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224
Fast image recovery using variable splitting and constrained optimization
 IEEE Trans. Image Process
, 2010
"... Abstract—We propose a new fast algorithm for solving one of the standard formulations of image restoration and reconstruction which consists of an unconstrained optimization problem where the objective includes an `2 datafidelity term and a nonsmooth regularizer. This formulation allows both wavele ..."
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Cited by 126 (10 self)
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Abstract—We propose a new fast algorithm for solving one of the standard formulations of image restoration and reconstruction which consists of an unconstrained optimization problem where the objective includes an `2 datafidelity term and a nonsmooth regularizer. This formulation allows both waveletbased (with orthogonal or framebased representations) regularization or totalvariation regularization. Our approach is based on a variable splitting to obtain an equivalent constrained optimization formulation, which is then addressed with an augmented Lagrangian method. The proposed algorithm is an instance of the socalled alternating direction method of multipliers, for which convergence has been proved. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is faster than the current state of the art methods. Index Terms—Augmented Lagrangian, compressive sensing, convex optimization, image reconstruction, image restoration,
Fast image deconvolution using hyperlaplacian priors, supplementary material
, 2009
"... The heavytailed distribution of gradients in natural scenes have proven effective priors for a range of problems such as denoising, deblurring and superresolution. These distributions are well modeled by a hyperLaplacian p(x) ∝ e−kxα), typically with 0.5 ≤ α ≤ 0.8. However, the use of sparse ..."
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Cited by 109 (2 self)
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The heavytailed distribution of gradients in natural scenes have proven effective priors for a range of problems such as denoising, deblurring and superresolution. These distributions are well modeled by a hyperLaplacian p(x) ∝ e−kxα), typically with 0.5 ≤ α ≤ 0.8. However, the use of sparse distributions makes the problem nonconvex and impractically slow to solve for multimegapixel images. In this paper we describe a deconvolution approach that is several orders of magnitude faster than existing techniques that use hyperLaplacian priors. We adopt an alternating minimization scheme where one of the two phases is a nonconvex problem that is separable over pixels. This perpixel subproblem may be solved with a lookup table (LUT). Alternatively, for two specific values of α, 1/2 and 2/3 an analytic solution can be found, by finding the roots of a cubic and quartic polynomial, respectively. Our approach (using either LUTs or analytic formulae) is able to deconvolve a 1 megapixel image in less than ∼3 seconds, achieving comparable quality to existing methods such as iteratively reweighted least squares (IRLS) that take ∼20 minutes. Furthermore, our method is quite general and can easily be extended to related image processing problems, beyond the deconvolution application demonstrated. 1
An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems
 IEEE Trans. Image Process
, 2011
"... Abstract—We propose a new fast algorithm for solving one of the standard approaches to illposed linear inverse problems (IPLIP), where a (possibly nonsmooth) regularizer is minimized under the constraint that the solution explains the observations sufficiently well. Although the regularizer and con ..."
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Cited by 92 (9 self)
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Abstract—We propose a new fast algorithm for solving one of the standard approaches to illposed linear inverse problems (IPLIP), where a (possibly nonsmooth) regularizer is minimized under the constraint that the solution explains the observations sufficiently well. Although the regularizer and constraint are usually convex, several particular features of these problems (huge dimensionality, nonsmoothness) preclude the use of offtheshelf optimization tools and have stimulated a considerable amount of research. In this paper, we propose a new efficient algorithm to handle one class of constrained problems (often known as basis pursuit denoising) tailored to image recovery applications. The proposed algorithm, which belongs to the family of augmented Lagrangian methods, can be used to deal with a variety of imaging IPLIP, including deconvolution and reconstruction from compressive observations (such as MRI), using either totalvariation or waveletbased (or, more generally, framebased) regularization. The proposed algorithm is an instance of the socalled alternating direction method of multipliers, for which convergence sufficient conditions are known; we show that these conditions are satisfied by the proposed algorithm. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is a strong contender for the stateoftheart. Index Terms—Convex optimization, frames, image reconstruction, image restoration, inpainting, totalvariation. A. Problem Formulation
TwoPhase Kernel Estimation for Robust Motion Deblurring
"... Abstract. We discuss a few new motion deblurring problems that are significant to kernel estimation and nonblind deconvolution. We found that strong edges do not always profit kernel estimation, but instead under certain circumstance degrade it. This finding leads to a new metric to measure the use ..."
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Cited by 92 (4 self)
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Abstract. We discuss a few new motion deblurring problems that are significant to kernel estimation and nonblind deconvolution. We found that strong edges do not always profit kernel estimation, but instead under certain circumstance degrade it. This finding leads to a new metric to measure the usefulness of image edges in motion deblurring and a gradient selection process to mitigate their possible adverse effect. We also propose an efficient and highquality kernel estimation method based on using the spatial prior and the iterative support detection (ISD) kernel refinement, which avoids hard threshold of the kernel elements to enforce sparsity. We employ the TVℓ1 deconvolution model, solved with a new variable substitution scheme to robustly suppress noise. 1
Motion Detail Preserving Optical Flow Estimation
"... We discuss the cause of a severe optical flow estimation problem that fine motion structures cannot always be correctly reconstructed in the commonly employed multiscale variational framework. Our major finding is that significant and abrupt displacement transition wrecks smallscale motion structur ..."
