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65
Bregman iterative algorithms for ℓ1minimization with applications to compressed sensing
 SIAM J. IMAGING SCI
, 2008
"... We propose simple and extremely efficient methods for solving the basis pursuit problem min{‖u‖1: Au = f,u ∈ R n}, which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number of 1 insta ..."
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Cited by 84 (15 self)
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We propose simple and extremely efficient methods for solving the basis pursuit problem min{‖u‖1: Au = f,u ∈ R n}, which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number of 1 instances of the unconstrained problem minu∈Rn μ‖u‖1 + 2 ‖Au−fk ‖ 2 2 for given matrix A and vector f k. We show analytically that this iterative approach yields exact solutions in a finite number of steps and present numerical results that demonstrate that as few as two to six iterations are sufficient in most cases. Our approach is especially useful for many compressed sensing applications where matrixvector operations involving A and A ⊤ can be computed by fast transforms. Utilizing a fast fixedpoint continuation solver that is based solely on such operations for solving the above unconstrained subproblem, we were able to quickly solve huge instances of compressed sensing problems on a standard PC.
Visual Modelling of
 Complex Business Processes with Trees, Overlays and DistortionBased Displays, Proc VLHCC’07, IEEE CS
"... evolution laws for thin crystalline films: ..."
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Analysis and generalizations of the linearized Bregman method
 SIAM J. IMAGING SCI
, 2010
"... This paper analyzes and improves the linearized Bregman method for solving the basis pursuit and related sparse optimization problems. The analysis shows that the linearized Bregman method has the exact regularization property; namely, it converges to an exact solution of the basis pursuit problem ..."
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Cited by 36 (9 self)
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This paper analyzes and improves the linearized Bregman method for solving the basis pursuit and related sparse optimization problems. The analysis shows that the linearized Bregman method has the exact regularization property; namely, it converges to an exact solution of the basis pursuit problem whenever its smooth parameter α is greater than a certain value. The analysis is based on showing that the linearized Bregman algorithm is equivalent to gradient descent applied to a certain dual formulation. This result motivates generalizations of the algorithm enabling the use of gradientbased optimization techniques such as line search, Barzilai–Borwein, limited memory BFGS (LBFGS), nonlinear conjugate gradient, and Nesterov’s methods. In the numerical simulations, the two proposed implementations, one using Barzilai–Borwein steps with nonmonotone line search and the other using LBFGS, gave more accurate solutions in much shorter times than the basic implementation of the linearized Bregman method with a socalled kicking technique.
Removing multiplicative noise by DouglasRachford splitting methods
 Journal of Mathematical Imaging and Vision
"... In this paper, we consider a variational restoration model consisting of the Idivergence as data fitting term and the total variation seminorm or nonlocal means as regularizer for removing multiplicative Gamma noise. Although the Idivergence is the typical data fitting term when dealing with Pois ..."
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Cited by 28 (1 self)
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In this paper, we consider a variational restoration model consisting of the Idivergence as data fitting term and the total variation seminorm or nonlocal means as regularizer for removing multiplicative Gamma noise. Although the Idivergence is the typical data fitting term when dealing with Poisson noise we substantiate why it is also appropriate for cleaning Gamma noise. We propose to compute the minimizers of our restoration functionals by applying DouglasRachford splitting techniques, resp. alternating direction methods of multipliers. For a particular splitting, we present a semiimplicit scheme to solve the involved nonlinear systems of equations and prove its Qlinear convergence. Finally, we demonstrate the performance of our methods by numerical examples. 1
Compressed Synthetic Aperture Radar
, 2010
"... In this paper, we introduce a new synthetic aperture radar (SAR) imaging modality which can provide a highresolution map of the spatial distribution of targets and terrain using a significantly reduced number of needed transmitted and/or received electromagnetic waveforms. This new imaging scheme, ..."
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Cited by 26 (3 self)
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In this paper, we introduce a new synthetic aperture radar (SAR) imaging modality which can provide a highresolution map of the spatial distribution of targets and terrain using a significantly reduced number of needed transmitted and/or received electromagnetic waveforms. This new imaging scheme, requires no new hardware components and allows the aperture to be compressed. It also presents many new applications and advantages which include strong resistance to countermesasures and interception, imaging much wider swaths and reduced onboard storage requirements.
Image cartoontexture decomposition and feature selection using the total variation regularized L 1 functional
, 2006
"... Abstract. This paper studies the model of minimizing total variation with an L 1norm fidelity term for decomposing a real image into the sum of cartoon and texture. This model is also analyzed and shown to be able to select features of an image according to their scales. 1 ..."
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Cited by 24 (4 self)
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Abstract. This paper studies the model of minimizing total variation with an L 1norm fidelity term for decomposing a real image into the sum of cartoon and texture. This model is also analyzed and shown to be able to select features of an image according to their scales. 1
Multiplicative noise removal using variable splitting and constrained optimization
 IEEE Transactions on Image Processing
, 2010
"... Abstract—Multiplicative noise (also known as speckle noise) models are central to the study of coherent imaging systems, such as synthetic aperture radar and sonar, and ultrasound and laser imaging. These models introduce two additional layers of difficulties with respect to the standard Gaussian ad ..."
