Vector Image Segmentation by Piecewise Continuous Approximation

  • Tobias Hanning
  • René Schöne
  • Georg Pisinger


In this article we present an approach to the segmentation problem by a piecewise approximation of the given image with continuous functions. Unlike the common approach of Mumford and Shah in our formulation of the problem the number of segments is a parameter, which can be estimated. The problem can be stated as: Compute the optimal segmentation with a fixed number of segments, then reduce the number of segments until the segmentation result fulfills a given suitability. This merging algorithm results in a multi-objective optimization, which is not only resolved by a linear combination of the contradicting error functions. To constrain the problem we use a finite dimensional vector space of functions in our approximation and we restrict the shape of the segments. Our approach results in a multi-objective optimization: On the one hand the number of segments is to be minimized, on the other hand the approximation error should also be kept minimal. The approach is sound theoretically and practically: We show that for L 2-images a Pareto-optimal solution exists and can be computed for the discretization of the image efficiently.


image segmentation piecewise constrained approximation multi-objective optimization 


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© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceUniversity of PassauPassau

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