A Convex Approach to K-Means Clustering and Image Segmentation

  • Laurent CondatEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10746)


A new convex formulation of data clustering and image segmentation is proposed, with fixed number K of regions and possible penalization of the region perimeters. So, this problem is a spatially regularized version of the K-means problem, a.k.a. piecewise constant Mumford–Shah problem. The proposed approach relies on a discretization of the search space; that is, a finite number of candidates must be specified, from which the K centroids are determined. After reformulation as an assignment problem, a convex relaxation is proposed, which involves a kind of \(l_{1,\infty }\) norm ball. A splitting of it is proposed, so as to avoid the costly projection onto this set. Some examples illustrate the efficiency of the approach.


Image segmentation Piecewise constant Mumford–Shah problem K-means Convex relaxation 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CNRS, GIPSA-LabUniv. Grenoble AlpesGrenobleFrance

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