Polynomial-time solutions to image segmentation

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

Research output: Chapter in Book/Report/Conference proceedingConference contribution

37 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 complexity of the segmentation problem, assuming that the object forms a connected region in an intensity image. We show that the optimization problem of separating a connected region in an n-pixel grid is NP-hard under the interclass variance, a criterion that is used in discriminant analysis. More importantly, we consider the basic case in which the object is separated by two x-monotone curves (i.e., the object itself is x-monotone), and present polynomial-time algorithms for computing exact and approximate optimal segmentation. Our main algorithm for exact optimal segmentation by two x-monotone curves runs in O(n2) 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 point set, and dynamic programming using fast matrix searching. Our efficient approximation scheme obtains an ϵ-approximate solution in O(ϵ-1n log L) time, where ϵ is any fixed constant with 1 > ϵ > 0, and L is the total sum of the absolute values of brightness levels of the image.

Original languageEnglish
Title of host publicationProceedings of the 7th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1996
PublisherAssociation for Computing Machinery
Number of pages10
VolumePart F129447
ISBN (Electronic)0898713668
Publication statusPublished - 1996 Jan 28
Externally publishedYes
Event7th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1996 - Atlanta, United States
Duration: 1996 Jan 281996 Jan 30


Other7th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1996
Country/TerritoryUnited States

ASJC Scopus subject areas

  • Software
  • Mathematics(all)


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