The tree of shapes of an image

Coloma Ballester; Vicent Caselles; P. Monasse

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 9, page 1-18
  • ISSN: 1292-8119

Abstract

top
In [CITE], Kronrod proves that the connected components of isolevel sets of a continuous function can be endowed with a tree structure. Obviously, the connected components of upper level sets are an inclusion tree, and the same is true for connected components of lower level sets. We prove that in the case of semicontinuous functions, those trees can be merged into a single one, which, following its use in image processing, we call “tree of shapes”. This permits us to solve a classical representation problem in mathematical morphology: to represent an image in such a way that maxima and minima can be computationally dealt with simultaneously. We prove the finiteness of the tree when the image is the result of applying any extrema killer (a classical denoising filter in image processing). The shape tree also yields an easy mathematical definition of adaptive image quantization.

How to cite

top

Ballester, Coloma, Caselles, Vicent, and Monasse, P.. "The tree of shapes of an image." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 1-18. <http://eudml.org/doc/90691>.

@article{Ballester2010,
abstract = { In [CITE], Kronrod proves that the connected components of isolevel sets of a continuous function can be endowed with a tree structure. Obviously, the connected components of upper level sets are an inclusion tree, and the same is true for connected components of lower level sets. We prove that in the case of semicontinuous functions, those trees can be merged into a single one, which, following its use in image processing, we call “tree of shapes”. This permits us to solve a classical representation problem in mathematical morphology: to represent an image in such a way that maxima and minima can be computationally dealt with simultaneously. We prove the finiteness of the tree when the image is the result of applying any extrema killer (a classical denoising filter in image processing). The shape tree also yields an easy mathematical definition of adaptive image quantization. },
author = {Ballester, Coloma, Caselles, Vicent, Monasse, P.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Image representation; mathematical morphology; tree structure; level sets.; image representation; level sets},
language = {eng},
month = {3},
pages = {1-18},
publisher = {EDP Sciences},
title = {The tree of shapes of an image},
url = {http://eudml.org/doc/90691},
volume = {9},
year = {2010},
}

TY - JOUR
AU - Ballester, Coloma
AU - Caselles, Vicent
AU - Monasse, P.
TI - The tree of shapes of an image
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 9
SP - 1
EP - 18
AB - In [CITE], Kronrod proves that the connected components of isolevel sets of a continuous function can be endowed with a tree structure. Obviously, the connected components of upper level sets are an inclusion tree, and the same is true for connected components of lower level sets. We prove that in the case of semicontinuous functions, those trees can be merged into a single one, which, following its use in image processing, we call “tree of shapes”. This permits us to solve a classical representation problem in mathematical morphology: to represent an image in such a way that maxima and minima can be computationally dealt with simultaneously. We prove the finiteness of the tree when the image is the result of applying any extrema killer (a classical denoising filter in image processing). The shape tree also yields an easy mathematical definition of adaptive image quantization.
LA - eng
KW - Image representation; mathematical morphology; tree structure; level sets.; image representation; level sets
UR - http://eudml.org/doc/90691
ER -

