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Graph Cuts and Approximation of the Euclidean Metric on Anisotropic Grids

Graph cuts can be used to find globally minimal contours and
surfaces in 2D and 3D space, respectively. To achieve this,
weights of the edges in the graph are set so the capacity of
the cut approximates the contour length or surface area under
chosen metric. Formulas giving good approximation in the case
of the Euclidean metric are known, however, they assume
isotropic resolution of the underlying grid of pixels or
voxels. Anisotropy has to be simulated using more general
Riemannian metrics. In this paper we show how to circumvent
this and obtain a good approximation of the Euclidean metric on
anisotropic grids directly by exploiting the well-known
Cauchy-Crofton formulas and Voronoi diagrams theory.
Furthermore, we show that our approach yields much smaller
metrication errors and most interestingly, it is in particular
situations better even in the isotropic case due to its
invariance to mirroring. Finally, we demonstrate an application
of the derived formulas to biomedical image segmentation.