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Imaging with KantorovichRubinstein discrepancy
, 2014
"... We propose the use of the KantorovichRubinstein norm from optimal transport in imaging problems. In particular, we discuss a variational regularisation model endowed with a KantorovichRubinstein discrepancy term and total variation regularization in the context of image denoising and cartoontextu ..."
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We propose the use of the KantorovichRubinstein norm from optimal transport in imaging problems. In particular, we discuss a variational regularisation model endowed with a KantorovichRubinstein discrepancy term and total variation regularization in the context of image denoising and cartoontexture decomposition. We point out connections of this approach to several other recently proposed methods such as total generalized variation and norms capturing oscillating patterns. We also show that the respective optimization problem can be turned into a convexconcave saddle point problem with simple constraints and hence, can be solved by standard tools. Numerical examples exhibit interesting features and favourable performance for denoising and cartoontexture decomposition. 1
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
IEEE TRANSACTIONS ON IMAGE PROCESSING, 2015 1 An Approach Towards Fast Gradientbased Image Segmentation
"... Abstract—In this paper we present and investigate an approach to fast multilabel color image segmentation using convex optimization techniques. The presented model is in some ways related to the wellknown MumfordShah model, but deviates in certain important aspects. The optimization problem has ..."
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Abstract—In this paper we present and investigate an approach to fast multilabel color image segmentation using convex optimization techniques. The presented model is in some ways related to the wellknown MumfordShah model, but deviates in certain important aspects. The optimization problem has been designed with two goals in mind: The objective function should represent fundamental concepts of image segmentation, such as incorporation of weighted curve length and variation of intensity in the segmented regions, while allowing transformation into a convex concave saddle point problem that is computationally inexpensive to solve. This paper introduces such a model, the nontrivial transformation of this model into a convexconcave saddle point problem, and the numerical treatment of the problem. We evaluate our approach by applying our algorithm to various images and show that our results are competitive in terms of quality at unprecedentedly low computation times. Our algorithm allows highquality segmentation of megapixel images in a few seconds and achieves interactive performance for low resolution images. Index Terms—unsupervised image segmentation, convex optimization I.