Skip to main content
SpringerLink
Log in

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

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • 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.

    Article  MathSciNet  Google Scholar 

  • Blake, A., & Zisserman, A. (1987). Visual Reconstruction. Cambridge: MIT Press.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

  • 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.

  • 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.

    Chapter  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • Caselles, V., Catté, F., Coll, T., & Dibos, F. (1993). A geometric model for active contours in image processing. Numerische Mathematik, 66, 1–31.

    Article  MATH  MathSciNet  Google Scholar 

  • Chan, T., & Vese, L. (2001). Active contours without edges. IEEE Transactions on Image Processing, 10(2), 266–277.

    Article  MATH  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

  • 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.

    Article  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Deriche, R. (1990). Fast algorithms for low-level vision. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12, 78–87.

    Article  Google Scholar 

  • 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.

    Article  MATH  Google Scholar 

  • 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 

  • Heiler, M., & Schnörr, C. (2005). Natural image statistics for natural image segmentation. International Journal of Computer Vision, 63(1), 5–19.

    Article  Google Scholar 

  • Ising, E. (1925). Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik, 31, 253–258.

    Article  Google Scholar 

  • Kass, M., Witkin, A., & Terzopoulos, D. (1988). Snakes: active contour models. International Journal of Computer Vision, 1, 321–331.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Lenz, W. (1920). Beitrag zum Verständnis der magnetischen Erscheinungen in festen Körpern. Physikalische Zeitschrift, 21, 613–615.

    Google Scholar 

  • Morel, J.-M., & Solimini, S. (1994). Variational methods in image segmentation. Basel: Birkhäuser.

    MATH  Google Scholar 

  • Mumford, D., & Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42, 577–685.

    Article  MATH  MathSciNet  Google Scholar 

  • Nielsen, M., Florack, L., & Deriche, R. (1994). Regularization and scale space (Technical Report 2352). INRIA, Sophia-Antipolis, France.

  • Nielsen, M., Florack, L., & Deriche, R. (1997). Regularization, scale-space and edge detection filters. Journal of Mathematical Imaging and Vision, 7, 291–307.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Parzen, E. (1962). On the estimation of a probability density function and the mode. Annals of Mathematical Statistics, 33, 1065–1076.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • Potts, R. (1952). Some generalized order-disorder transformation. Proceedings of the Cambridge Philosophical Society, 48, 106–109.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

  • 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 

  • 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.

    Article  MATH  Google Scholar 

  • Yuille, A., & Grzywacz, N. M. (1988). A computational theory for the perception of coherent visual motion. Nature, 333, 71–74.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Vision Group, University of Bonn, Römerstr. 164, 53117, Bonn, Germany

    Thomas Brox & Daniel Cremers

Authors
  1. Thomas Brox
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Cremers
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Brox.

Additional information

We thank the anonymous reviewers for comments leading to improvements of the manuscript, and we gratefully acknowledge funding by the German Research Foundation (DFG) under the project Cr250/1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brox, T., Cremers, D. On Local Region Models and a Statistical Interpretation of the Piecewise Smooth Mumford-Shah Functional. Int J Comput Vis 84, 184–193 (2009). https://doi.org/10.1007/s11263-008-0153-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11263-008-0153-5

Keywords

Search

Navigation