Computing Geodesics and Minimal Surfaces via Graph Cuts

  • Authors:
  • Yuri Boykov;Vladimir Kolmogorov

  • Affiliations:
  • -;-

  • Venue:
  • ICCV '03 Proceedings of the Ninth IEEE International Conference on Computer Vision - Volume 2
  • Year:
  • 2003

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Abstract

Geodesic active contours and graph cuts are two standard imagesegmentation techniques. We introduce a new segmentation methodcombining some of their benefits. Our main intuition is that anycut on a graph embedded in some continuous space can be interpretedas a contour (in 2D) or a surface (in 3D). We show how to build agrid graph and set its edge weights so that the cost of cuts isarbitrarily close to the length (area) of the correspondingcontours(surfaces) for any anisotropic Riemannian metric. There aretwo interesting consequences of this technical result. First, graphcut algorithms can be used to find globally minimum geodesiccontours (minimal surfaces in 3D) under arbitrary Riemannian metricfor a given set of boundary conditions. Second, we show how tominimize metrication artifacts in existing graph-cut based methodsin vision. Theoretically speaking, our work provides an interestinglink between several branches of mathematics - differentialgeometry, integral geometry, and combinatorial optimization. Themain technical problem is solved using Cauchy-Crofton formula fromintegral geometry.