Computing geodesics and minimal surfaces via graph cuts

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Computing geodesics and minimal surfaces via graph cuts (2003)

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by Yuri Boykov

Venue:	in International Conference on Computer Vision
Citations:	250 - 26 self

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@INPROCEEDINGS{Boykov03computinggeodesics,
    author = {Yuri Boykov},
    title = {Computing geodesics and minimal surfaces via graph cuts},
    booktitle = {in International Conference on Computer Vision},
    year = {2003},
    pages = {26--33}
}

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Abstract

Geodesic active contours and graph cuts are two standard image segmentation techniques. We introduce a new segmentation method combining some of their benefits. Our main intuition is that any cut on a graph embedded in some continuous space can be interpreted as a contour (in 2D) or a surface (in 3D). We show how to build a grid graph and set its edge weights so that the cost of cuts is arbitrarily close to the length (area) of the corresponding contours (surfaces) for any anisotropic Riemannian metric. There are two interesting consequences of this technical result. First, graph cut algorithms can be used to find globally minimum geodesic contours (minimal surfaces in 3D) under arbitrary Riemannian metric for a given set of boundary conditions. Second, we show how to minimize metrication artifacts in existing graph-cut based methods in vision. Theoretically speaking, our work provides an interesting link between several branches of mathematics-differential geometry, integral geometry, and combinatorial optimization. The main technical problem is solved using Cauchy-Crofton formula from integral geometry. 1.

Keyphrases

graph cut minimal surface integral geometry mathematics-differential geometry several branch technical result geodesic active contour edge weight minimum geodesic contour metrication artifact continuous space cauchy-crofton formula interesting consequence main technical problem interesting link graph cut algorithm combinatorial optimization corresponding contour boundary condition grid graph main intuition new segmentation method standard image segmentation technique

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