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Linearized Alternating Direction Method with Gaussian Back Substitution for Separable Convex Programming
, 2011
"... Abstract. Recently, we have proposed to combine the alternating direction method (ADM) with a Gaussian back substitution procedure for solving the convex minimization model with linear constraints and a general separable objective function, i.e., the objective function is the sum of many functions w ..."
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Cited by 35 (3 self)
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Abstract. Recently, we have proposed to combine the alternating direction method (ADM) with a Gaussian back substitution procedure for solving the convex minimization model with linear constraints and a general separable objective function, i.e., the objective function is the sum of many functions without coupled variables. In this paper, we further study this topic and show that the decomposed subproblems in the ADM procedure can be substantially alleviated by linearizing the involved quadratic terms arising from the augmented Lagrangian penalty on the model’s linear constraints. When the resolvent operators of the separable functions in the objective have closedform representations, embedding the linearization into the ADM subproblems becomes necessary to yield easy subproblems with closedform solutions. We thus show theoretically that the blend of ADM, Gaussian back substitution and linearization works effectively for the separable convex minimization model under consideration.
Constrained total variation deblurring models and fast algorithms based on alternating direction method of multipliers
 SIAM Journal on Imaging Sciences
"... Abstract. The total variation (TV) model is attractive for being able to preserve sharp attributes in images. However, the restored images from TVbased methods do not usually stay in a given dynamic range, and hence projection is required to bring them back into the dynamic range for visual present ..."
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Cited by 7 (0 self)
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Abstract. The total variation (TV) model is attractive for being able to preserve sharp attributes in images. However, the restored images from TVbased methods do not usually stay in a given dynamic range, and hence projection is required to bring them back into the dynamic range for visual presentation or for storage in digital media. This will affect the accuracy of the restoration as the projected image will no longer be the minimizer of the given TV model. In this paper, we show that one can get much more accurate solutions by imposing box constraints on the TV models and solving the resulting constrained models. Our numerical results show that for some images where there are many pixels with values lying on the boundary of the dynamic range, the gain can be as great as 10.28dB in peak signaltonoise ratio. One traditional hinderance of using the constrained model is that it is difficult to solve. However, in this paper, we propose to use the alternating direction method of multipliers (ADMM) to solve the constrained models. This leads to a fast and convergent algorithm that is applicable for both Gaussian and impulse noise. Numerical results show that our ADMM algorithm is better than some stateoftheart algorithms for unconstrained models both in terms of accuracy and robustness with respect to the regularization parameter. Key words. Total variation, deblurring, alternating direction method of multipliers, box constraint AMS subject classifications. 68U10, 65J22, 65K10, 65T50, 90C25
Linearized alternating direction method for constrained linear leastsquares problem
, 2011
"... In this paper, we apply the alternating direction method (ADM) to solve a constrained linear leastsquares problem where the objective function is a sum of two leastsquares terms and the constraints are box constraints. Using ADM, we decompose the original problem into two easier leastsquares sub ..."
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Cited by 2 (2 self)
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In this paper, we apply the alternating direction method (ADM) to solve a constrained linear leastsquares problem where the objective function is a sum of two leastsquares terms and the constraints are box constraints. Using ADM, we decompose the original problem into two easier leastsquares subproblems at each iteration. To speed up the inner iteration, we linearize the subproblems whenever their closedform solutions do not exist. We prove the convergence of the resulting algorithm and apply it to solve some image deblurring problems. We show the efficiency of our algorithm by comparing it with Newtontype methods.
6 POSITIVELY CONSTRAINED TOTAL VARIATION PENALIZED IMAGE RESTORATION
, 1793
"... The total variation (TV) minimization models are widely used in image processing, mainly due to their remarkable ability in preserving edges. There are many methods for solving the TV model. These methods, however, seldom consider the positivity constraint one should impose on imageprocessing probl ..."
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Cited by 1 (0 self)
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The total variation (TV) minimization models are widely used in image processing, mainly due to their remarkable ability in preserving edges. There are many methods for solving the TV model. These methods, however, seldom consider the positivity constraint one should impose on imageprocessing problems. In this paper we develop and implement a new approach for TV image restoration. Our method is based on the multiplicative iterative algorithm originally developed for tomographic image reconstruction. The advantages of our algorithm are that it is very easy to derive and implement under different image noise models and it respects the positivity constraint. Our method can be applied to various noise models commonly used in image restoration, such as the Gaussian noise model, the Poisson noise model, and the impulsive noise model. In the numerical tests, we apply our algorithm to deblur images corrupted by Gaussian noise. The results show that our method give better restored images than the forward–backward splitting algorithm.
Customized proximal point algorithms for linearly constrained convex minimization and saddlepoint problems: a uniform approach
, 2011
"... Abstract. This paper takes a uniform look at the customized applications of proximal point algorithm (PPA) to two classes of problems: the linearly constrained convex minimization problem with a generic or separable objective function and a saddlepoint problem. We model these two classes of problem ..."
