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Application of the Level-Set Model with Constraints in Image Segmentation

Published online by Cambridge University Press:  15 February 2016

Vladimír Klement
Affiliation:
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Praha 2, 120 00, Czech Republic
Tomáš Oberhuber
Affiliation:
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Praha 2, 120 00, Czech Republic
Daniel Ševčovič*
Affiliation:
Department of Applied Mathematics and Statistics, Comenius University, 842 48 Bratislava, Slovakia
*
*Corresponding author. Email addresses: tomas.oberhuber@fjfi.cvut.cz (T. Oberhuber), vladimir.klement@fjfi.cvut.cz (V. Klement), sevcovic@fmph.uniba.sk (D. Ševčovič)
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Abstract

We propose and analyze a constrained level-set method for semi-automatic image segmentation. Our level-set model with constraints on the level-set function enables us to specify which parts of the image lie inside respectively outside the segmented objects. Such a-priori information can be expressed in terms of upper and lower constraints prescribed for the level-set function. Constraints have the same conceptual meaning as initial seeds of the popular graph-cuts based methods for image segmentation. A numerical approximation scheme is based on the complementary-finite volumes method combined with the Projected successive over-relaxation method adopted for solving constrained linear complementarity problems. The advantage of the constrained level-set method is demonstrated on several artificial images as well as on cardiac MRI data.

Type
Research Article
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Beneš, M., Kimura, M., Pauš, P., Ševčovič, D., Tsujikawa, T., Yazaki, S., Application of a curvature adjusted method in image segmentation, Bulletin of Inst. of Mathematics, Academia Sinica, New Series 3 (2008), 509524.Google Scholar
[2]Beneš, M., Máca, R., Application of a degenerate diffusion method in 3D medical image processing, Proceedings of Agoritmy 2012, Handlovičová, A., Minarechová, Z. and Ševčovič, D. (ed.), (2012), 416426.Google Scholar
[3]Boykov, Y., Funka-Lea, G., Graph cuts and efficient n-d image segmentation, International Journal of Computer Vision 70 (2006), 109131.CrossRefGoogle Scholar
[4]Boykov, Y., Jolly, M. P., Interactive graph cuts for optimal boundary & region segmentation of objects in n-D images. In: Proceedings of International Conference on Computer Vision, Vancouver, Canada, July 2001, Vol. 1, 105112, 2001.Google Scholar
[5]Boykov, Y., Kolmogorov, V, An experimental comparison of min-cut/max- flow algorithms for energy minimization in vision, Pattern Analysis and Machine Intelligence 26 (2004), 11241137.CrossRefGoogle ScholarPubMed
[6]Brézis, H., Problèmes unilatéraux, J. Math. Pures Appl. 51 (1972), 1168.Google Scholar
[7]Caselles, V., Catté, F., Coll, T., Dibos, F., A geometric model for active contours in image processing, Numerische Mathematik 66 (1993), 131.CrossRefGoogle Scholar
[8]Caselles, V., R. Kimmel, , Sapiro, G., Geodesic active contours, in: Proceedings International Conference on Computer Vision' 95, Boston, pp. 694699.Google Scholar
[9]Chan, T., Zhu, W., Level set based shape prior segmentation, in: Computer Vision and Pattern Recognition, IEEE Computer Society Conference on, 2, (2005), 11641170.Google Scholar
[10]Chung, G., Vese, L. A., Image segmentation using a multilayer level-set approach, Computing and Visualization in Science, 12(6) (2009), 267285.CrossRefGoogle Scholar
[11]Cremers, D., Rousson, M., Deriche, R., A Review of Statistical Approaches to Level Set Segmentation: Integrating Color, Texture, Motion and Shape, Internationa Journal of Computer Vision, 72(2) (2007), 195215.CrossRefGoogle Scholar
[12]Droske, M., Rumpf, M., A level set formulation for Willmore flow, Interfaces Free Boundaries, 6 (3) (2004), 361378.CrossRefGoogle Scholar
[13]Elliott, C. M., Ockendon, J. R., Weak and Variational Methods for Moving Boundary Problems, Vol. 59, Research Notes in Mathematics, Pitman, Boston, Mass., 1982.Google Scholar
[14]Gurholt, T. P., Xuecheng, Tai, 3D Multiphase Piecewise Constant Level Set Method Based on Graph Cut Minimization, Numer. Math. Theor. Meth. Appl., 2 (2009), 403420.Google Scholar
[15]Handlovičová, A., Mikula, K., Sgallari, F., Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution, Numerische Mathematik 93 (2003), 675695.CrossRefGoogle Scholar
[16]He, L., Peng, Z., Everding, B., Wang, X., Han, Ch. Y., Weiss, K. L., Wee, W. G., A comparative study of deformable contour methods on medical image segmentation, Image and Vision Computing 26 (2008), 141163.CrossRefGoogle Scholar
[17]Jiang, Y., Wang, M., Xu, H., A Survey for Region-based Level Set Image Segmentation, in: IEEE Proceedings of 11th International Symposium on Distributed Computing and Applications to Business, Engineering & Science (DCABES), 19-22 Oct. 2012, 413416.CrossRefGoogle Scholar
[18]Kass, M., Witkin, A., Terzopoulos, D., Snakes: Active contour models, International Journal of Computer Vision 1 (1987), 321331.CrossRefGoogle Scholar
[19]Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A., Jr., Conformal curvature flows: from phase transitions to active vision, Arch. Rational Mech. Anal. 134 (1996), 275301.CrossRefGoogle Scholar
[20]Loucký, J., Oberhuber, T., Graph cuts in segmentation of a left ventricle from MRI data, Proceedings of Czech-Japanese Seminar in Applied Mathematics 2010, ed. Beneš, M., Kimura, M., Yazaki, S., COE Lecture Note, 36 (2012), 4654.Google Scholar
[21]Mangasarian, O. L., Solution of symmetric linear complementarity problems by iterative methods, J. Optimization Theory Applicat. 22 (1977), 465485.CrossRefGoogle Scholar
[22]Mikula, K., Sarti, A., Parallel co-volume subjective surface method for 3D medical image segmentation, in: Parametric and Geometric Deformable Models: An application in Bio-materials and Medical Imagery, Volume-II, Springer Publishers, (Eds. Jasjit Suri, S. and Farag, Aly), 2007, pp. 123160.CrossRefGoogle Scholar
[23]Mikula, K., Ševčovič, D., Computational and qualitative aspects of evolution of curves driven by curvature and external force, Comput. Vis. Sci. 6 (2004), 211225.CrossRefGoogle Scholar
[24]Oberhuber, T., Complementary finite volume scheme for the anisotropic surface diffusion flow, in: Proceedings of Algoritmy 2009, (2009), 153164.Google Scholar
[25]Geometric Level Set Methods in Imaging, Vision, and Graphics, Osher, S., Paragios, N. (Eds.), Springer, New-York, 2003.Google Scholar
[26]Paragios, N., Deriche, R., Geodesic active contours and level sets for the detection and tracking of moving objects, in: Pattern Analysis and Machine Intelligence, IEEE Transactions on 22,(3) (2000), 266280.Google Scholar
[27]Quarteroni, A., Sacco, R., Saleri, F., Numerical Mathematics, Springer, 2000Google Scholar
[28]Sethian, J. A., Level Set Methods, Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, New York, 1996.Google Scholar