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28
A primaldual algorithmic framework for constrained convex minimization,” arXiv preprint:1406.5403
, 2014
"... We present a primaldual algorithmic framework to obtain approximate solutions to a prototypical constrained convex optimization problem, and rigorously characterize how common structural assumptions affect the numerical efficiency. Our main analysis technique provides a fresh perspective on Nester ..."
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We present a primaldual algorithmic framework to obtain approximate solutions to a prototypical constrained convex optimization problem, and rigorously characterize how common structural assumptions affect the numerical efficiency. Our main analysis technique provides a fresh perspective on Nesterov’s excessive gap technique in a structured fashion and unifies it with smoothing and primaldual methods. For instance, through the choices of a dual smoothing strategy and a center point, our framework subsumes decomposition algorithms, augmented Lagrangian as well as the alternating direction methodofmultipliers methods as its special cases, and provides optimal convergence rates on the primal objective residual as well as the primal feasibility gap of the iterates for all.
A TWOSTAGE IMAGE SEGMENTATION METHOD USING A CONVEX VARIANT OF THE MUMFORDSHAH MODEL AND THRESHOLDING
"... Abstract. The MumfordShah model is one of the most important image segmentation models, and has been studied extensively in the last twenty years. In this paper, we propose a twostage segmentation method based on the MumfordShah model. The first stage of our method is to find a smooth solution g ..."
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Abstract. The MumfordShah model is one of the most important image segmentation models, and has been studied extensively in the last twenty years. In this paper, we propose a twostage segmentation method based on the MumfordShah model. The first stage of our method is to find a smooth solution g to a convex variant of the MumfordShah model. Once g is obtained, then in the second stage, the segmentation is done by thresholding g into different phases. The thresholds can be given by the users or can be obtained automatically using any clustering methods. Because of the convexity of the model, g can be solved efficiently by techniques like the splitBregman algorithm or the ChambollePock method. We prove that our method is convergent and the solution g is always unique. In our method, there is no need to specify the number of segments K (K ≥ 2) before finding g. We can obtain any Kphase segmentations by choosing (K −1) thresholds after g is found in the first stage; and in the second stage there is no need to recompute g if the thresholds are changed to reveal different segmentation features in the image. Experimental results show that our twostage method performs better than many standard twophase or multiphase segmentation methods for very general images, including antimass, tubular, MRI, noisy, and blurry images. Key words. Image segmentation, MumfordShah model, splitBregman, total variation. AMS subject classifications. 52A41, 65D15, 68W40, 90C25, 90C90
An iteratively reweighted Algorithm for Nonsmooth Nonconvex Optimization in Computer Vision
, 2014
"... Natural image statistics indicate that we should use nonconvex norms for most regularization tasks in image processing and computer vision. Still, they are rarely used in practice due to the challenge of optimization. Recently, iteratively reweighed `1 minimization (IRL1) has been proposed as a way ..."
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Natural image statistics indicate that we should use nonconvex norms for most regularization tasks in image processing and computer vision. Still, they are rarely used in practice due to the challenge of optimization. Recently, iteratively reweighed `1 minimization (IRL1) has been proposed as a way to tackle a class of nonconvex functions by solving a sequence of convex `2`1 problems. We extend the problem class to the sum of a convex function and a (nonconvex) nondeceasing function applied to another convex function. The proposed algorithm sequentially optimizes suitably constructed convex majorizers. Convergence to a critical point is proved when the Kurdyka Lojasiewicz property and additional mild restrictions hold for the objective function. The efficiency of the algorithm and the practical importance of the algorithm is demonstrated in computer vision tasks such as image denoising and optical flow. Most applications seek smooth results with sharp discontinuities. This is achieved by combining nonconvexity
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.
Image Segmentation Methods Based on Tightframe and MumfordShah Model
"... Image segmentation is a very important topic in image processing. It is the process of identifying object outlines within images. There are quite a few efficient algorithms for segmentation such as the model based approaches, pattern recognition techniques, trackingbased approaches, artificial inte ..."
