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International Journal of Computer Vision

, Volume 92, Issue 1, pp 112–129 | Cite as

Global Minimization for Continuous Multiphase Partitioning Problems Using a Dual Approach

  • Egil BaeEmail author
  • Jing Yuan
  • Xue-Cheng Tai
Open Access
Article

Abstract

This paper is devoted to the optimization problem of continuous multi-partitioning, or multi-labeling, which is based on a convex relaxation of the continuous Potts model. In contrast to previous efforts, which are tackling the optimal labeling problem in a direct manner, we first propose a novel dual model and then build up a corresponding duality-based approach. By analyzing the dual formulation, sufficient conditions are derived which show that the relaxation is often exact, i.e. there exists optimal solutions that are also globally optimal to the original nonconvex Potts model. In order to deal with the nonsmooth dual problem, we develop a smoothing method based on the log-sum exponential function and indicate that such a smoothing approach leads to a novel smoothed primal-dual model and suggests labelings with maximum entropy. Such a smoothing method for the dual model also yields a new thresholding scheme to obtain approximate solutions. An expectation maximization like algorithm is proposed based on the smoothed formulation which is shown to be superior in efficiency compared to earlier approaches from continuous optimization. Numerical experiments also show that our method outperforms several competitive approaches in various aspects, such as lower energies and better visual quality.

Keywords

Convex relaxation Image segmentation Primal-dual methods Total variation 

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Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BergenBergenNorway
  2. 2.Department of Computer ScienceUniversity of Western OntarioLondonCanada
  3. 3.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore

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