Skip to main content

Level Lines Selection with Variational Models for Segmentation and Encoding


This paper discusses the interest of the Tree of Shapes of an image as a region oriented image representation. The Tree of Shapes offers a compact and structured representation of the family of level lines of an image. This representation has been used for many processing tasks such as filtering, registration, or shape analysis. In this paper we show how this representation can be used for segmentation, rate distortion optimization, and encoding. We address the problem of segmentation and rate distortion optimization using Guigues algorithm on a hierarchy of partitions constructed using the simplified Mumford-Shah multiscale energy. To segment an image, we minimize the simplified Mumford-Shah energy functional on the set of partitions represented in this hierarchy. The rate distortion problem is also solved in this hierarchy of partitions. In the case of encoding, we propose a variational model to select a family of level lines of a gray level image in order to obtain a minimal description of it. Our energy functional represents the cost in bits of encoding the selected level lines while controlling the maximum error of the reconstructed image. In this case, a greedy algorithm is used to minimize the corresponding functional. Some experiments are displayed.

This is a preview of subscription content, access via your institution.


  1. M.D. Adams, “The jpeg-2000 still image compression standard.” Available at, 2001.

  2. C. Ballester, E. Cubero-Castan, M. Gonzalez, and J.M. Morel, “Image intersection and applications to satellite imaging,” Preprint, C.M.L.A., Ecole Normale Supérieure de Cachan, 1998.

  3. C. Ballester, V. Caselles, and P. Monasse, “The tree of shapes of an image,” ESAIM: Control, Opt. and Calc. of Variations, Vol. 9, pp. 1–18, 2003.

    MathSciNet  Google Scholar 

  4. T. Berger, Rate Distortion Theory: A Mathematical Basis for Data Compression. Prentice Hall, 1971.

  5. L. Breiman, J.H. Friedman, R.A. Olsen, and C. J. Stone, Classification and Regression Trees. Wadsworth, Belmont, California, 1984.

    MATH  Google Scholar 

  6. F. Cao, P. Musé, and F. Sur, “Extracting meaningful curves in images,” To appear in Journal of Mathematical Imaging and Vision, 2005.

  7. F. Cao, Y. Gousseau, J.M. Morel, P. Musé, and F. Sur, “An a contrario decision method for shape element recognition,” Preprint CMLA, No. 16, 2004.

  8. V. Caselles, B. Coll, and J.M. Morel, “Topographic maps and local contrast changes in natural images,” Int. J. Comp. Vision, Vol. 33, pp. 5–27, 1999.

    Article  Google Scholar 

  9. V. Caselles, J.L. Lisani, J.M. Morel, and G. Sapiro, “Shape preserving histogram modification,” IEEE Transactions on Image Processing, Vol. 8, No. 2, pp. 220–230, 1999.

    Article  Google Scholar 

  10. V. Caselles, J.M. Morel, and C. Sbert, “An axiomatic approach to image interpolation,” IEEE Transactions on Image Processing, Vol. 7, pp. 376–386, 1998.

    Article  MathSciNet  Google Scholar 

  11. V. Caselles, and P. Monasse, “Morse theories of the Topographic map,” In preparation.

  12. V. Caselles, G. Sapiro, A. Solé, and C. Ballester, “Morse description and morphological encoding of continuous data,” SIAM Journal on Multiscale Modeling and Simulation, Vol. 2, pp. 179–209, 2004.

    Article  Google Scholar 

  13. R. Chiariglioni, “MPEG and multimedia communitations,” IEEE Transactions on Circuits and Systems for Video Technology, Vol. 7, pp. 5–18, 1997.

    Article  Google Scholar 

  14. P.A. Chou, T. Lookabaugh, and R.M. Gray, “Optimal pruning with applications to tree-structured source coding and modeling,” IEEE Trans. Inform. Theory, Vol. 35, pp. 299–315, 1989.

    Article  MathSciNet  Google Scholar 

  15. A. Desolneux, L. Moisan, and J.M. Morel, “Edge detection by Helmholtz principle,” Journal of Mathematical Imaging and Vision, Vol. 14, pp. 271–284, 2001.

    Article  Google Scholar 

  16. M. Eden and M. Kocher, “On the performance of a contour coding algorithm in the context of image coding. Part I: Contour segment coding,” Signal Processing, Vol. 8, pp. 381–386, 1985.

