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220
Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging
 MAGNETIC RESONANCE IN MEDICINE 58:1182–1195
, 2007
"... The sparsity which is implicit in MR images is exploited to significantly undersample kspace. Some MR images such as angiograms are already sparse in the pixel representation; other, more complicated images have a sparse representation in some transform domain–for example, in terms of spatial finit ..."
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Cited by 538 (11 self)
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The sparsity which is implicit in MR images is exploited to significantly undersample kspace. Some MR images such as angiograms are already sparse in the pixel representation; other, more complicated images have a sparse representation in some transform domain–for example, in terms of spatial finitedifferences or their wavelet coefficients. According to the recently developed mathematical theory of compressedsensing, images with a sparse representation can be recovered from randomly undersampled kspace data, provided an appropriate nonlinear recovery scheme is used. Intuitively, artifacts due to random undersampling add as noiselike interference. In the sparse transform domain the significant coefficients stand out above the interference. A nonlinear thresholding scheme can recover the sparse coefficients, effectively recovering the image itself. In this article, practical incoherent undersampling schemes are developed and analyzed by means of their aliasing interference. Incoherence is introduced by pseudorandom variabledensity undersampling of phaseencodes. The reconstruction is performed by minimizing the ℓ1 norm of a transformed image, subject to data fidelity constraints. Examples demonstrate improved spatial resolution and accelerated acquisition for multislice fast spinecho brain imaging and 3D contrast enhanced angiography.
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 427 (36 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
Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit
, 2006
"... Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NPhard in general. We show here that for systems with ‘typical’/‘random ’ Φ, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra. Our pr ..."
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Cited by 274 (22 self)
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Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NPhard in general. We show here that for systems with ‘typical’/‘random ’ Φ, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra. Our proposal, Stagewise Orthogonal Matching Pursuit (StOMP), successively transforms the signal into a negligible residual. Starting with initial residual r0 = y, at the sth stage it forms the ‘matched filter ’ Φ T rs−1, identifies all coordinates with amplitudes exceeding a speciallychosen threshold, solves a leastsquares problem using the selected coordinates, and subtracts the leastsquares fit, producing a new residual. After a fixed number of stages (e.g. 10), it stops. In contrast to Orthogonal Matching Pursuit (OMP), many coefficients can enter the model at each stage in StOMP while only one enters per stage in OMP; and StOMP takes a fixed number of stages (e.g. 10), while OMP can take many (e.g. n). StOMP runs much faster than competing proposals for sparse solutions, such as ℓ1 minimization and OMP, and so is attractive for solving largescale problems. We use phase diagrams to compare algorithm performance. The problem of recovering a ksparse vector x0 from (y, Φ) where Φ is random n × N and y = Φx0 is represented by a point (n/N, k/n)
An iterative regularization method for total variationbased image restoration
 MULTISCALE MODEL. SIMUL.
, 2005
"... We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods by using total variation regu ..."
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Cited by 195 (29 self)
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We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods by using total variation regularization. We obtain rigorous convergence results and effective stopping criteria for the general procedure. The numerical results for denoising appear to give significant improvement over standard models, and preliminary results for deblurring/denoising are very encouraging.
Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions
, 2004
"... In this paper, we develop a robust uncertainty principle for finite signals in C N which states that for nearly all choices T, Ω ⊂ {0,..., N − 1} such that T  + Ω  ≍ (log N) −1/2 · N, there is no signal f supported on T whose discrete Fourier transform ˆ f is supported on Ω. In fact, we can mak ..."
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Cited by 181 (17 self)
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In this paper, we develop a robust uncertainty principle for finite signals in C N which states that for nearly all choices T, Ω ⊂ {0,..., N − 1} such that T  + Ω  ≍ (log N) −1/2 · N, there is no signal f supported on T whose discrete Fourier transform ˆ f is supported on Ω. In fact, we can make the above uncertainty principle quantitative in the sense that if f is supported on T, then only a small percentage of the energy (less than half, say) of ˆ f is concentrated on Ω. As an application of this robust uncertainty principle (QRUP), we consider the problem of decomposing a signal into a sparse superposition of spikes and complex sinusoids f(s) = � α1(t)δ(s − t) + � α2(ω)e i2πωs/N / √ N. t∈T We show that if a generic signal f has a decomposition (α1, α2) using spike and frequency locations in T and Ω respectively, and obeying ω∈Ω T  + Ω  ≤ Const · (log N) −1/2 · N, then (α1, α2) is the unique sparsest possible decomposition (all other decompositions have more nonzero terms). In addition, if T  + Ω  ≤ Const · (log N) −1 · N, then the sparsest (α1, α2) can be found by solving a convex optimization problem. Underlying our results is a new probabilistic approach which insists on finding the correct uncertainty relation or the optimally sparse solution for nearly all subsets but not necessarily all of them, and allows to considerably sharpen previously known results [9, 10]. In fact, we show that the fraction of sets (T, Ω) for which the above properties do not hold can be upper bounded by quantities like N −α for large values of α. The QRUP (and the application to finding sparse representations) can be extended to general pairs of orthogonal bases Φ1, Φ2 of C N. For nearly all choices Γ1, Γ2 ⊂ {0,..., N − 1} obeying Γ1  + Γ2  ≍ µ(Φ1, Φ2) −2 · (log N) −m, where m ≤ 6, there is no signal f such that Φ1f is supported on Γ1 and Φ2f is supported on Γ2 where µ(Φ1, Φ2) is the mutual coherence between Φ1 and Φ2.
