The tree of shapes of an image
ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 1-18.

In [30], 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.

DOI : https://doi.org/10.1051/cocv:2002069
Classification : 68U10,  O5C05
Mots clés : image representation, mathematical morphology, tree structure, level sets
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Ballester, Coloma; Caselles, Vicent; Monasse, P. The tree of shapes of an image. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 1-18. doi : 10.1051/cocv:2002069. http://www.numdam.org/articles/10.1051/cocv:2002069/

[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. | MR 1225209 | Zbl 0788.68153

[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs (2000). | MR 1857292 | Zbl 0957.49001

[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. | MR 1812124 | Zbl 0981.49024

[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. | MR 1876918 | Zbl 0991.54013

[6] V. Caselles and P. Monasse, Grain filters. J. Math. Imaging Vision (to appear). | MR 1945474 | Zbl 1028.68140

[7] V. Caselles, B. Coll and J.M. Morel, Topographic Maps and Local Contrast Changes in Natural Images. Int. J. Comput. Vision 33 (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. | Zbl 0852.68114

[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. Lincei 8 (1988) 199-210. | EuDML 287202 | Zbl 0715.49014

[12] A. Desolneux, L. Moisan and J.M. Morel, Edge Detection by Helmholtz Principle. J. Math. Imaging Vision 14 (2001) 271-284. | Zbl 0988.68819

[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). | MR 2071628

[16] S. Durand, F. Malgouyres and B. Rougé, Image Deblurring, Spectrum Interpolation and Application to Satellite Imaging. Math. Model. Numer. Anal. (to appear). | Numdam | MR 1789371 | Zbl 0946.68150

[17] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. Studies in Advanced Math., CRC Press (1992). | MR 1158660 | Zbl 0804.28001

[18] J. Froment, A compact and multiscale image model based on level sets1999) 152-163.

[19] M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities. Springer-Verlag (1973). | MR 341518 | Zbl 0294.58004

[20] L. Alvarez, Y. Gousseau and J.M. Morel, Scales in natural images and a consequence on their BV norm1999) 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). | MR 1296876 | Zbl 0849.01044

[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 Vision 18 (2000). | Zbl 1073.68774

[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 Representation 1 (1990) 21-46.

[28] J.M. Morel and S. Solimini, Variational methods in image processing. Birkhäuser (1994). | Zbl 0827.68111

[29] J. Milnor, Morse Theory. Princeton University Press, Annals Math. Studies 51 (1963). | MR 163331 | Zbl 0108.10401

[30] A.S. Kronrod, On functions of two variables. Uspehi Mathematical Sciences (NS) 5 (1950) 24-134. | MR 34826 | Zbl 0040.31603

[31] S. Mallat, A Wavelet Tour of Signal Processing. Academic Press, New York (1998). | MR 1614527 | Zbl 0937.94001

[32] G. Matheron, Random Sets and Integral Geometry. John Wiley, NY (1975). | MR 385969 | Zbl 0321.60009

[33] F. Meyer and P. Maragos, Morphological scale-space representation with levelings1999) 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 Representation 11 (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. | MR 997568 | Zbl 0691.49036

[38] M.H.A. Newman, Elements of the Topology of Plane Sets of Points. Dover Publications, New York (1992). | MR 1160354 | Zbl 0845.54002

[39] L.I. Rudin, S. Osher and E. Fatemi, Nonlinear Total Variation Based Noise Removal Algorithms. Physica D 60 (1990) 259-269. | Zbl 0780.49028

[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). | MR 753649 | Zbl 0565.92001

[46] J. Serra, Image analysis and mathematical morphology. Volume 2: Theoretical Advances. Academic Press (1988). | MR 949918 | Zbl 0565.92001

[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. | MR 1283596

[48] G. Sapiro and A. Tannenbaum, On affine plane curve evolution. J. Funct. Anal. 119 (1994) 79-120. | MR 1255274 | Zbl 0801.53008

[49] G. Sapiro and A. Tannenbaum, Affine invariant scale space. Int. J. Comput. Vision 11 (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 applications1993) 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). | MR 1375276 | Zbl 0847.65083

[55] M. Wertheimer, Untersuchungen zur Lehre der Gestalt, II. Psychologische Forschung 4 (1923) 301-350.

[56] L.P. Yaroslavsky and M. Eden, Fundamentals of digital optics. Birkhäuser, Boston (1996). | Zbl 0877.94006

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