# GRAPH CUTS AND APPROXIMATION OF THE EUCLIDEAN METRIC ON ANISOTROPIC GRIDS

### Ondřej Daněk, Pavel Matula

#### Abstract

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 that 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.

#### References

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#### Paper Citation

#### in Harvard Style

Daněk O. and Matula P. (2010). **GRAPH CUTS AND APPROXIMATION OF THE EUCLIDEAN METRIC ON ANISOTROPIC GRIDS** . In *Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 2: VISAPP, (VISIGRAPP 2010)* ISBN 978-989-674-029-0, pages 68-73. DOI: 10.5220/0002833000680073

#### in Bibtex Style

@conference{visapp10,

author={Ondřej Daněk and Pavel Matula},

title={GRAPH CUTS AND APPROXIMATION OF THE EUCLIDEAN METRIC ON ANISOTROPIC GRIDS},

booktitle={Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 2: VISAPP, (VISIGRAPP 2010)},

year={2010},

pages={68-73},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0002833000680073},

isbn={978-989-674-029-0},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 2: VISAPP, (VISIGRAPP 2010)

TI - GRAPH CUTS AND APPROXIMATION OF THE EUCLIDEAN METRIC ON ANISOTROPIC GRIDS

SN - 978-989-674-029-0

AU - Daněk O.

AU - Matula P.

PY - 2010

SP - 68

EP - 73

DO - 10.5220/0002833000680073