Annals of Mathematical Sciences and Applications

Volume 3 (2018)

Number 1

Special issue in honor of Professor David Mumford, dedicated to the memory of Jennifer Mumford

Guest Editors: Stuart Geman, David Gu, Stanley Osher, Chi-Wang Shu, Yang Wang, and Shing-Tung Yau

New region force for variational models in image segmentation and high dimensional data clustering

Pages: 255 – 286



Ke Wei (School of Data Science, Fudan University, Shanghai, China)

Ke Yin (Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, China)

Xue-Cheng Tai (Department of Mathematics, Hong Kong Baptist University, Hong Kong)

Tony F. Chan (Office of President, Hong Kong University of Science and Technology, Hong Kong)


We propose an effective framework for multi-phase image segmentation and semi-supervised data clustering by introducing a novel region force term into the Potts model. Assuming the probability that a pixel or a data point belongs to each class is known a priori, we show that the corresponding indicator function obeys the Bernoulli distribution and the new region force function can be computed as the negative log-likelihood function under the Bernoulli distribution.We solve the Potts model by the primal-dual hybrid gradient method and the augmented Lagrangian method, which are based on two different dual problems of the same primal problem. Empirical evaluations of the Potts model with the new region force function on benchmark problems show that it is competitive with existing variational methods in both image segmentation and semi-supervised data clustering.

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The work of Ke Wei was supported by the National Science Foundation under grant number DTRA-DMS 1322393.

Xue-Cheng Tai acknowledges the support from Norwegian Research Council through ISP-Matematikk (Project no. 239033/F20).

Received 26 April 2017

Published 27 March 2018