctional, J(f, λ0; X, Y) = infu+v= ..."
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Cited by 71 (10 self)
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We propose a new multiscale image decomposition which offers a hierarchical, adaptive representation for the different features in general images. The starting point is a variational decomposition of an image, f = u0 { + v0, where [u0,v0] is the minimizer of a Jfunctional, J(f, λ0; X, Y) = infu+v=f ‖u‖X + λ0‖v ‖ p} Y. Such minimizers are standard tools for image manipulations
Upper Bounds on Coarsening Rates
"... We consider two standard models of surfaceenergydriven coarsening: a constantmobility CahnHilliard equation, whose largetime behavior corresponds to MullinsSekerka dynamics; and a degeneratemobility CahnHilliard equation, whose largetime behavior corresponds to motion by surface diusion. ..."
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Cited by 50 (10 self)
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We consider two standard models of surfaceenergydriven coarsening: a constantmobility CahnHilliard equation, whose largetime behavior corresponds to MullinsSekerka dynamics; and a degeneratemobility CahnHilliard equation, whose largetime behavior corresponds to motion by surface diusion. Arguments based on scaling suggest that the typical length scale should behave as `(t) t in the rst case and `(t) t in the second. We prove a weak, onesided version of this assertion  showing, roughly speaking, that no solution can coarsen faster than the expected rate. Our result constrains the behavior in a timeaveraged sense rather than pointwise in time, and it constrains not the physical length scale but rather the perimeter per unit volume. The argument is simple and robust, combining the basic dissipation relations with an interpolation inequality and an ODE argument.
Stable image reconstruction using total variation minimization
 SIAM Journal on Imaging Sciences
, 2013
"... This article presents nearoptimal guarantees for accurate and robust image recovery from undersampled noisy measurements using total variation minimization, and our results may be the first of this kind. In particular, we show that from O(s log(N)) nonadaptive linear measurements, an image can be ..."
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Cited by 48 (2 self)
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This article presents nearoptimal guarantees for accurate and robust image recovery from undersampled noisy measurements using total variation minimization, and our results may be the first of this kind. In particular, we show that from O(s log(N)) nonadaptive linear measurements, an image can be reconstructed to within the best sterm approximation of its gradient, up to a logarithmic factor. Along the way, we prove a strengthened Sobolev inequality for functions lying in the null space of a suitably incoherent matrix. 1
Bandelet Image Approximation and Compression
 SIAM JOURNAL OF MULTISCALE MODELING AND SIMULATION
, 2005
"... Finding efficient geometric representations of images is a central issue to improving image compression and noise removal algorithms. We introduce bandelet orthogonal bases and frames that are adapted to the geometric regularity of an image. Images are approximated by finding a best bandelet basis o ..."
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Cited by 36 (4 self)
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Finding efficient geometric representations of images is a central issue to improving image compression and noise removal algorithms. We introduce bandelet orthogonal bases and frames that are adapted to the geometric regularity of an image. Images are approximated by finding a best bandelet basis or frame that produces a sparse representation. For functions that are uniformly regular outside a set of edge curves that are geometrically regular, the main theorem proves that bandelet approximations satisfy an optimal asymptotic error decay rate. A bandelet image compression scheme is derived. For computational applications, a fast discrete bandelet transform algorithm is introduced, with a fast best basis search which preserves asymptotic approximation and coding error decay rates.
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”.
Bandelet image approximation and compression
 SIAM J. Multiscale Model. Simul
, 2005
"... Abstract. Finding efficient geometric representations of images is a central issue to improve image compression and noise removal algorithms. We introduce bandelet orthogonal bases and frames that are adapted to the geometric regularity of an image. Images are approximated by finding a best bandelet ..."
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Cited by 20 (0 self)
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Abstract. Finding efficient geometric representations of images is a central issue to improve image compression and noise removal algorithms. We introduce bandelet orthogonal bases and frames that are adapted to the geometric regularity of an image. Images are approximated by finding a best bandelet basis or frame that produces a sparse representation. For functions that are uniformly regular outside a set of edge curves that are geometrically regular, the main theorem proves that bandelet approximations satisfy an optimal asymptotic error decay rate. A bandelet image compression scheme is derived. For computational applications, a fast discrete bandelet transform algorithm is introduced, with a fast best basis search which preserves asymptotic approximation and coding error decay rates.
Correspondences between wavelet shrinkage and nonlinear diffusion
 in: L.D. Griffin and M. Lillholm (Eds.), ScaleSpace Methods in Computer Vision, Lecture Notes in Computer Science
, 2003
"... Abstract. We study the connections between discrete onedimensional schemes for nonlinear diffusion and shiftinvariant Haar wavelet shrinkage. We show that one step of (stabilised) explicit discretisation of nonlinear diffusion can be expressed in terms of wavelet shrinkage on a single spatial leve ..."
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Cited by 19 (5 self)
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Abstract. We study the connections between discrete onedimensional schemes for nonlinear diffusion and shiftinvariant Haar wavelet shrinkage. We show that one step of (stabilised) explicit discretisation of nonlinear diffusion can be expressed in terms of wavelet shrinkage on a single spatial level. This equivalence allows a fruitful exchange of ideas between the two fields. In this paper we derive new wavelet shrinkage functions from existing diffusivity functions, and identify some previously used shrinkage functions as corresponding to well known diffusivities. We demonstrate experimentally that some of the diffusioninspired shrinkage functions are among the best for translationinvariant multiscale wavelet shrinkage denoising. 1