Efficient algorithms for optimization-based image segmentation

Tetsuo Asano, Danny Z. Chen, Naoki Katoh, Takeshi Tokuyama

Research output: Contribution to journalArticlepeer-review

31 Citations (Scopus)


Separating an object in an image from its background is a central problem (called segmentation) in pattern recognition and computer vision. In this paper, we study the computational complexity of the segmentation problem, assuming that the sought object forms a connected region in an intensity image. We show that the optimization problem of separating a connected region in a grid of N x N pixels is NP-hard under the interclass variance, a criterion that is often used in discriminant analysis. More importantly, we consider the basic case in which the object is bounded by two x-monotone curves (i.e., the object itself is x-monotone), and present polynomial-time algorithms for computing the optimal segmentation. Our main algorithm for exact optimal segmentation by two x-monotone curves runs in O(N4) time; this algorithm is based on several techniques such as a parametric optimization formulation, a hand-probing algorithm for the convex hull of an unknown planar point set, and dynamic programming using fast matrix searching. Our efficient approximation scheme obtains an ∈-approximate solution in O(∈-1 N2 log L) time, where ∈ is any fixed constant with 1 > ∈ > 0, and L is the total sum of the absolute values of the brightness levels of the image.

Original languageEnglish
Pages (from-to)145-166
Number of pages22
JournalInternational Journal of Computational Geometry and Applications
Issue number2
Publication statusPublished - 2001 Apr
Externally publishedYes


  • Approximation scheme
  • Computer vision
  • Dynamic programming
  • Fast matrix searching
  • Hand probing
  • Polynomial-time algorithm
  • Region segmentation

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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