A Multiclass Anisotropic Mumford-Shah Functional for Segmentation of D-dimensional Vectorial Images

J. F. Garamendi, E. Schiavi

Abstract

We present a general model for multi-class segmentation of multi-channel digital images. It is based on the minimization of an anisotropic version of the Mumford-Shah energy functional in the class of piecewise constant functions. In the framework of geometric measure theory we use the concept of common interphases between regions (classes) and the value of the jump discontinuities of the (weak) solution between adjacent regions in order to define a minimal partition energy functional. The resulting problem is non-smooth and non-convex. Non-smoothness is dealt with highlighting the relationship of the proposed model with the well known Rudin, Osher and Fatemi model for image denoising when piecewise constant solutions (i.e partitions) are considered. Non-convexity is tackled with an optimal threshold of the ROF solution which we which generalize to multi-channel images through a probabilistic clustering. The optimal solution is then computed with a fixed point iteration. The resulting algorithm is described and results are presented showing the successful application of the method to Light Field (LF) images.

References

  1. Ambrosio, L., Fusco, N., and Pallara, D. (2000). Functions of Bounded Variation and free discontinuity problems. Clarendon Press, Oxford University.
  2. Bresson, X. and Chan, T. (2008). Fast dual minimization of the vectorial total variation norm and applications to color image processing. Inverse Problems and Imaging, 2(4):455-484.
  3. Brown, E. S., Chan, T. F., and Bresson, X. (2011). Completely Convex Formulation of the Chan-Vese Image Segmentation Model. International Journal of Computer Vision, 98(1):103-121.
  4. Chambolle, A. (2004). An Algorithm for Total Variation Minimization and Applications. Journal of Mathematical Imaging and Vision, 20(1/2):89-97.
  5. Chambolle, A. and Darbon, J. (2008). On Total Variation Minimization and Surface Evolution using Parametric Maximum Flows Antonin Chambolle and Jéroˆme Darbon. International Journal of Computer Vision, 84(April):288-307.
  6. Chambolle, A. and Lions, P.-L. (1997). Image recovery via total variation minimization and related problems. Numerische Mathematik, 76(2):167-188.
  7. Chambolle, A. and Pock, T. (2011). A first-order primaldual algorithm for convex problems with applications to imaging. In J. Math. Imaging Vision, volume 40, pages 120-145.
  8. Chan, T. F. and Vese, L. a. (2001). Active contours without edges. IEEE transactions on image processing : a publication of the IEEE Signal Processing Society, 10(2):266-77.
  9. Garamendi, J. F., Gaspar, F. J., Malpica, N., and Schiavi, E. (2013). Box relaxation schemes in staggered discretizations for the dual formulation of total variation minimization. IEEE transactions on image processing, 22(5):2030-43.
  10. Klann, E. and Ramlau, R. (2013). Regularization Properties of Mumford-Shah-Type Functionals with Perimeter and Norm Constraints for Linear Ill-Posed Problems. SIAM Journal on Imaging Sciences, 6(1):413--436.
  11. Lippman, R. (1908). La photographie intégrale. Acadéemie des sciences, pages 446-551.
  12. Mumford, D. and Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42(5):577-685.
  13. Ng, R. (2006). Digital light field photography. PhD thesis.
  14. Ng, R., Levoy, M., Duval, G., Horowitz, M., and Hanrahan, P. (2005). Light Field Photography with a Hand-held Plenoptic Camera. Informational, pages 1-11.
  15. Nikolova, M., Esedoglu, S., Chan, T. F., Esedo, S., and Glu, . (2006). Algorithms for Finding Global Minimizers of Image Segmentation and Denoising Models. SIAM Journal on Applied Mathematics, 66(5):1632-1648.
  16. Osher, S. and Paragios, N. (2003). Geometric level set methods in imaging, vision, and graphics.
  17. Perwass, C. and Wietzke, L. (2010). www.raytrix.de.
  18. Pock, T., Chambolle, A., Cremers, D., and Bischof, H. (2009). A convex relaxation approach for computing minimal partitions. In 2009 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, CVPR Workshops 2009, pages 810- 817.
  19. Reddy, D., Bai, J., and Ramamoorthi, R. (2013). External mask based depth and light field camera. Workshop Consumer Depth Cameras for Vision.
  20. Rousson, M. and Deriche, R. (2002). A variational framework for active and adaptative segmentation of vector valued images. In Proceedings - Workshop on Motion and Video Computing, MOTION 2002, pages 56-61. Institute of Electrical and Electronics Engineers Inc.
  21. Rudin, L., Osher, S., and Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, 60(1-4):259-268.
  22. Wanner, S., Meister, S., and Goldluecke, B. (2013a). Datasets and Benchmarks for Densely Sampled 4D Light Fields. In Vision, Modeling \& Visualization, pages 225--226.
  23. Wanner, S., Straehle, C., and Goldluecke, B. (2013b). Globally Consistent Multi-label Assignment on the Ray Space of 4D Light Fields. 2013 IEEE Conference on Computer Vision and Pattern Recognition, pages 1011-1018.
Download


Paper Citation


in Harvard Style

Garamendi J. and Schiavi E. (2017). A Multiclass Anisotropic Mumford-Shah Functional for Segmentation of D-dimensional Vectorial Images . In Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 4: VISAPP, (VISIGRAPP 2017) ISBN 978-989-758-225-7, pages 468-475. DOI: 10.5220/0006127804680475


in Bibtex Style

@conference{visapp17,
author={J. F. Garamendi and E. Schiavi},
title={A Multiclass Anisotropic Mumford-Shah Functional for Segmentation of D-dimensional Vectorial Images},
booktitle={Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 4: VISAPP, (VISIGRAPP 2017)},
year={2017},
pages={468-475},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0006127804680475},
isbn={978-989-758-225-7},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 4: VISAPP, (VISIGRAPP 2017)
TI - A Multiclass Anisotropic Mumford-Shah Functional for Segmentation of D-dimensional Vectorial Images
SN - 978-989-758-225-7
AU - Garamendi J.
AU - Schiavi E.
PY - 2017
SP - 468
EP - 475
DO - 10.5220/0006127804680475