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[OLS 94] O LSEN L, Random geometrically graph directed self-similar multifractals , in Pitman Research Notes in Mathematics, Series 307, 1994 [OLS 02] O LSEN L, A multifractal formalism , Advances in Mathematics, 2002 [RIE 94] R IEDI R, Explicit bounds for the Hausdorff dimension of certain self-similar sets , in N OVAK MM (Ed), Fractals in the Natural and Applied Sciences, North-Holland, IFIP Transactions, p 313 324, 1994 [RIE 95] R IEDI R, M ANDELBROT BB, Multifractal formalism for in nite multinomial measures , Advances in Applied Mathematics, vol 16, p 132 150, 1995 [TRIC 82] T RICOT C, Two de nitions of fractional dimension , Math Proc Camb Phil Soc, vol 91, p 54 74, 1982 [TRIC 93] T RICOT C, Courbes et dimension fractale, Springer-Verlag, 1993 [TRIE 78] T RIEBEL H, Spaces of Besov-Hardy-Sobolev Type, Teubner, Texte zur Mathematik, 1978 [VOJ 95] VOJAK R, L VY V HEL J, Higher order multifractal analysis , SIAM Journal on Mathematical Analysis, 1995 [VRS 90] V RSCAY ER, Moment and collage methods for the inverse problem of fractal construction with iterated function systems , in Proceedings of the Fractal 90 Conference (Lisbon, Portugal), 1990 [VRS 91a] V RSCAY ER, Iterated function systems: Theory, applications, and the inverse problem , in Fractal Geometry and Analysis, p 405 468, 1991 [VRS 91b] V RSCAY ER, W EIL D, Missing moment and perturbative methods for polynomial iterated function systems , Physica D, vol 50, p 478 492, 1991 [WEI 95] W EIERSTRASS K, On continuous function of a real argument that do not have a well-de ned differential quotient , Mathematische Werke, p 71 74, 1895
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101 Introduction An iterated function system (IFS) makes it possible to generate fractal images from a set of contracting transformations [BARN 88] It is also possible to rely on a simple fractal image to determine the parameters of contracting transformations, thereby allowing us to synthesize it (in this case, such a process is referred to as the inverse problem) The rst part of this chapter is devoted to the review of some basic concepts necessary for the comprehension of IFS theory An adaptation of this theory, initially proposed in [JACQ 92], makes it possible to generate non-fractal images of natural scenes from a set of local contracting transformations The purpose of the second part of this chapter is to introduce this method, which can be used for analyzing or coding images We then describe the principles of natural image coding by fractals, which consists of the automatic calculation of the transformation parameters that enables us to generate a given image Finally, we present various solutions which allow us to speed up the automatic calculation of the contracting transformations and to improve the quality of reconstructed images 102 Iterated transformation systems In this section, we recall the main assets of the iterated transformation systems theory, which enables the coding and synthesis of binary fractal images
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and of gray-level images The concept of IFS is also discussed, from a broader perspective, in 9 for the generation of the continuous function graphs and the characterization of H lder functions 1021 Contracting transformations and iterated transformation systems 10211 Lipschitzian transformation Let : R2 R2 be a transformation de ned on the metric space (R2 , d) The symbol d indicates the distance between two points of R2 The transformation is called Lipschitzian with strictly real positive Lipschitz factor s, if: d (x), (y) 10212 Contracting transformation Let : R2 R2 be a transformation de ned on the metric space (R2 , d) The transformation is said to be contracting with real contraction factor s, 0 < s < 1, if: d (x), (y) 10213 Fixed point A contracting transformation possesses a single xed point xf R2 , such that (xf ) = xf Let us note by on ( ) the application of reiterated n times For any point x element of R2 , the sequence { on (x) : n = 0, 1, 2, } converges towards xf :
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