TY - JOUR

T1 - Efficient algorithms for optimization-based image segmentation

AU - Asano, Tetsuo

AU - Chen, Danny Z.

AU - Katoh, Naoki

AU - Tokuyama, Takeshi

N1 - Funding Information:
Received 15 July 1999 Revised 9 November 1999 Communicated by J. S. B. Mitchell ABSTRACT 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 a>monotone curves (i.e., *This paper is the full version of the extended abstract [T. Asano, D.Z. Chen, N. Katoh, and T. Tokuyama, "Polynomial-Time Solutions to Image Segmentation," in Proc. of the 7th Annual ACM-SIAM Symposium on Discrete Algorithms, 1996, pp. 104-113]. +The research of the first, third, and fourth authors was supported in part by the Grant in Aid for Scientific Research of the Ministry of Education, Science and Cultures of Japan. The research of the second author was supported in part by the National Science Foundation under Grants CCR-9623585 and CCR-9988468.

PY - 2001/4

Y1 - 2001/4

N2 - 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.

AB - 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.

KW - Approximation scheme

KW - Computer vision

KW - Dynamic programming

KW - Fast matrix searching

KW - Hand probing

KW - Polynomial-time algorithm

KW - Region segmentation

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U2 - 10.1142/S0218195901000420

DO - 10.1142/S0218195901000420

M3 - Article

AN - SCOPUS:0035590607

VL - 11

SP - 145

EP - 166

JO - International Journal of Computational Geometry and Applications

JF - International Journal of Computational Geometry and Applications

SN - 0218-1959

IS - 2

ER -