A review of curvelets and recent applications
 IEEE Signal Processing Magazine
, 2009
"... Multiresolution methods are deeply related to image processing, biological and computer vision, scientific computing, etc. The curvelet transform is a multiscale directional transform which allows an almost optimal nonadaptive sparse representation of objects with edges. It has generated increasing ..."
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Cited by 128 (10 self)
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Multiresolution methods are deeply related to image processing, biological and computer vision, scientific computing, etc. The curvelet transform is a multiscale directional transform which allows an almost optimal nonadaptive sparse representation of objects with edges. It has generated increasing interest in the community of applied mathematics and signal processing over the past years. In this paper, we present a review on the curvelet transform, including its history beginning from wavelets, its logical relationship to other multiresolution multidirectional methods like contourlets and shearlets, its basic theory and discrete algorithm. Further, we consider recent applications in image/video processing, seismic exploration, fluid mechanics, simulation of partial different equations, and compressed sensing.
CurveletWavelet Regularized Split Bregman Iteration for Compressed Sensing
"... Compressed sensing is a new concept in signal processing. Assuming that a signal can be represented or approximated by only a few suitably chosen terms in a frame expansion, compressed sensing allows to recover this signal from much fewer samples than the ShannonNyquist theory requires. Many images ..."
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Cited by 119 (6 self)
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Compressed sensing is a new concept in signal processing. Assuming that a signal can be represented or approximated by only a few suitably chosen terms in a frame expansion, compressed sensing allows to recover this signal from much fewer samples than the ShannonNyquist theory requires. Many images can be sparsely approximated in expansions of suitable frames as wavelets, curvelets, wave atoms and others. Generally, wavelets represent pointlike features while curvelets represent linelike features well. For a suitable recovery of images, we propose models that contain weighted sparsity constraints in two different frames. Given the incomplete measurements f = Φu + ɛ with the measurement matrix Φ ∈ R K×N, K<<N, we consider a jointly sparsityconstrained optimization problem of the form argmin{‖ΛcΨcu‖1 + ‖ΛwΨwu‖1 + u 1 2‖f − Φu‖22}. Here Ψcand Ψw are the transform matrices corresponding to the two frames, and the diagonal matrices Λc, Λw contain the weights for the frame coefficients. We present efficient iteration methods to solve the optimization problem, based on Alternating Split Bregman algorithms. The convergence of the proposed iteration schemes will be proved by showing that they can be understood as special cases of the DouglasRachford Split algorithm. Numerical experiments for compressed sensing based Fourierdomain random imaging show good performances of the proposed curveletwavelet regularized split Bregman (CWSpB) methods,whereweparticularlyuseacombination of wavelet and curvelet coefficients as sparsity constraints.
A New Hybrid Method for Image Approximation using the Easy Path Wavelet Transform
"... The Easy Path Wavelet Transform (EPWT) 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 functi ..."
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Cited by 111 (4 self)
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The Easy Path Wavelet Transform (EPWT) 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 exploits the local correlations of the given data in a simple appropriate manner. However, the EPWT suffers from its adaptivity costs that arise from the storage of path vectors. In this paper, we propose a new hybrid method for image compression that exploits the advantages of the usual tensor product wavelet transform for the representation of smooth images and uses the EPWT for an efficient representation of edges and texture. Numerical results show the efficiency of this procedure. Key words. sparse data representation, tensor product wavelet transform, easy path wavelet transform, linear diffusion, smoothing filters, adaptive wavelet bases, Nterm approximation AMS Subject classifications. 41A25, 42C40, 68U10, 94A08 1
Featureenhanced synthetic aperture radar image formation based on nonquadratic regularization",
 IEEE Trans. Image Processing
, 2001
"... ABSTRACT Nonquadratic regularization based image formation is a recently proposed framework for featureenhanced radar imaging. Specific image formation techniques in this framework have so far focused on enhancing one type of feature, such as strong point scatterers, or smooth regions. However, m ..."
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Cited by 97 (9 self)
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ABSTRACT Nonquadratic regularization based image formation is a recently proposed framework for featureenhanced radar imaging. Specific image formation techniques in this framework have so far focused on enhancing one type of feature, such as strong point scatterers, or smooth regions. However, many scenes contain a number of such features. We develop an image formation technique that simultaneously enhances multiple types of features by posing the problem as one of sparse signal representation based on overcomplete dictionaries. Due to the complexvalued nature of the reflectivities in SAR, our new approach is designed to sparsely represent the magnitude of the complexvalued scattered field in terms of multiple features, which turns the image reconstruction problem into a joint optimization problem over the representation of the magnitude and the phase of the underlying field reflectivities. We formulate the mathematical framework needed for this method and propose an iterative solution for the corresponding joint optimization problem. We demonstrate the effectiveness of this approach on various SAR images.