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logarithmic factors) of the trajectories Indeed, FBM is a good candidate for modeling phenomena which, using a statistical processing, are found to have C H trajectories and are supposed, for theoretical reasons, to be self-similar The importance of the identi cation of H, starting from the samples of the phenomenon necessarily taken in discrete time, is thus crucial At the same time, it is necessary to remember that FBMs are processes with stationary increments, which simpli es the spectral study of the process but is too restrictive for certain applications Indeed, in many elds (when we want to simulate textures in an image), it is expected, a priori, that order H depends on the point at which the process is observed For example, if, using a random eld, we want to model the aerial photographing of a forest, we would like to have a model where the parameter of self-similarity around the point x, noted by h(x), depends on the geological nature of the ground in the vicinity of x However, a spatial modulation of the eld law is generally incompatible with the property of self-similarity, which is the overall property Consequently, the problem consists of arriving at a concept of a suf ciently exible, locally self-similar eld, so that the parameters which de ne the eld law could vary with the position, yet be simple enough to enable the identi cation of these parameters Unfortunately, the simple approach, which consists of reproducing H by a function h(x) in the formula giving the covariance of a FBM, is not satisfactory: generally, we can show that there does not exist any Gaussian eld having this generalized covariance We will thus have to introduce the mathematical tools which will make it possible to build models generalizing FBMs and also to identify the functional parameters of these models These theoretical recaps will be dealt with in section 62, where we will consider each time the relevance of the concept introduced for an example of fractional Brownian motion In this context, we discuss traditional techniques for the study of Gaussian elds and also the tools of analysis in wavelets Using this theoretical framework, in section 63 we will formally de ne the property of local self-similarity and present two examples which form the base of all the later models Having established that these models are not suf ciently general for the applications, we will penetrate into the multifractional world in section 64 In each preceding model, speci c attention will be given to the regularity of the trajectories and, in section 65, we shall develop the statistical methods which make it possible o estimate this regularity This is what we call model identi ability At this point, it is necessary to clarify that the term elds is used for families 1 In applications, the most of random variables indexed by groups of d for d interesting cases correspond to d > 1 (for example, d = 2, for the images) However, certain statements, particularly those concerning identi cation, will relate only to processes (ie, elds where we have d = 1).

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When 0 < h(t) < 1 is a suf ciently regular deterministic function, we describe the process X as multifractional or, when it is Gaussian, as locally self-similar, ie, that locally around t X(t) possesses similarities with a FBM of parameter H = h(t) (for more details, see 6) The local regularity is no longer a uniform or global quantity along the sample path but, on the contrary, it varies in time, according to h(t), which therefore makes it possible to model time variations of the roughness When h(t) is itself a strong irregular function, possibly a random process, in the sense that, with t xed, h(t) depends on the observed realization of X, the process X is said to be multifractal The variability uctuations are no longer described by h(t), but by a multifractal spectrum D(h) which characterizes the Hausdorff dimension of the set of points t where h(t) = h (see 1 and 3) One of the major consequences of multifractality in the processes is the fact that quantities usually called partition functions, behave according to power laws in the small scale limit: 1 n.

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In this scenario, the function n( ) is no longer xed a priori to be log , as much as the function H(q) is no longer a priori linear according to qH The concept of an in nitely divisible cascade was initially introduced by Castaing in the context of turbulence [CAS 90, CAS 96] The complete de nition of this notion is beyond the scope of this chapter and can be found in [VEI 00] It is nonetheless important to indicate that a quantity, called the propagator of the cascade, plays an important role here: it links the probability densities of process increments with two different scales and The in nite divisibility formally translates the notion of the absence of any preferred time scale and demands this propagator be constituted of an elementary function G0 , convoluted with itself a number of times dependent only on the scales and , and therefore with the following functional form: G , (log ) = [G0 (log )] (n( ) n( .

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Locally Self-similar Fields. Recognizing UPC Symbol In Visual . UPC A Maker In VS .NET Using Barcode generator for .The comparison of this de nition with that of MBM is instructive: if it is supposed that functions (hn ) of the multifractional sequence are all equal to a function h verifying Hypothesis 64, then it is noted that the GMBM is a MBM with near normalization: 1 Y(h) (t) = Bh (t) v h(t) It is also noted that the law of a GMBM depends on the whole of the multifractional sequence and not only on the generalized multifractional function limit In fact, writing the GMBM in series form, let us uess that the nth function of the multifractional sequence governs the behavior of the GMBM at scale 1/2n 1 To clarify this idea, let us suppose that the white noise which intervenes in formula (654) is W (d ) and let us consider the development in series of the GMBM which is deduced from it:.Related:

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