    Article  Google Scholar 

  17. J. Froment, “A functional analysis model for natural images permitting structured compression,” ESAIM: Control, Optimization and Calculus of Variation, Vol. 4, pp. 473–495, 1999.

    Article  MathSciNet  Google Scholar 

  18. L. Garrido, “Hierarchical region based processing of images and video sequences: Application to filtering, sementation and information retrieval,” PhD thesis, Universitat Politécnica de Catalunya, 2002.

  19. F. Guichard and J.M. Morel, Image Iterative Smoothing and P.D.E.’s’. Book in preparation, 2000.

  20. L. Guigues, “Modèles Multi-Échelles pour la Segmentation d’Images,” PhD thesis, Université de Cergy-Pontoise, 2003.

  21. A. Gersho and R.M. Gray, Vector Quantization and Signal Compression, Kluwer Academic Publishers, 1992.

  22. J.M. Hyman, “Accurate monotonicity preserving cubic interpolation.” SIAM J. Sci. Stat. Comp, Vol. 4, pp. 645–654, 1983.

    Article  MathSciNet  Google Scholar 

  23. L. Igual, L. Garrido, and V. Caselles, “A contrast invariant approach to motion estimation, validation and motion segmentation,”

  24. G. Koepfler, C. Lopez, and J.M. Morel, “A multiscale algorithm for image segmentation by variational method,” SIAM J. Numer. Anal, Vol. 31, pp. 282–299, 1994.

    Article  MathSciNet  Google Scholar 

  25. C. Kuratowski, Topologie I, II. Editions J. Gabay: Paris, 1992.

  26. Y. G. Leclerc, “Constructing simple stable descriptions for image partitioning,” International Journal of Computer Vision, Vol. 3, pp. 73–102, 1989.

    Article  Google Scholar 

  27. F. Meyer, “The dynamics of minima and contours,” in Methematical Morphology and Its Applications to Image Processing, Atlanta (GA), USA, pp. 329–336, 1996.

  28. F. Meyer and S. Beucher, “Morphological segmentation,” J. of Visual Communication and Image Representation, Vol. 1, pp. 21–46, 1990.

    Article  Google Scholar 

  29. P. Monasse, “Représentation morphologique d’images numériques et application au recalage,” PhD thesis, Université de Paris-Dauphine, 2000.

  30. P. Monasse, “Contrast invariant image registration,” in: Proc. of International Conference on Acoustics, Speech and Signal Processing, Vol. 6, pp. 3221–3224, 1999.

  31. P. Monasse and F. Guichard, “Fast computation of a contrast invariant image representation,” IEEE Transactions on Image Processing, Vol. 9, pp. 860–872, 2000.

    Article  Google Scholar 

  32. J.M. Morel and S. Solimini, Variational Methods in Image Processing. Birkhäuser Verlag: Basel, 1994.

    MATH  Google Scholar 

  33. D. Mumford and J. Shah, “Optimal approximations by piecewise smooth functions and variational problems,” Communications on Pure and Applied Mathematics, Vol. 42, No. 5, pp. 577–685, 1988.

    MathSciNet  Google Scholar 

  34. J. Rissanen, “Minimum-Description-Length principle,” in Encyclopedia of Statistical Sciences. John Wiley: New York, 1987, Vol. 5, pp. 523–527.

  35. P. Salembier, “Morphological multiscale segmentation for image coding,” IEEE Transactions on Signal Processing, Vol. 38, No. 3, pp. 359–386, 1994.

    Google Scholar 

  36. P. Salembier, P. Brigger, J. R. Casas, and M. Pardàs, “Morphological operators for image and video compression,” IEEE Transactions on Image Processing, Vol. 5, pp. 881–897, 1996.

    Article  Google Scholar 

  37. P. Salembier and L. Garrido, “Binary partition tree as an efficient representation for image processing, segmentation, and information retrieval,” IEEE Transactions on Image Processing, Vol. 9, No. 4, pp. 561–576, 2000.

    Article  Google Scholar 

  38. P. Salembier, F. Marqués, M. Pardás, R. Morros, I. Corset, S. Jeannin, L. Bouchard, F. Meyer, and B. Marcotegui, “Segmentation-based video coding system allowing the manipulation of objects,” IEEE Transactions on Circuits and Systems for Video Technology, Vol. 7, pp. 60–73, 1997.

