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21
Elad M 2003 Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization
 Proc. Natl Acad. Sci. USA 100 2197–202
"... Given a ‘dictionary ’ D = {dk} of vectors dk, we seek to represent a signal S as a linear combination S = ∑ k γ(k)dk, with scalar coefficients γ(k). In particular, we aim for the sparsest representation possible. In general, this requires a combinatorial optimization process. Previous work considere ..."
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Cited by 368 (32 self)
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Given a ‘dictionary ’ D = {dk} of vectors dk, we seek to represent a signal S as a linear combination S = ∑ k γ(k)dk, with scalar coefficients γ(k). In particular, we aim for the sparsest representation possible. In general, this requires a combinatorial optimization process. Previous work considered the special case where D is an overcomplete system consisting of exactly two orthobases, and has shown that, under a condition of mutual incoherence of the two bases, and assuming that S has a sufficiently sparse representation, this representation is unique and can be found by solving a convex optimization problem: specifically, minimizing the ℓ1 norm of the coefficients γ. In this paper, we obtain parallel results in a more general setting, where the dictionary D can arise from two or several bases, frames, or even less structured systems. We introduce the Spark, ameasure of linear dependence in such a system; it is the size of the smallest linearly dependent subset (dk). We show that, when the signal S has a representation using less than Spark(D)/2 nonzeros, this representation is necessarily unique.
Stable recovery of sparse overcomplete representations in the presence of noise
 IEEE TRANS. INFORM. THEORY
, 2006
"... Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes t ..."
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Cited by 291 (20 self)
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Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system. Considering an ideal underlying signal that has a sufficiently sparse representation, it is assumed that only a noisy version of it can be observed. Assuming further that the overcomplete system is incoherent, it is shown that the optimally sparse approximation to the noisy data differs from the optimally sparse decomposition of the ideal noiseless signal by at most a constant multiple of the noise level. As this optimalsparsity method requires heavy (combinatorial) computational effort, approximation algorithms are considered. It is shown that similar stability is also available using the basis and the matching pursuit algorithms. Furthermore, it is shown that these methods result in sparse approximation of the noisy data that contains only terms also appearing in the unique sparsest representation of the ideal noiseless sparse signal.
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
New Multiscale Transforms, Minimum Total Variation Synthesis: Applications to EdgePreserving Image Reconstruction
, 2001
"... This paper describes newly invented multiscale transforms known under the name of the ridgelet [6] and the curvelet transforms [9, 8]. These systems combine ideas of multiscale analysis and geometry. Inspired by some recent work on digital Radon transforms [1], we then present very effective and acc ..."
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Cited by 76 (11 self)
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This paper describes newly invented multiscale transforms known under the name of the ridgelet [6] and the curvelet transforms [9, 8]. These systems combine ideas of multiscale analysis and geometry. Inspired by some recent work on digital Radon transforms [1], we then present very effective and accurate numerical implementations with computational complexities of at most N log N. In the second part of the paper, we propose to combine these new expansions with the Total Variation minimization principle for the reconstruction of an object whose curvelet coefficients are known only approximately: quantized, thresholded, noisy coefficients, etc. We set up a convex optimization problem and seek a reconstruction that has minimum Total Variation under the constraint that its coefficients do not exhibit a large discrepancy from the the data available on the coefficients of the unknown object. We will present a series of numerical experiments which clearly demonstrate the remarkable potential of this new methodology for image compression, image reconstruction and image ‘denoising.’
Beamlets and Multiscale Image Analysis
 in Multiscale and Multiresolution Methods
, 2001
"... We describe a framework for multiscale image analysis in which line segments play a role analogous to the role played by points in wavelet analysis. ..."
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Cited by 54 (16 self)
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We describe a framework for multiscale image analysis in which line segments play a role analogous to the role played by points in wavelet analysis.
Sparse Representations are Most Likely to be the Sparsest Possible
 EURASIP Journal on Applied Signal Processing, Paper No. 96247
, 2004
"... and a full rank matrix D with N < L, we define the signal's overcomplete representations as all # satisfying S = D#. Among all the possible solutions, we have special interest in the sparsest one  the one minimizing 0 . Previous work has established that a representation is unique if i ..."
