Advertisement

On Local Region Models and a Statistical Interpretation of the Piecewise Smooth Mumford-Shah Functional

  • Thomas BroxEmail author
  • Daniel Cremers
Article

Abstract

The Mumford-Shah functional is a general and quite popular variational model for image segmentation. In particular, it provides the possibility to represent regions by smooth approximations. In this paper, we derive a statistical interpretation of the full (piecewise smooth) Mumford-Shah functional by relating it to recent works on local region statistics. Moreover, we show that this statistical interpretation comes along with several implications. Firstly, one can derive extended versions of the Mumford-Shah functional including more general distribution models. Secondly, it leads to faster implementations. Finally, thanks to the analytical expression of the smooth approximation via Gaussian convolution, the coordinate descent can be replaced by a true gradient descent.

Keywords

Segmentation Variational methods Statistical methods Regularization 

References

  1. Ambrosio, L., & Tortorelli, V. (1990). Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Communications on Pure and Applied Mathematics, XLIII, 999–1036. CrossRefMathSciNetGoogle Scholar
  2. Blake, A., & Zisserman, A. (1987). Visual Reconstruction. Cambridge: MIT Press. Google Scholar
  3. Boykov, Y., Veksler, O., & Zabih, R. (2001). Fast approximate energy minimization via graph cuts. IEEE Transactions on Pattern Analysis and Machine Intelligence, 23(11), 1222–1239. CrossRefGoogle Scholar
  4. Brox, T. (2005). From pixels to regions: partial differential equations in image analysis. Ph.D. thesis, Faculty of Mathematics and Computer Science, Saarland University, Germany. Google Scholar
  5. Brox, T., & Cremers, D. (2007a). On local region models and the statistical interpretation of the piecewise smooth Mumford-Shah functional. Supplementary online material. Available at http://www-cvpr.iai.uni-bonn.de/pub/pub/brox_ijcv08_sup.pdf.
  6. Brox, T., & Cremers, D. (2007b). On the statistical interpretation of the piecewise smooth Mumford-Shah functional. In F. Sgallari, A. Murli, & N. Paragios (Eds.), LNCS : Vol. 4485. Scale space and variational methods in computer vision (pp. 203–213). Berlin: Springer. CrossRefGoogle Scholar
  7. Brox, T., & Weickert, J. (2006). A TV flow based local scale estimate and its application to texture discrimination. Journal of Visual Communication and Image Representation, 17(5), 1053–1073. CrossRefGoogle Scholar
  8. Brox, T., Rosenhahn, B., & Weickert, J. (2005). Three-dimensional shape knowledge for joint image segmentation and pose estimation. In W. Kropatsch, R. Sablatnig, & A. Hanbury (Eds.), LNCS : Vol. 3663. Pattern recognition (pp. 109–116). Berlin: Springer. CrossRefGoogle Scholar
  9. Caselles, V., Catté, F., Coll, T., & Dibos, F. (1993). A geometric model for active contours in image processing. Numerische Mathematik, 66, 1–31. zbMATHCrossRefMathSciNetGoogle Scholar
  10. Chan, T., & Vese, L. (2001). Active contours without edges. IEEE Transactions on Image Processing, 10(2), 266–277. zbMATHCrossRefGoogle Scholar
  11. Chan, T., Esedoḡlu, S., & Nikolova, M. (2006). Algorithms for finding global minimizers of image segmentation and denoising models. SIAM Journal on Applied Mathematics, 66(5), 1632–1648. zbMATHCrossRefMathSciNetGoogle Scholar
  12. Cremers, D., & Rousson, M. (2007). Efficient kernel density estimation of shape and intensity priors for level set segmentation. In J.S. Suri, & A. Farag (Eds.), Parametric and Geometric Deformable Models: An application in Biomaterials and Medical Imagery. Berlin: Springer. Google Scholar
  13. Cremers, D., Tischhäuser, F., Weickert, J., & Schnörr, C. (2002). Diffusion snakes: introducing statistical shape knowledge into the Mumford-Shah functional. International Journal of Computer Vision, 50(3), 295–313. zbMATHCrossRefGoogle Scholar