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Cited by 21 (1 self)
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Abstract—Multiplicative noise (also known as speckle noise) models are central to the study of coherent imaging systems, such as synthetic aperture radar and sonar, and ultrasound and laser imaging. These models introduce two additional layers of difficulties with respect to the standard Gaussian additive noise scenario: 1) the noise is multiplied by (rather than added to) the original image; 2) the noise is not Gaussian, with Rayleigh and Gamma being commonly used densities. These two features of multiplicative noise models preclude the direct application of most stateoftheart algorithms, which are designed for solving unconstrained optimization problems where the objective has two terms: a quadratic data term (loglikelihood), reflecting the additive and Gaussian nature of the noise, plus a convex (possibly nonsmooth) regularizer (e.g., a total variation or waveletbased regularizer/prior). In this paper, we address these difficulties by: 1) converting the multiplicative model into an additive one by taking logarithms, as proposed by some other authors; 2) using variable splitting to obtain an equivalent constrained problem; and 3) dealing with this optimization problem using the augmented Lagrangian framework. A set of experiments shows that the proposed method, which we name MIDAL (multiplicative image denoising by augmented Lagrangian), yields stateoftheart results both in terms of speed and denoising performance. Index Terms—Augmented Lagrangian, Douglas–Rachford splitting, multiplicative noise, speckled images, synthetic aperture
Error estimation for Bregman iterations and inverse scale space methods in image restoration
 Computing
, 2008
"... In this paper we consider error estimation for image restoration problems based on generalized Bregman distances. This error estimation technique has been used to derive convergence rates of variational regularization schemes for linear and nonlinear inverse problems by the authors before (cf. [4, 2 ..."
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Cited by 21 (12 self)
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In this paper we consider error estimation for image restoration problems based on generalized Bregman distances. This error estimation technique has been used to derive convergence rates of variational regularization schemes for linear and nonlinear inverse problems by the authors before (cf. [4, 21, 22]), but so far it was not applied to image restoration in a systematic way. Due to the flexibility of the Bregman distances, this approach is particularly attractive for imaging tasks, where often singular energies (nondifferentiable, not strictly convex) are usedto achieve certain tasks such as preservation of edges. Besides the discussion of the variational image restoration schemes, our main goal in this paper is to extend the error estimation approach to iterative regularization schemes (and timecontinuous flows) that have emerged recently as multiscale restoration techniques and could improve some shortcomings of the variational schemes. We derive error estimates between the iterates and the exact image both in the case of clean and noisy data, the latter also giving indications on the choice of termination criteria. The error estimates are applied to various image restoration approaches such as denoising and decomposition by total variation and wavelet methods. We shall see that interesting results for various restoration approaches can be deduced from our general results by just exploring the structure of subgradients.
Multiplicative Noise Removal Using L1 Fidelity on Frame Coefficients
, 2008
"... We address the denoising of images contaminated with multiplicative noise, e.g. speckle noise. Classical ways to solve such problems are filtering, statistical (Bayesian) methods, variational methods, and methods that convert the multiplicative noise into additive noise (using a logarithmic function ..."
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Cited by 17 (4 self)
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We address the denoising of images contaminated with multiplicative noise, e.g. speckle noise. Classical ways to solve such problems are filtering, statistical (Bayesian) methods, variational methods, and methods that convert the multiplicative noise into additive noise (using a logarithmic function), shrinkage of the coefficients of the logimage data in a wavelet basis or in a frame, and transform back the result using an exponential function. We propose a method composed of several stages: we use the logimage data and apply a reasonable underoptimal hardthresholding on its curvelet transform; then we apply a variational method where we minimize a specialized criterion composed of an ℓ 1 datafitting to the thresholded coefficients and a Total Variation regularization (TV) term in the image domain; the restored image is an exponential of the obtained minimizer, weighted in a way that the mean of the original image is preserved. Our restored images combine the advantages of shrinkage and variational methods and avoid their main drawbacks. For the minimization stage, we propose a properly adapted fast minimization scheme based on DouglasRachford splitting. The existence of a minimizer of our specialized criterion being proven, we demonstrate the convergence of the minimization scheme. The obtained numerical results outperform the main alternative methods.
Error estimates for general fidelities
 Electronic Transactions on Numerical Analysis
"... Abstract. Appropriate error estimation for regularization methods in imaging and inverse problems is of enormous importance for controlling approximation properties and understanding types of solutions that are particularly favoured. In the case of linear problems, i.e., variational methods with qua ..."
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Cited by 10 (4 self)
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Abstract. Appropriate error estimation for regularization methods in imaging and inverse problems is of enormous importance for controlling approximation properties and understanding types of solutions that are particularly favoured. In the case of linear problems, i.e., variational methods with quadratic fidelity and quadratic regularization, the error estimation is wellunderstood under socalled source conditions. Significant progress for nonquadratic regularization functionals has been made recently after the introduction of the Bregman distance as an appropriate error measure. The other important generalization, namely for nonquadratic fidelities, has not been analyzed so far. In this paper we develop a framework for the derivation of error estimates in the case of rather general fidelities and highlight the importance of duality for the shape of the estimates. We then specialize the approach for several important fidelities in imaging (L 1, KullbackLeibler).