![](bg6.png)
which corresponds to piecewise-constant specializa-
tion of the well-known Mumford-Shah energy func-
tional. In simple terms, it segments the image into
two regions trying to minimize the length of the fron-
tier between them and their intensity variance. This
functional can be minimized using graph cuts (Zeng
et al., 2006) and as it tries to minimize the boundary
length it obviously depends on the Euclidean metric
approximation.
To test the improvement of our approximation
over the previous approach also in 3D we plugged
the derived formulas into the algorithm and used it to
segment low-quality volumetric images of cell clus-
ters acquired by an optical microscope. The yz cross-
sections of the segmented images are depicted in
Fig.4a. The dimensions of the images are 280×360×
50, with resolution in the xy plane being about 4.5
times the resolution in the z direction. We used 26-
neighbourhood to segment the images.
In Fig. 4b is the Chan-Vese segmentation com-
puted using level-sets. This technique was much
slower than the graph cuts, however, it does not suf-
fer from the metrication errors so we used its results
as the ground truth. Figure 4c shows the graph cut
based segmentation when the anisotropy is ignored.
The results obtained using the Riemannian metric and
our weights are depicted in Fig. 4d and Fig. 4e, re-
spectively. Clearly, our method gives a result closest
to the level-sets. On the other hand, the segmenta-
tion based on the Riemannian metric seems too flat
or chopped. Based on the results from the previous
subsection it could be probably greatly improved us-
ing a larger neighbourhood, but at the cost of higher
computational demands.
5 CONCLUSIONS AND FUTURE
WORK
In this paper we addressed the problem of approxima-
tion of the Euclidean metric on 2D and 3D anisotropic
grids via graph cuts. We derived the required formu-
las and showed that our approach has a significantly
smaller metrication error than the previous one and
that it is invariant to image mirroring. Using the pre-
sented results it is possible to exploit graph cut based
energy minimization dependent on contour length or
surface area over images with anisotropic resolution
directly without the need to resample them or to use
large neighbourhoods for better precision. A possi-
ble application of the results was demonstrated on a
biomedical image segmentation.
As explained in Section 4.1 anisotropic grids cor-
respond to a special case of the Riemannian met-
ric with a constant metric tensor with eigenvectors
aligned with the coordinate system. However, the
general case of this metric is also being widely used
in several fields including image segmentation. Tak-
ing into account the relatively high error of the current
formulas we would like to make use of the presented
results and focus on better approximations of the gen-
eral case of the Riemannian metric in our future work.
ACKNOWLEDGEMENTS
This work has been supported by the Ministry of Ed-
ucation of the Czech Republic (Projects No. MSM-
0021622419, No. LC535 and No. 2B06052).
REFERENCES
Boykov, Y. and Funka-Lea, G. (2006). Graph cuts and effi-
cient n-d image segmentation (review). International
Journal of Computer Vision, 70(2):109–131.
Boykov, Y. and Kolmogorov, V. (2003). Computing geo-
desics and minimal surfaces via graph cuts. In ICCV
’03: Proceedings of the Ninth IEEE International
Conference on Computer Vision, page 26.
Boykov, Y. and Kolmogorov, V. (2004). An experi-
mental comparison of min-cut/max-flow algorithms
for energy minimization in vision. IEEE Transac-
tions on Pattern Analysis and Machine Intelligence,
26(9):1124–1137.
Boykov, Y. and Veksler, O. (2006). Graph cuts in vision and
graphics: Theories and applications. In Handbook of
Mathematical Models in Computer Vision, pages 79–
96. Springer-Verlag.
Brown, K. Q. (1979). Geometric transforms for fast geo-
metric algorithms. PhD thesis, Pittsburgh, PA, USA.
Chan, T. F. and Vese, L. A. (2001). Active contours with-
out edges. IEEE Transactions on Image Processing,
10(2):266–277.
Kolmogorov, V. and Boykov, Y. (2005). What metrics can
be approximated by geo-cuts, or global optimization
of length/area and flux. In ICCV ’05: Proceedings
of the Tenth IEEE International Conference on Com-
puter Vision, pages 564–571.
Kolmogorov, V. and Zabih, R. (2004). What energy func-
tions can be minimized via graph cuts? IEEE Trans-
actions on Pattern Analysis and Machine Intelligence,
26(2):147–159.
Zeng, Y., Chen, W., and Peng, Q. (2006). Efficiently solv-
ing the piecewise constant mumford-shah model using
graph cuts. Technical report, Dept. of Computer Sci-
ence, Zhejiang University, P.R. China.
GRAPH CUTS AND APPROXIMATION OF THE EUCLIDEAN METRIC ON ANISOTROPIC GRIDS
73