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a functional analysis model for natural images permitting structured compression

Published online by Cambridge University Press:  15 August 2002

Jacques Froment
Jacques Froment*
Affiliation:
PRISME, UFR de Math. et Info., Université R. Descartes Paris 5, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France. CMLA, École Normale Supérieure de Cachan, 61 avenue du Président Wilson, 94235 Cachan Cedex, France.
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Abstract

This paper describes a compact perceptual image model intended for morphological representation of the visual information contained in natural images. We explain why the total variation can be a criterion to split the information between the two main visual structures, which are the sketch and the microtextures. We deduce a morphological decomposition scheme, based on a segmentation where the borders of the regions correspond to the location of the topological singularities of a topographic map. This leads to propose a new and morphological definition of edges. The sketch is computed by approximating the image with a piecewise smooth non-oscillating function, using a Lipshitz interpolant given as the solution of a PDE. The data needed to reconstruct the sketch image are very compact, so that an immediate outcome of this image model is the design of a progressive, and artifact-free, image compression scheme.

Keywords

Perceptual image modeltotal variationmathematical morphologysegmentationimage compression.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 473 - 495
Copyright
© EDP Sciences, SMAI, 1999

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References

J.P. D'Alès, J. Froment and J.M. Morel, Reconstruction visuelle et généricité, Second European Workshop on Image Processing and Mean Curvature Motion, UIB, Spain (1995) 1-21.
Alvarez, L., Guichard, F., Lions, P.L. and Morel, J.M., Axioms and fundamental equations of image processing. Arch. Rational Mechanics and Anal. 16 (1993) 200-257.
F. Cao, Absolutely minimizing Lipschitz extension with discontinuous boundary data. C.R. Acad. Sci. Paris 327, Série I (1998) 563-568.
Canny, J., A computational approach to edge detection. IEEE Trans. Pattern Analysis and Machine Intelligence 8 (1986) 679-698. CrossRef
Carlsson, S., Sketch based coding of grey level images. Signal Processing North-Holland 15 (1998) 57-83. CrossRef
J.R. Casas, Morphological Interpolation for Image Coding, 12th International Conference on Analysis and Optimization of Systems. Images, Wavelets and PDEs. M.O. Berger, R. Deriche, I. Herlin, J. Jaffré, J.M. Morel, Eds., Springer, Paris (1996).
Caselles, V., Coll, B. and Morel, J.M., Kanizsa, A programme. Progress in Nonlinear Differential Equs. and their Applications 25 (1996) 35-55.
Caselles, V., Morel, J.M. and Sbert, C., Axiomatic Approach, An to Image Interpolation. IEEE Trans. on Image Processing 7 (1998) 376-386. CrossRefPubMed
V. Caselles, B. Coll and J.M. Morel, Topographic maps and local contrast changes in natural images. Internat. J. Comput. Vision, to appear.
V. Caselles, B. Coll and J.M. Morel, The Connected Components of Sets of Finite Perimeter in the Plane, Preprint (1998).
Chambolle, A. and Lions, P.L., Image Recovery via Total Variation Minimization and Related Problems. Numerische Mathematik 76 (1997) 167-188. CrossRef
A. Cohen, I. Daubechies and J.C. Feauveau, Biorthogonal bases of compactly supported wavelets. Comm. in Pure & Appl. Math. 45 (1992).
M.G. Crandall, H. Ishii and P.L. Lions, User's Guide to Viscosity Solution of Second Order Partial Differential Equations. Bull. Amer. Math. Soc. 27 (1992) 1-67.
F. Dibos and G. Koepfler, Global Total Variation Minimization, TR 9801, CEREMADE, Université Paris-Dauphine, France (1998).
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press Inc. (1992).
J. Froment and S. Mallat, Second Generation Compact Image Coding with Wavelets, Wavelets - A Tutorial in Theory and Applications, C.K. Chui, Ed., Academic Press (1992) 655-678.
J. Froment, Vol. 1: MegaWave2 User's Guide, Vol. 2: MegaWave2 System Library, Vol. 3: MegaWave2 User's Modules Library, The Preliminary Guides to the MegaWave2 Software (1998). A version is available at http://www.cmla.ens-cachan.fr/Cmla/Megawave/ into the software package.
R. Hummel and R. Moniot, Reconstruction from zero-crossings in scale-space. IEEE Trans. on Acoustic, Speech and Signal Processing 37 (1989).
G. Kanisza, Grammatica del Vedere, Il Mulino, Bologna (1980).
G. Kanisza, Vedere e pensare, Il Mulino, Bologna (1991).
M. Kunt, M. Bénard and R. Leonardi, Recent Results in High-Compression Image Coding. IEEE Trans. on Circuits and Systems (1987) 1306-1336.
Li, J., Cheng, P.Y. and Kuo, C.C.J., An embedded wavelet packed transform technique for texture compression. SPIE 2569 (1995) 602-613.
Mallat, S. and Zhong, S., Characterization of signals from multiscale edges. IEEE Trans. Pattern Recognition and Machine Intelligence 14 (1992) 710-732. CrossRef
D. Marr, Vision, W.H. Freeman and Co. (1982).
S. Masnou and J.M. Morel, Level lines based disocclusion, ICIP'98 IEEE Int. Conf. on Image Processing Chicago (1998).
G. Matheron, Random Sets and Integral Geometry, John Wiley, N.Y. (1975).
F.G. Meyer, A.Z. Averbuch, J-O. Strömberg and R.R. Coifman, Multi-layered image representation: Application to image compression, ICIP'98 IEEE Int. Conf. on Image Processing, Chicago (1998).
Y. Meyer, Wavelets: Algorithms and applications, SIAM (1993).
P. Monasse and F. Guichard, Fast computation of a contrast-invariant image representation, TR 9815, CMLA, ENS Cachan, France, IEEE Trans. on image processing, submitted.
J.M. Morel, Perception visuelle et traitement d'images (I) et (II), Revue du Palais de la Découverte (novembre 1996 et Février 1997).
J.M. Morel and S. Solimini, Variational methods in image processing, Birkhäuser (1994).
L.I. Rudin and S. Osher, Total Variation based image restoration with free local constraints, in Proc. IEEE Int. Conf. on Image Processing 1 (1994) 31-35.
J. Serra, Image analysis and mathematical morphology, Academic Press (1982).
Shapiro, J., Embedded Image Coding Using Zerotrees of Wavelet Coefficients. IEEE Trans. on Signal Processing 41 (1993) 3445-3462. CrossRef
M. Wertheimer, Untersuchungen zur Lehre der Gestalt, Psychologische Forschung, IV (1923) 301-350.