# a functional analysis model for natural images permitting structured compression

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

- Volume: 4, page 473-495
- ISSN: 1292-8119

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topFroment, Jacques. "a functional analysis model for natural images permitting structured compression ." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 473-495. <http://eudml.org/doc/197315>.

@article{Froment2010,

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.
},

author = {Froment, Jacques},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Perceptual image model; total variation; mathematical
morphology; segmentation; image compression.; perceptual image model; total vatiation; mathematical morphology; image compression},

language = {eng},

month = {3},

pages = {473-495},

publisher = {EDP Sciences},

title = {a functional analysis model for natural images permitting structured compression },

url = {http://eudml.org/doc/197315},

volume = {4},

year = {2010},

}

TY - JOUR

AU - Froment, Jacques

TI - a functional analysis model for natural images permitting structured compression

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 4

SP - 473

EP - 495

AB -
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.

LA - eng

KW - Perceptual image model; total variation; mathematical
morphology; segmentation; image compression.; perceptual image model; total vatiation; mathematical morphology; image compression

UR - http://eudml.org/doc/197315

ER -

## References

top- 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.
- L. Alvarez, F. Guichard, P.L. Lions and J.M. Morel, Axioms and fundamental equations of image processing. Arch. Rational Mechanics and Anal.16 (1993) 200-257. Zbl0788.68153
- F. Cao, Absolutely minimizing Lipschitz extension with discontinuous boundary data. C.R. Acad. Sci. Paris327, Série I (1998) 563-568. Zbl0990.35048
- J. Canny, A computational approach to edge detection. IEEE Trans. Pattern Analysis and Machine Intelligence8 (1986) 679-698.
- S. Carlsson, Sketch based coding of grey level images. Signal Processing North-Holland15 (1998) 57-83.
- 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).
- V. Caselles, B. Coll and J.M. Morel, A Kanizsa programme. Progress in Nonlinear Differential Equs. and their Applications25 (1996) 35-55.
- V. Caselles, J.M. Morel and C. Sbert, An Axiomatic Approach to Image Interpolation. IEEE Trans. on Image Processing7 (1998) 376-386. Zbl0993.94504
- 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).
- A. Chambolle and P.L. Lions, Image Recovery via Total Variation Minimization and Related Problems. Numerische Mathematik76 (1997) 167-188. Zbl0874.68299
- A. Cohen, I. Daubechies and J.C. Feauveau, Biorthogonal bases of compactly supported wavelets. Comm. in Pure & Appl. Math.45 (1992). Zbl0776.42020
- 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). Zbl0957.49002
- L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press Inc. (1992). Zbl0804.28001
- 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 Processing37 (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.
- J. Li, P.Y. Cheng and C.C.J. Kuo, An embedded wavelet packed transform technique for texture compression. SPIE2569 (1995) 602-613.
- S. Mallat and S. Zhong, Characterization of signals from multiscale edges. IEEE Trans. Pattern Recognition and Machine Intelligence14 (1992) 710-732.
- 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). Zbl0321.60009
- 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). Zbl0821.42018
- 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). Zbl0827.68111
- L.I. Rudin and S. Osher, Total Variation based image restoration with free local constraints, in Proc. IEEE Int. Conf. on Image Processing1 (1994) 31-35.
- J. Serra, Image analysis and mathematical morphology, Academic Press (1982). Zbl0565.92001
- J. Shapiro, Embedded Image Coding Using Zerotrees of Wavelet Coefficients. IEEE Trans. on Signal Processing41 (1993) 3445-3462. Zbl0841.94020
- M. Wertheimer, Untersuchungen zur Lehre der Gestalt, Psychologische Forschung, IV (1923) 301-350.

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