References

top
  1. L. Alvarez, F. Guichard, P.L. Lions and J.M. Morel, Axioms and fundamental equations of image processing. Arch. Rational Mech. Anal.16 (1993) 200-257.  
  2. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs (2000).  
  3. L. Ambrosio, V. Caselles, S. Masnou and J.M. Morel, The Connected Components of Sets of Finite Perimeter. Eur. J. Math.3 (2001) 39-92.  
  4. C. Ballester, E. Cubero-Castan, M. Gonzalez and J.M. Morel, Image intersection and applications to satellite imaging, Preprint. C.M.L.A., École Normale Supérieure de Cachan (1998).  
  5. C. Ballester and V. Caselles, The M-components of level sets of continuous functions in WBV. Publ. Mat.45 (2001) 477-527.  
  6. V. Caselles and P. Monasse, Grain filters. J. Math. Imaging Vision (to appear).  
  7. V. Caselles, B. Coll and J.M. Morel, Topographic Maps and Local Contrast Changes in Natural Images. Int. J. Comput. Vision33 (1999) 5-27.  
  8. V. Caselles, J.L. Lisani, J.M. Morel and G. Sapiro, Shape Preserving Histogram Modification. IEEE Trans. Image Process.8 (1999).  
  9. T. Chan, G. Golub and P. Mulet, A Nonlinear Primal-Dual Method for TV-Based Image Restoration, in ICAOS'96, 12th International Conference on Analysis and Optimization of Systems: Images, Wavelets and PDE's, edited by M. Berger,R. Deriche, I. Herlin, J. Jaffre and J.M. Morel, Lecture Notes in Control and Inform. Sci.219 (1996) 241-252.  
  10. J.L. Cox and D.B. Karron, Digital Morse Theory (1998), available at http://www.casi.net  
  11. E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del Calcolo delle Variazioni. Atti Accad. Naz. Lincei8 (1988) 199-210.  
  12. A. Desolneux, L. Moisan and J.M. Morel, Edge Detection by Helmholtz Principle. J. Math. Imaging Vision14 (2001) 271-284.  
  13. F. Dibos and G. Koepfler, Total Variation Minimization by the Fast Level Sets Transform, in Proc. of IEEE Workshop on Variational and Level Sets Methods in Computer Vision (2001).  
  14. F. Dibos, G. Koepfler and P. Monasse, Total Variation Minimization: Application to Gray-scale, Color Images and Optical Flow Regularization, in Geometric Level Sets Methods in Imaging, Vision and Graphics, edited by S. Osher and N. Paragios (to appear).  
  15. F. Dibos, G. Koepfler and P. Monasse, Image Registration, in Geometric Level Sets Methods in Imaging, Vision and Graphics, edited by S. Osher and N. Paragios (to appear).  
  16. S. Durand, F. Malgouyres and B. Rougé, Image Deblurring, Spectrum Interpolation and Application to Satellite Imaging. Math. Model. Numer. Anal. (to appear).  
  17. L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. Studies in Advanced Math., CRC Press (1992).  
  18. J. Froment, A compact and multiscale image model based on level sets, in Proc. of the 2nd Workshop on Scale-Space Theories in Computer Vision. Corfu, Greece (1999) 152-163.  
  19. M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities. Springer-Verlag (1973).  
  20. L. Alvarez, Y. Gousseau and J.M. Morel, Scales in natural images and a consequence on their BV norm, in Proc. of the 2nd Workshop on Scale-Space Theories in Computer Vision. Corfu, Greece (1999) 247-258.  
  21. Y. Gousseau and J.M. Morel, Texture Synthesis through Level Sets. Preprint CMLA (2000).  
  22. F. Guichard and J.M. Morel, Image iterative smoothing and P.D.E.'s (in preparation).  
  23. C. Kuratowski, Topologie I, II. Editions J. Gabay (1992).  
  24. J.-L. Lisani, L. Moisan, P. Monasse and J.M. Morel, Affine Invariant Mathematical Morphology Applied to a Generic Shape Recognition Algorithm. Comp. Imaging and Vision18 (2000).  
  25. D. Marr, Vision. Freeman and Co. (1981).  
  26. S. Masnou, Filtrage et désocclusion d'images par méthodes d'ensembles de niveau, Ph.D. Thesis. Ceremade, Université Paris-Dauphine (1998).  
  27. F. Meyer and S. Beucher, Morphological Segmentation. J. Visual Commun. Image Representation1 (1990) 21-46.  