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Cited by 72 (1 self)
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We discuss the cause of a severe optical flow estimation problem that fine motion structures cannot always be correctly reconstructed in the commonly employed multiscale variational framework. Our major finding is that significant and abrupt displacement transition wrecks smallscale motion structures in the coarsetofine refinement. A novel optical flow estimation method is proposed in this paper to address this issue, which reduces the reliance of the flow estimates on their initial values propagated from the coarser level and enables recovering many motion details in each scale. The contribution of this paper also includes adaption of the objective function and development of a new optimization procedure. The effectiveness of our method is borne out by experiments for both large and smalldisplacement optical flow estimation.
Alternating direction augmented Lagrangian methods for semidefinite programming
, 2009
"... We present an alternating direction method based on an augmented Lagrangian framework for solving semidefinite programming (SDP) problems in standard form. At each iteration, the algorithm, also known as a twosplitting scheme, minimizes the dual augmented Lagrangian function sequentially with resp ..."
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Cited by 68 (2 self)
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We present an alternating direction method based on an augmented Lagrangian framework for solving semidefinite programming (SDP) problems in standard form. At each iteration, the algorithm, also known as a twosplitting scheme, minimizes the dual augmented Lagrangian function sequentially with respect to the Lagrange multipliers corresponding to the linear constraints, then the dual slack variables and finally the primal variables, while in each minimization keeping the other variables fixed. Convergence is proved by using a fixedpoint argument. A multiplesplitting algorithm is then proposed to handle SDPs with inequality constraints and positivity constraints directly without transforming them to the equality constraints in standard form. Finally, numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems are presented to demonstrate the robustness and efficiency of our algorithm.
Deconvolutional networks
 In CVPR
, 2010
"... Building robust low and midlevel image representations, beyond edge primitives, is a longstanding goal in vision. Many existing feature detectors spatially pool edge information which destroys cues such as edge intersections, parallelism and symmetry. We present a learning framework where features ..."
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Cited by 63 (4 self)
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Building robust low and midlevel image representations, beyond edge primitives, is a longstanding goal in vision. Many existing feature detectors spatially pool edge information which destroys cues such as edge intersections, parallelism and symmetry. We present a learning framework where features that capture these midlevel cues spontaneously emerge from image data. Our approach is based on the convolutional decomposition of images under a sparsity constraint and is totally unsupervised. By building a hierarchy of such decompositions we can learn rich feature sets that are a robust image representation for both the analysis and synthesis of images. 1.
Dequantizing compressed sensing: When oversampling and nongaussian constraints combine
, 2009
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Geometric Applications of the Split Bregman Method: Segmentation and Surface Reconstruction
, 2009
"... Variational models for image segmentation have many applications, but can be slow to compute. Recently, globally convex segmentation models have been introduced which are very reliable, but contain TVregularizers, making them difficult to compute. The previously introduced Split Bregman method is a ..."
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Cited by 58 (7 self)
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Variational models for image segmentation have many applications, but can be slow to compute. Recently, globally convex segmentation models have been introduced which are very reliable, but contain TVregularizers, making them difficult to compute. The previously introduced Split Bregman method is a technique for fast minimization of L1 regularized functionals, and has been applied to denoising and compressed sensing problems. By applying the Split Bregman concept to image segmentation problems, we build fast solvers which can outperform more conventional schemes, such as duality based methods and graphcuts. We also consider the related problem of surface reconstruction from unorganized data points, which is used for constructing level set representations in 3 dimensions.
Restoration of Poissonian images using alternating direction optimization
 IEEE Trans. Image Process
, 2010
"... Abstract—Much research has been devoted to the problem of restoring Poissonian images, namely for medical and astronomical applications. However, the restoration of these images using stateoftheart regularizers (such as those based upon multiscale representations or total variation) is still an a ..."
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Cited by 53 (5 self)
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Abstract—Much research has been devoted to the problem of restoring Poissonian images, namely for medical and astronomical applications. However, the restoration of these images using stateoftheart regularizers (such as those based upon multiscale representations or total variation) is still an active research area, since the associated optimization problems are quite challenging. In this paper, we propose an approach to deconvolving Poissonian images, which is based upon an alternating direction optimization method. The standard regularization [or maximum a posteriori (MAP)] restoration criterion, which combines the Poisson loglikelihood with a (nonsmooth) convex regularizer (logprior), leads to hard optimization problems: the loglikelihood is nonquadratic and nonseparable, the regularizer is nonsmooth, and there is a nonnegativity constraint. Using standard convex analysis tools, we present sufficient conditions for existence and uniqueness of solutions of these optimization problems, for several types of regularizers: totalvariation, framebased analysis, and framebased synthesis. We attack these problems with an instance of the alternating direction method of multipliers (ADMM), which belongs to the family of augmented Lagrangian algorithms. We study sufficient conditions for convergence and show that these are satisfied, either under totalvariation or framebased (analysis and synthesis) regularization. The resulting algorithms are shown to outperform alternative stateoftheart methods, both in terms of speed and restoration accuracy. Index Terms—Alternating direction methods, augmented Lagrangian, convex optimization, image deconvolution, image restoration, Poisson images. I.