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.
|ジャーナル||International Journal of Computational Geometry and Applications|
|出版ステータス||Published - 2001 4月|
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