Figure 1123 (a) Local H lder exponents image and (b) changes detected using multifractal analysis

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Scaling, Fractals and Wavelets

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1110 Image reconstruction In [TUR 02], the authors describe an interesting method for image reconstruction that mixes multifractal analysis and more traditional diffusion-like techniques The image I (or, more precisely, the modulus of its gradient) is rst modeled as a so-called log-Poisson multifractal measure This means that its spectrum f admits the following parametric form: f ( ) = f + 1 log (2 f ) ,

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where is the lowest observed exponent, f the associated spectrum value and := log[1 + /(2 f )] The justi cation for considering such a model is that it seems to describe many natural images with reasonable accuracy For most images, it is observed that f 1, and this is the value chosen for the rest of the method The points in the image having exponent are the most singular ones In many cases, the set comprising all these points, which is called the most singular manifold (MSM), is strongly related to the edges of the images This observation and the fact that the whole spectrum may be computed once is known suggest that the MSM contains the most relevant information, and that the whole image may be reconstructed from it In order to implement this idea, a linear operator is applied to the MSM This operator must satisfy the following natural constraints: it should be translation-invariant, isotropic, and allow us to recover the original power (Fourier) spectrum of the image Under these constraints, we may show that there is essentially exactly one possible operator, and that the image I may be reconstructed through the following formula: c(f ) = where c is the Fourier transform of c := I I0 , I0 is the mean of I, f denotes frequency, v0 (x) = c(x) if x belongs to the MSM, and v0 (x) = 0 otherwise Reconstruction of real-world images, such as outdoor scenes or natural textures, is surprisingly good considering the simplicity of the method if v0 (f ) , f2

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Local Regularity and Multifractal Methods for Image and Signal Analysis

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1111 Bibliography

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[AMA 99] A MARAL LAN, G OLDBERGER AL, I VANOV PC, S TANLEY HE Modeling heart rate variability by stochastic feedback , Computer Phys Comm, vol 121-122, p 126 128, 1999 [AYA 99] AYACHE A, L VY V HEL J, Generalized multifractional Brownian motion: de nition and preliminary results , in D EKKING M, L EVY V EHEL J, L UTTON E and T RICOT C (Eds), Fractals: Theory and Applications in Engineering, Springer, 1999 [AYA 00a] AYACHE A, The generalized multifractional eld: a nice tool for the study of the generalized multifractional Brownian motion , Journal of Fourier Analysis and Applications, vol 8, p 581 601, 2000 [AYA 00b] AYACHE A, L VY V HEL J, The generalized multifractional Brownian motion , Statistical Inference for Stochastic Processes 3, pp7 18, 2000 [BAR 07] BARRI RE O, Synth se et estimation de mouvements Browniens multifractionnaires et autres processus r gularit prescrite De nition du processus auto-regul multifractionnaire et applications, PhD Thesis, University of Nantes, 2007 [BIR 97] B IRG L, M ASSART P, From model selection to adaptive estimation , in Torgersen E, Pollard D, Yang G (Eds), Festschrift for Lucien Le Cam, Springer, New York, p 55 87, 1997 [CAN 96] C ANUS C, L VY V HEL J, Change detection in sequences of images by multifractal analysis , in Proc ICASSP-96, May 7-10, Atlanta, 1996 [DEV 92] D EVORE RA, L UCIER B, Fast wavelet techniques for near optimal image processing , IEEE Military Communications Conference, vol 2, no 12, 1992 [POP 88] D EVORE RA, P OPOV VA, Interpolation of Besov spaces , Transactions of the American Mathematical Society, vol 305, no 1, p 397 414, 1988 [DO 01] D O MT, Z AHOUANI H, Frottement pneumatique/chauss e, in uence de la microtexture des surfaces de chauss e, JIFT, 2001 [DON 94] D ONOHO DL, De-noising by soft-thresholding , IEEE Trans Inf Theory, vol 41, no 3, p 613 627, 1994 [ECH 07] E CHELARD A, Analyse 2-microlocale et application au d bruitage , PhD Thesis, University of Nantes, December 2007 [FracLab] FracLab: a software toolbox http://apissaclayinriafr/FracLab/ for fractal processing of signals,

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[GOL 02] G OLDBERGER AL, A MARAL LAN, H AUSDORFF JM, I VANOV PC, P ENG CK, S TANLEY HE, Fractal dynamics in physiology: alterations with disease and aging , PNAS, vol 99, p 2466 2472, 2002 [GUG 98] G UGLIELMI M, L VY V HEL J, Analysis and simulation of road pro le by means of fractal model, Conference on Advances in Vehicle Control and Safety (AVCS 98), Amiens, 1998 [HAR 98] H RDLE W, K ERKYACHARIAN G, P ICARD D, T SYBAKOV A, Wavelets, Approximation and Statistical Applications, Lecture Notes in Statistics, Springer, 1998

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