Kybernetika 49 no. 2, 301-318, 2013

Segmentation of MRI data by means of nonlinear diffusion

Radomír Chabiniok, Radek Máca, Michal Beneš and Jaroslav Tintěra


\noindent The article focuses on the application of the segmentation algorithm based on the numerical solution of the Allen-Cahn non-linear diffusion partial differential equation. This equation is related to the motion of curves by mean curvature. It exhibits several suitable mathematical properties including stable solution profile. This allows the user to follow accurately the position of the segmentation curve by bringing it quickly to the vicinity of the segmented object and by approaching the details of the segmentation curve. The purpose of the article is to indicate how the algorithm parameters are set up and to show how the algorithm behaves when applied to the particular class of medical data. In detail we describe the algorithm parameters influencing the segmentation procedure. The left ventricle volume estimated by the segmentation of scanned slices is evaluated through the cardiac cycle. Consequently, the ejection fraction is evaluated. The described approach allows the user to process cardiac cine MR images in an automated way and represents, therefore, an alternative to other commonly used methods. Based on the physical and mathematical background, the presented algorithm exhibits the stable behavior in the segmentation of MRI test data, it is computationally efficient and allows the user to perform various implementation improvements.


image segmentation, degenerate diffusion, Allen-Cahn equation, magnetic resonance imaging


80A22, 82C26, 35A40, 68U10


  1. S. Allen and J. W. Cahn: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall 27 (1979), 1084-1095.   CrossRef
  2. M. Beneš: Mathematical analysis of phase-field equations with numerically efficient coupling terms. Interfaces and Free Boundaries 3 (2001), 201-221.   CrossRef
  3. M. Beneš, V. Chalupecký and K. Mikula: Geometrical image segmentation by the Allen-Cahn equation. Appl. Numer. Math. 51 (2004), 2, 187-205.   CrossRef
  4. M. Beneš, R. Chabiniok, M. Kimura and K. Mikula: Nonlinear Gauss-Seidel scheme for Allen-Cahn type systems. In: MAGIA 2007 (Mathematics, Geometry and Their Applications)(M. Vajs{á}blov{á} and P. Struk, eds.), Publishing House of Slovak Technical University, Bratislava 2008, pp. 29-35.   CrossRef
  5. J. Bogaert, S. Dymarkowski and A. M. Taylor: Clinical Cardiac MRI. Springer, Berlin - Heidelberg 2005.   CrossRef
  6. A. Bovik (ed): Handbook of Image and Video Processing. Academic Press, San Diego 1990.   CrossRef
  7. Y. Boykov, O. Veksler and R. Zabih: Fast approximate energy minimization via graph cuts. IEEE Trans. Pattern Analysis and Machine Intelligence 23 (2001), 1222-1239.   CrossRef
  8. M. D. Cerqueira, N. J. Weissman and V. Dilsizian et al.: Standardized myocardial segmentation and nomenclature for tomographic imaging of the heart: A statement of healthcare professionals from the Cardiac Imaging Comittee of the Council on Clinical Cardiology of the American Heart Association. Circulation 105 (2002), 539-542.   CrossRef
  9. Y. Cheng: Mean shift, mode deeking, and clustering. IEEE Trans. on Pattern Analysis and Machine Intelligence 17 (1995), 790-799.   CrossRef
  10. M. G. Crandall, H. Ishii and P. L. Lions: User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992), 1-67.   CrossRef
  11. S. Geman and D. Geman: Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence 6 (1984), 721-741.   CrossRef
  12. E. Heiberg, L. Wigstrom, M. Carlsson, A. F. Bolger and M. Karlsson: Time resolved three-dimensional automated segmentation of the left ventricle. In: Proc. IEEE Computers in Cardiology 2005 (32), Lyon 2005, pp. 599-602.   CrossRef
  13. E. Heiberg, J. Sj{ö}gren, M. Ugander, M. Carlsson, H. Engblom and H. Arheden: Design and validation of SEGMENT - a freely available software for cardiovascular image analysis. BMC Medical Imaging 10 (2010), 1.   CrossRef
  14. M. Kass, A. Witkin and D. Terzopoulos: Snakes: Active contour models. Internat. J. Computer Vision 1 (1988), 321-331.   CrossRef
  15. R. M{á}ca, M. Beneš and J. Tintěra: Degenerate diffusion methods in computer image processing and application. J. Math-for-Industry 3 (2011), 33-40.   CrossRef
  16. R. Malladi, J. A. Sethian and B. Vemuri: Shape modeling with front propagation: A level set approach. IEEE Trans. on Pattern Analysis and Machine Intelligence 17 (1995), 2, 158-175.   CrossRef
  17. K. Mikula, A. Sarti. and F. Sgallari: Co-volume level set method in subjective surface based medical image segmentation. In: Handbook of Medical Image Analysis: Segmentation and Registration Models (J. Suri et al., eds.), Springer, New York 2005, pp. 583-626.   CrossRef
  18. D. Mumford and J. Shah: Boundary detection by minimizing functionals. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition 1985, pp. 22-26.   CrossRef
  19. S. Osher and R. Fedkiw: Level Set Methods and Dynamic Implicit Surfaces. Springer Verlag, New York 2003.   CrossRef
  20. N. Paragios, Y. Chen and O. Faugeras: Handbook of Mathematical Models of Computer Vision. Springer, New York 2005.   CrossRef
  21. P. Perona and J. Malik: Scale space and edge detection using anisotropic diffusion. IEEE Trans. on Pattern Analysis and Machine Intelligence 12 (1990), 629-639.   CrossRef
  22. H. K. Zhao, S. Osher, T. Chan and B. Merriman: A variational level set approach to multiphase motion. J. Comput. Phys. 127 (1996), 179-195.   CrossRef
  23. S. Zhu and A. Yuille: Region competition: Unifying snakes, region growing, and Bayes/Mdl for multiband image segmentation. IEEE Trans. on Pattern Analysis and Machine Intelligence 18 (1996), 884-900.   CrossRef