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Image segmentation is a very important topic in image processing. It is the process of identifying object outlines within images. There are quite a few efficient algorithms for segmentation such as the model based approaches, pattern recognition techniques, trackingbased approaches, artificial intelligencebased approaches, etc. In this thesis, we mainly study two kinds of image segmentation problems. More precisely, one kind problem is the vessel segmentation problem in medical imaging, the other is the generic image segmentation problem, i.e., twophase and multiphase image segmentation for very general images, for example medical, noisy, and blurry images, etc. In Part I of this thesis, we focus on the vessel segmentation problem in medical Images, and our tightframe based vessel segmentation algorithm will be proposed. Tightframe, a generalization of orthogonal wavelets, has been used successfully in various problems in image processing, including inpainting, impulse noise removal, superresolution image restoration, etc. In this part, we propose to apply the tightframe approach to automatically identify tubelike
DOI 10.1007/s108510120347x
, 2012
"... Convex relaxation of a class of vertex penalizing functionals ..."
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INERTIAL PROXIMAL ADMM FOR LINEARLY CONSTRAINED SEPARABLE CONVEX OPTIMIZATION
"... Abstract. The alternating direction method of multipliers (ADMM) is a popular and efficient firstorder method that has recently found numerous applications, and the proximal ADMM is an important variant of it. The main contributions of this paper are the proposition and the analysis of a class of i ..."
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Abstract. The alternating direction method of multipliers (ADMM) is a popular and efficient firstorder method that has recently found numerous applications, and the proximal ADMM is an important variant of it. The main contributions of this paper are the proposition and the analysis of a class of inertial proximal ADMMs, which unify the basic ideas of the inertial proximal point method and the proximal ADMM, for linearly constrained separable convex optimization. This class of methods are of inertial nature because at each iteration the proximal ADMM is applied to a point extrapolated at the current iterate in the direction of last movement. The recently proposed inertial primaldual algorithm [1, Algorithm 3] and the inertial linearized ADMM [2, Eq. (3.23)] are covered as special cases. The proposed algorithmic framework is very general in the sense that the weighting matrices in the proximal terms are allowed to be only positive semidefinite, but not necessarily positive definite as required by existing methods of the same kind. By setting the two proximal terms to zero, we obtain an inertial variant of the classical ADMM, which is new to the best of our knowledge. We carry out a unified analysis for the entire class of methods under very mild assumptions. In particular, convergence, as well as asymptotic o(1/ k) and nonasymptotic O(1/ k) rates of convergence, are established for the best primal function value and feasibility residues, where k denotes the iteration counter. The global iterate convergence of the generated sequence is established under an additional assumption. We also
A TwoStage Image Segmentation Method Using a Convex Variant of the Mumford–Shah Model and Thresholding∗
"... Abstract. The Mumford–Shah model is one of the most important image segmentation models and has been studied extensively in the last twenty years. In this paper, we propose a twostage segmentation method based on the Mumford–Shah model. The first stage of our method is to find a smooth solution g t ..."
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Abstract. The Mumford–Shah model is one of the most important image segmentation models and has been studied extensively in the last twenty years. In this paper, we propose a twostage segmentation method based on the Mumford–Shah model. The first stage of our method is to find a smooth solution g to a convex variant of the Mumford–Shah model. Once g is obtained, then in the second stage the segmentation is done by thresholding g into different phases. The thresholds can be given by the users or can be obtained automatically using any clustering methods. Because of the convexity of the model, g can be solved efficiently by techniques like the splitBregman algorithm or the Chambolle–Pock method. We prove that our method is convergent and that the solution g is always unique. In our method, there is no need to specify the number of segments K (K ≥ 2) before finding g. We can obtain any Kphase segmentations by choosing (K − 1) thresholds after g is found in the first stage, and in the second stage there is no need to recompute g if the thresholds are changed to reveal different segmentation features in the image. Experimental results show that our twostage method performs better than many standard twophase or multiphase segmentation methods for very general images, including antimass, tubular, MRI, noisy, and blurry images.