  14. Cremers, D., Rousson, M., & Deriche, R. (2007). A review of statistical approaches to level set segmentation: integrating color, texture, motion and shape. International Journal of Computer Vision, 72(2), 195–215. CrossRefGoogle Scholar
  15. Deriche, R. (1990). Fast algorithms for low-level vision. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12, 78–87. CrossRefGoogle Scholar
  16. Geman, S., & Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741. zbMATHCrossRefGoogle Scholar
  17. Greig, D., Porteous, B., & Seheult, A. (1989). Exact maximum a posteriori estimation for binary images. Journal of the Royal Statistical Society B, 51(2), 271–279. Google Scholar
  18. Heiler, M., & Schnörr, C. (2005). Natural image statistics for natural image segmentation. International Journal of Computer Vision, 63(1), 5–19. CrossRefGoogle Scholar
  19. Ising, E. (1925). Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik, 31, 253–258. CrossRefGoogle Scholar
  20. Kass, M., Witkin, A., & Terzopoulos, D. (1988). Snakes: active contour models. International Journal of Computer Vision, 1, 321–331. CrossRefGoogle Scholar
  21. Kim, J., Fisher, J., Yezzi, A., Cetin, M., & Willsky, A. (2005). A nonparametric statistical method for image segmentation using information theory and curve evolution. IEEE Transactions on Image Processing, 14(10), 1486–1502. CrossRefMathSciNetGoogle Scholar
  22. Lenz, W. (1920). Beitrag zum Verständnis der magnetischen Erscheinungen in festen Körpern. Physikalische Zeitschrift, 21, 613–615. Google Scholar
  23. Morel, J.-M., & Solimini, S. (1994). Variational methods in image segmentation. Basel: Birkhäuser. zbMATHGoogle Scholar
  24. Mumford, D., & Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42, 577–685. zbMATHCrossRefMathSciNetGoogle Scholar
  25. Nielsen, M., Florack, L., & Deriche, R. (1994). Regularization and scale space (Technical Report 2352). INRIA, Sophia-Antipolis, France. Google Scholar
  26. Nielsen, M., Florack, L., & Deriche, R. (1997). Regularization, scale-space and edge detection filters. Journal of Mathematical Imaging and Vision, 7, 291–307. CrossRefMathSciNetGoogle Scholar
  27. Paragios, N., & Deriche, R. (2002). Geodesic active regions: a new paradigm to deal with frame partition problems in computer vision. Journal of Visual Communication and Image Representation, 13(1/2), 249–268. CrossRefGoogle Scholar
  28. Parzen, E. (1962). On the estimation of a probability density function and the mode. Annals of Mathematical Statistics, 33, 1065–1076. zbMATHCrossRefMathSciNetGoogle Scholar
  29. Piovano, J., Rousson, M., & Papadopoulo, T. (2007). Efficient segmentation of piecewise smooth images. In F. Sgallari, A. Murli, & N. Paragios (Eds.), LNCS : Vol. 4485. Scale space and variational methods in computer vision (pp. 709–720). Berlin: Springer. CrossRefGoogle Scholar
  30. Potts, R. (1952). Some generalized order-disorder transformation. Proceedings of the Cambridge Philosophical Society, 48, 106–109. zbMATHCrossRefMathSciNetGoogle Scholar
  31. Rousson, M., & Deriche, R. (2002). A variational framework for active and adaptive segmentation of vector-valued images. In Proc. IEEE workshop on motion and video computing (pp. 56–62), Orlando, Florida. Google Scholar
  32. Taron, M., Paragios, N., & Jolly, M.-P. (2004). Border detection on short axis echocardiographic views using an ellipse driven region-based framework. In LNCS : Vol. 3216. Medical image computing and computer assisted interventions (pp. 443–450). Berlin: Springer. Google Scholar
  33. Tsai, A., Yezzi, A., & Willsky, A. (2001). Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification. IEEE Transactions on Image Processing, 10(8), 1169–1186. zbMATHCrossRefGoogle Scholar
  34. Yuille, A., & Grzywacz, N. M. (1988). A computational theory for the perception of coherent visual motion. Nature, 333, 71–74. CrossRefGoogle Scholar
  35. Zhu, S.-C., & Yuille, A. (1996). Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18(9), 884–900. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Computer Vision GroupUniversity of BonnBonnGermany

Personalised recommendations