    Article  Google Scholar 

  39. P. Salembier and J. Serra, “Flat zones filtering, connected operators and filters by reconstruction,” IEEE Transactions Image Processing, Vol. 4, No. 8, pp. 1153–1160, 1995.

    Article  Google Scholar 

  40. G.M. Schuster and A.K. Katsaggelos, Rate-Distortion Based Video Compression. Kluwer Academic Publishers, 1997.

  41. Y. Shoham and A. Gersho, “Efficient bit allocation for an arbitrary set of quantizers.” IEEE Trans. on Acoust., Speech, Signal Processing, Vol. 36, No. 9, pp. 1445–1453, 1988.

    Article  Google Scholar 

  42. J. Serra, Image Analysis and Mathematical Morphology, Academic Press: New York, 1982.

    MATH  Google Scholar 

  43. T. Sikora, “The MPEG-7 visual standard for content description - an overview.” IEEE Transactions on Circuits and Systems for Video Technology, Vol. 11, pp. 696–702, 2001.

    Article  Google Scholar 

  44. P. Soille, Morphological Image Analysis. Springer Verlag, 2003.

  45. A. Solé, “Geometric image coding, filtering and restoration,” PhD thesis, Universitat Pompeu Fabra, 2002.

  46. L. Vincent, “Morphological area openings and closings for grey-scale images,” in Proceedings of the Workshop Shape in Picture: Mathematical Description of Shape in Gray-Level Images. Driebergen, The Netherlands, pp. 197–208, 1994.

  47. L. Vincent and P. Soille, “Watersheds in digital spaces: An efficient algorithm based on Immersion simulations,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 13, No. 6, pp. 583–598, 1991.

    Article  Google Scholar 

  48. M.J. Weinberger, G. Seroussi, and G. Sapiro, “The LOCO-I lossless image compression algorithm: Principles and standardization into JPEG-LS,” IEEE Transactions on Image Processing Vol. 9, No. 8, pp. 1309–1324, 2000.

    Article  Google Scholar 

  49. M. Wertheimer, “Untersuchungen zur Lehre der Gestalt, II,” Psychologische Forschung, Vol. 4, pp. 301–350, 1923.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Coloma Ballester.

Additional information

Coloma Ballester received the Licenciatura degree in mathematics from Barcelona University (UAB) and the Ph.D. degree in computer science from the University of Illes Balears, Spain, in 1995. Currently, she is an associate professor at the Pompeu Fabra University in Barcelona (Spain). Her research interests include image processing and computer vision.

Vicent Caselles received the Licenciatura and Ph.D. degrees in mathematics from Valencia University, Spain, in 1982 and 1985, respectively. Currently, he is professor at the Pompeu Fabra University (Barcelona). He is an associate member of IEEE. His research interests include image processing, computer vision, and the applications of geometry and partial differential equations to both previous fields.

Laura Igual received the degree of Mathematics from the University of Valencia in 2000 and the Ph.D. degree from the Pompeu Fabra University in January 2006. She has been working as research assistant at the Universitat Pompeu Fabra (Barcelona) from 2000 to 2005. Her research interests are several subjects of image processing: image segmentation and compression, motion estimation, and data interpolation on surfaces.

Luis Garrido received the degree of Telecommunication Engineering from the Telecommunication School of the Polytechnic University of Catalonia (UPC), Barcelona, Spain, in 1996. He joined afterwards the Image Processing Group at the UPC to work on his Ph.D. The topic of the work was the study of tree structures for region-based analysis of images and video sequences. Luis Garrido obtained the Ph.D. degree in June 2002.

In January 2003 he joined the Image Processing Group at the Universitat Pompeu Fabra (UPF), Barcelona, Spain. He currently has a Ramon y Cajal contract. His current research interests are contrast invariant motion estimation, tree based image representations and image segmentation.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ballester, C., Caselles, V., Igual, L. et al. Level Lines Selection with Variational Models for Segmentation and Encoding. J Math Imaging Vis 27, 5–27 (2007).

Download citation

  • Published:

  • Issue Date:

  • DOI:


  • mathematical morphology
  • tree structure
  • segmentation
  • rate distortion
  • morphological encoding
  • minimal description length