  28. J.M. Morel and S. Solimini, Variational methods in image processing. Birkhäuser (1994).  
  29. J. Milnor, Morse Theory. Princeton University Press, Annals Math. Studies51 (1963).  
  30. A.S. Kronrod, On functions of two variables. Uspehi Mathematical Sciences (NS) 5 (1950) 24-134.  
  31. S. Mallat, A Wavelet Tour of Signal Processing. Academic Press, New York (1998).  
  32. G. Matheron, Random Sets and Integral Geometry. John Wiley, NY (1975).  
  33. F. Meyer and P. Maragos, Morphological scale-space representation with levelings, in Proc. of the 2nd Workshop on Scale-Space Theories in Computer Vision (1999) 187-198.  
  34. P. Monasse, Contrast invariant image registration, in Proc. of International Conference on Acoustics. Speech and Signal Process.6 (1999) 3221-3224.  
  35. P. Monasse and F. Guichard, Fast computation of a contrast-invariant image representation. IEEE Trans. Image Process.9 (2000) 860-872.  
  36. P. Monasse and F. Guichard, Scale-space from a level lines tree. J. Visual Commun. Image Representation11 (2000) 224-236.  
  37. D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and variational problems. Comm. Pure Appl. Math.42 (1988) 577-685.  
  38. M.H.A. Newman, Elements of the Topology of Plane Sets of Points. Dover Publications, New York (1992).  
  39. L.I. Rudin, S. Osher and E. Fatemi, Nonlinear Total Variation Based Noise Removal Algorithms. Physica D60 (1990) 259-269.  
  40. P. Salembier, Morphological multiscale segmentation for image coding. IEEE Trans. Signal Process.38 (1994) 359-386.  
  41. P. Salembier, Region-based filtering of images and video sequences: A morphological viewpoint. Preprint (2000).  
  42. P. Salembier and J. Serra, Flat zones filtering, connected operators and filters by reconstruction. IEEE Trans. Image Process.4 (1995) 1153-1160.  
  43. P. Salembier, P. Brigger, J.R. Casas and M. Pardàs, Morphological Operators for Image and Video Compression. IEEE Trans. Image Process.5 (1996) 881-897.  
  44. P. Salembier and L. Garrido, Binary partition tree as an efficient representation for image processing, segmentation, and information retrieval. IEEE Trans. Image Process.9 (2000).  
  45. J. Serra, Image analysis and mathematical morphology. Academic Press (1982).  
  46. J. Serra, Image analysis and mathematical morphology. Volume 2: Theoretical Advances. Academic Press (1988).  
  47. J. Serra and P. Salembier, Connected operators and pyramids, in Proc. SPIE Image Algebra Math. Morphology. San Diego, CA, SPIE 2030 (1993) 65-76.  
  48. G. Sapiro and A. Tannenbaum, On affine plane curve evolution. J. Funct. Anal.119 (1994) 79-120.  
  49. G. Sapiro and A. Tannenbaum, Affine invariant scale space. Int. J. Comput. Vision11 (1993) 24-44.  
  50. C. Vachier, Valuation of image extrema using alterning filters by reconstruction, in Proc. SPIE, Image Algebra and Morphological Processing (1995).  
  51. L. Vincent, Grayscale area openings and closings, their efficient implementation and applications, in Proc. of the 1st Workshop on Mathematical Morphology and its Applications to Signal Processing, edited by J. Serra and Ph. Salembrier. Barcelona, Spain (1993) 22-27.  
  52. L. Vincent, Morphological area openings and closings for grey-scale images, in Proc. of the Workshop Shape in Picture: Mathematical Description of Shape in Gray-Level Images. Driebergen, The Netherlands (1994) 197-208.  
  53. L. Vincent and P. Soille, Watersheds in digital spaces: An efficient algorithm based on immersion simulations. IEEE Trans. Pattern Anal. Machine Intell.13 (1991) 583-598.  
  54. C.R. Vogel and M.E. Oman, Iterative Methods for Total Variation Denoising. SIAM J. Sci. Comput. (to appear).  
  55. M. Wertheimer, Untersuchungen zur Lehre der Gestalt, II. Psychologische Forschung4 (1923) 301-350.  
  56. L.P. Yaroslavsky and M. Eden, Fundamentals of digital optics. Birkhäuser, Boston (1996).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.