# MINIMALLY OVERLAPPING PATHS SETS FOR CLOSED CONTOUR EXTRACTION

### Julien Mille, Sébastien Bougleux, Laurent Cohen

#### Abstract

Active contours and minimal paths have been extensively studied theoretical tools for image segmentation. The recent geodesically linked active contour model, which basically consists in a set of vertices connected by paths of minimal cost, blend the benefits of both concepts. This makes up a closed piecewise-defined curve, over which an edge or region energy functional can be formulated. As an important shortcoming, the geodesically linked active contour model in its initial formulation does not guarantee to represent a simple curve, consistent with respect to the purpose of segmentation. In this paper, we propose to extract a similarly piecewise-defined curve from a set of possible paths, such that the resulting structure is guaranteed to represent a relevant closed curve. Toward this goal, we introduce a global constraint penalizing excessive overlap between paths.

#### References

- Amini, A., Weymouth, T., and Jain, R. (1990). Using dynamic programming for solving variational problems in vision. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(9):855-867.
- Benmansour, F. and Cohen, L. (2009). Fast object segmentation by growing minimal paths from a single point on 2D or 3D images. Journal of Mathematical Imaging and Vision, 33(2):209-221.
- Benmansour, F. and Cohen, L. (2011). Tubular structure segmentation based on minimal path method and anisotropic enhancement. International Journal of Computer Vision, 92(2):192-210.
- Brox, T. and Cremers, D. (2009). On local region models and a statistical interpretation of the piecewise smooth Mumford-Shah functional. International Journal of Computer Vision, 84(2):184-193.
- Chan, T. and Vese, L. (2001). Active contours without edges. IEEE Transactions on Image Processing, 10(2):266-277.
- Cohen, L. (1991). On active contour models and balloons. Computer Vision, Graphics, and Image Processing: Image Understanding, 53(2):211-218.
- Cohen, L. and Kimmel, R. (1997). Global minimum for active contour models: a minimal path approach. International Journal of Computer Vision, 24(1):57-78.
- Crandall, M., Ishii, H., and Lions, P.-L. (1992). User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc., 27:1-67.
- Dubuisson, M.-P. and Jain, A. (1994). A modified Hausdorff distance for object matching. In 12th International Conference on Pattern Recognition (ICPR), pages 566-568, Jerusalem, Israel.
- Eppstein, D. (1998). Finding the k shortest paths. SIAM Journal of Computing, 28(2):652-673.
- Kaul, V., Tsai, Y., and Yezzi, A. (2010). Detection of curves with unknown endpoints using minimal path techniques. In British Machine Vision Conference (BMVC), pages 1-12, Aberystwyth, UK.
- Lankton, S. and Tannenbaum, A. (2008). Localizing regionbased active contours. IEEE Transactions on Image Processing, 17(11):2029-2039.
- Mille, J. (2009). Narrow band region-based active contours and surfaces for 2D and 3D segmentation. Computer Vision and Image Understanding, 113(9):946-965.
- Mille, J. and Cohen, L. (2009). Geodesically linked active contours: evolution strategy based on minimal paths.
- In 2nd International Conference on Scale Space and Variational Methods in Computer Vision (SSVM), volume 5567 of LNCS, pages 163-174, Voss, Norway.
- Peyré, G., Pechaud, M., Keriven, R., and Cohen, L. (2010). Geodesic methods in computer vision and graphics. Foundations and Trends in Computer Graphics and Vision, 5(3-4):197-397.
- Sagiv, C., Sochen, N., and Zeevi, Y. (2006). Integrated active contours for texture segmentation. IEEE Transactions on Image Processing, 15(6):1633-1646.
- Sethian, J. (1996). A fast marching level set method for monotonically advancing fronts. Proceedings of the National Academy of Science, 93(4):1591-1595.
- Sethian, J. (1999). Level Sets Methods and Fast Marching Methods. Cambridge University Press, 2nd edition.
- Tsitsiklis, J. (1995). Efficient algorithms for globally optimal trajectories. IEEE Transactions on Automatic Control, 40(9):1528-1538.
- Williams, D. and Shah, M. (1992). A fast algorithm for active contours and curvature estimation. Computer Vision, Graphics, and Image Processing: Image Understanding, 55(1):14-26.
- Yen, J. (1971). Finding the K shortest loopless paths in a network. Management Science, 17(11):712-716.

#### Paper Citation

#### in Harvard Style

Mille J., Bougleux S. and Cohen L. (2012). **MINIMALLY OVERLAPPING PATHS SETS FOR CLOSED CONTOUR EXTRACTION** . In *Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2012)* ISBN 978-989-8565-03-7, pages 259-268. DOI: 10.5220/0003843102590268

#### in Bibtex Style

@conference{visapp12,

author={Julien Mille and Sébastien Bougleux and Laurent Cohen},

title={MINIMALLY OVERLAPPING PATHS SETS FOR CLOSED CONTOUR EXTRACTION},

booktitle={Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2012)},

year={2012},

pages={259-268},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0003843102590268},

isbn={978-989-8565-03-7},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2012)

TI - MINIMALLY OVERLAPPING PATHS SETS FOR CLOSED CONTOUR EXTRACTION

SN - 978-989-8565-03-7

AU - Mille J.

AU - Bougleux S.

AU - Cohen L.

PY - 2012

SP - 259

EP - 268

DO - 10.5220/0003843102590268