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Cited by 12 (2 self)
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and a full rank matrix D with N < L, we define the signal's overcomplete representations as all # satisfying S = D#. Among all the possible solutions, we have special interest in the sparsest one  the one minimizing 0 . Previous work has established that a representation is unique if it is sparse enough, requiring 0 < Spark(D)/2.
Fast Xray and beamlet transforms for threedimensional data
 in Modern Signal Processing
, 2002
"... Abstract. Threedimensional volumetric data are becoming increasingly available in a wide range of scientific and technical disciplines. With the right tools, we can expect such data to yield valuable insights about many important phenomena in our threedimensional world. In this paper, we develop t ..."
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Cited by 12 (8 self)
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Abstract. Threedimensional volumetric data are becoming increasingly available in a wide range of scientific and technical disciplines. With the right tools, we can expect such data to yield valuable insights about many important phenomena in our threedimensional world. In this paper, we develop tools for the analysis of 3D data which may contain structures built from lines, line segments, and filaments. These tools come in two main forms: (a) Monoscale: the Xray transform, offering the collection of line integrals along a wide range of lines running through the image — at all different orientations and positions; and (b) Multiscale: the (3D) beamlet transform, offering the collection of line integrals along line segments which, in addition to ranging through a wide collection of locations and positions, also occupy a wide range of scales. We describe different strategies for computing these transforms and several basic applications, for example in finding faint structures buried in noisy data. 1.
Multiscale Geometric Image Processing
 in Proceedings of the SPIE
, 2003
"... Since their introduction a little more than 10 years ago, wavelets have revolutionized image processing. Wavelet based algorithms define the stateoftheart for applications including image coding (JPEG2000), restoration, and segmentation. Despite their success, wavelets have significant shortcomin ..."
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Cited by 8 (2 self)
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Since their introduction a little more than 10 years ago, wavelets have revolutionized image processing. Wavelet based algorithms define the stateoftheart for applications including image coding (JPEG2000), restoration, and segmentation. Despite their success, wavelets have significant shortcomings in their treatment of edges. Wavelets do not parsimoniously capture even the simplest geometrical structure in images, and wavelet based processing algorithms often produce images with ringing around the edges.
Wavelets and Imaging Informatics: A Review of the Literature
"... Modern medicine is a field that has been revolutionized by the emergence of computer and imaging technology. It is increasingly difficult, however, to manage the ever growing enormous mount of medical imaging information available in digital formats. Numerous techniques have been developed to mak ..."
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Cited by 7 (0 self)
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Modern medicine is a field that has been revolutionized by the emergence of computer and imaging technology. It is increasingly difficult, however, to manage the ever growing enormous mount of medical imaging information available in digital formats. Numerous techniques have been developed to make the imaging information more easily accessible and to perform analysis automatically. Among these techniques, wavelet transforms have proven prominently useful not only for biomedical imaging but also for signal and image processing in general. Wavelet transforms decompose a signal into frequency bands, the width of which are determined by a dyadic scheme. This particular way of dividing frequency bands matches the statistical properties of most images very well. During the past decade, there has been active research in applying wavelets to various aspects of imaging informatics, including compression, enhancements, analysis, classification, and retrieval. This review represents a survey of the most significant practical and theoretical advances in the field of waveletbased imaging informatics.
Greedy Basis Pursuit
, 2006
"... We introduce Greedy Basis Pursuit (GBP), a new algorithm for computing signal representations using overcomplete dictionaries. GBP is rooted in computational geometry and exploits an equivalence between minimizing the ℓ 1norm of the representation coefficients and determining the intersection of th ..."
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Cited by 7 (0 self)
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We introduce Greedy Basis Pursuit (GBP), a new algorithm for computing signal representations using overcomplete dictionaries. GBP is rooted in computational geometry and exploits an equivalence between minimizing the ℓ 1norm of the representation coefficients and determining the intersection of the signal with the convex hull of the dictionary. GBP unifies the different advantages of previous algorithms: like standard approaches to Basis Pursuit, GBP computes representations that have minimum ℓ 1norm; like greedy algorithms such as Matching Pursuit, GBP builds up representations, sequentially selecting atoms. We describe the algorithm, demonstrate its performance, and provide code. Experiments show that GBP can provide a fast alternative to standard linear programming approaches to Basis Pursuit.