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Abstract. This paper takes a uniform look at the customized applications of proximal point algorithm (PPA) to two classes of problems: the linearly constrained convex minimization problem with a generic or separable objective function and a saddlepoint problem. We model these two classes of problems uniformly by a mixed variational inequality, and show how PPA with customized proximal parameters can yield favorable algorithms, which are able to exploit the structure of the models fully. Our customized PPA revisit turns out to be a uniform approach in designing a number of efficient algorithms, which are competitive with, or even more efficient than some benchmark methods in the existing literature such as the augmented Lagrangian method, the alternating direction method, the split inexact Uzawa method, and a class of primaldual methods, etc. From the PPA perspective, the global convergence and the O(1/t) convergence rate for this series of algorithms are established in a uniform way.
IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT 1 Phoenix: A Weightbased Network Coordinate System Using Matrix Factorization
"... Abstract—Network coordinate (NC) systems provide a lightweight and scalable way for predicting the distances, i.e., roundtrip latencies among Internet hosts. Most existing NC systems embed hosts into a low dimensional Euclidean space. Unfortunately, the persistent occurrence of Triangle Inequality ..."
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Abstract—Network coordinate (NC) systems provide a lightweight and scalable way for predicting the distances, i.e., roundtrip latencies among Internet hosts. Most existing NC systems embed hosts into a low dimensional Euclidean space. Unfortunately, the persistent occurrence of Triangle Inequality Violation (TIV) on the Internet largely limits the distance prediction accuracy of those NC systems. Some alternative systems aim at handling the persistent TIV, however, they only achieve comparable prediction accuracy with Euclidean distance based NC systems. In this paper, we propose an NC system, socalled Phoenix, which is based on the matrix factorization model. Phoenix introduces a weight to each reference NC and trusts the NCs with higher weight values more than the others. The weightbased mechanism can substantially reduce the impact of the error propagation. Using the representative aggregate data sets and the newly measured dynamic data set collected from the Internet, our simulations show that Phoenix achieves significantly higher prediction accuracy than other NC systems. We also show that Phoenix quickly converges to steady state, performs well under host churn, handles the drift of the NCs successfully by using regularization, and is robust against measurement anomalies. Phoenix achieves a scalable yet accurate endtoend distances monitoring. In addition, we study how well an NC system can characterize the TIV property on the Internet by introducing two new quantitative metrics, socalled RERP L and AERP L. We show that Phoenix is able to characterize TIV better than other existing NC systems.
Phoenix: A WeightBased Network Coordinate System Using Matrix Factorization
"... Abstract—Network coordinate (NC) systems provide a lightweight and scalable way for predicting the distances, i.e., roundtrip latencies among Internet hosts. Most existing NC systems embed hosts into a low dimensional Euclidean space. Unfortunately, the persistent occurrence of Triangle Inequality ..."
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Abstract—Network coordinate (NC) systems provide a lightweight and scalable way for predicting the distances, i.e., roundtrip latencies among Internet hosts. Most existing NC systems embed hosts into a low dimensional Euclidean space. Unfortunately, the persistent occurrence of Triangle Inequality Violation (TIV) on the Internet largely limits the distance prediction accuracy of those NC systems. Some alternative systems aim at handling the persistent TIV, however, they only achieve comparable prediction accuracy with Euclidean distance based NC systems. In this paper, we propose an NC system, socalled Phoenix, which is based on the matrix factorization model. Phoenix introduces a weight to each reference NC and trusts the NCs with higher weight values more than the others. The weightbased mechanism can substantially reduce the impact of the error propagation. Using the representative aggregate data sets and the newly measured dynamic data set collected from the Internet, our simulations show that Phoenix achieves significantly higher prediction accuracy than other NC systems. We also show that Phoenix quickly converges to steady state, performs well under host churn, handles the drift of the NCs successfully by using regularization, and is robust against measurement anomalies. Phoenix achieves a scalable yet accurate endtoend distances monitoring. In addition, we study how well an NC system can characterize the TIV property on the Internet by introducing two new quantitative metrics, socalled
IEEE TRANSACTIONS ON IMAGE PROCESSING 1
"... A multiplicative iterative algorithm for boxconstrained penalized likelihood image restoration Raymond H. Chan and Jun Ma Abstract—Image restoration is a computationally intensive problem as a large number of pixel values have to be determined. Since the pixel values of digital images can attain on ..."
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A multiplicative iterative algorithm for boxconstrained penalized likelihood image restoration Raymond H. Chan and Jun Ma Abstract—Image restoration is a computationally intensive problem as a large number of pixel values have to be determined. Since the pixel values of digital images can attain only a finite number of values (e.g. 8bit images can have only 256 gray levels), one would like to recover an image within some dynamic range. This leads to the imposition of box constraints on the pixel values. The traditional gradient projection methods for constrained optimization can be used to impose box constraints, but they may suffer from either slow convergence or repeated searching for active sets in each iteration. In this paper, we develop a new boxconstrained multiplicative iterative (BCMI) algorithm for boxconstrained image restoration. The BCMI algorithm just requires pixelwise updates in each iteration, and there is no need to invert any matrices. We give the convergence proof of this algorithm and apply it to TV image restoration problems where the observed blurry images contain Poisson, Gaussian or saltandpepper noises.
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"... be inserted by the editor) On efficiency of nonmonotone Armijotype line searches ..."
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be inserted by the editor) On efficiency of nonmonotone Armijotype line searches