Multifractal measures

Baveye, P., Boast, C. W., Gaspard, S., and Tarquis, A. M. (2008). Introduction to fractal geometry, fragmentation processes and multifractal measures Theory and operational aspects of their application to natural systems In Bio-Physical Chemistry of Fractal Structures and Process in Environmental Systems, Senesi, N., and Wilkinson, K., eds. IUPAC Series on Analytical and Physical Chemistry of Environmental Systems. Vol. 11, John Wiley Sons, Chichester, pp. 11-67. 

Introduction to Fractal Geometry, Fragmentation Processes and Multifractal Measures Theory and Operational Aspects of their Application to Natural Systems... 

A first objective of this chapter, therefore, is to attempt to fill these gaps and, in particular, to make more explicit the connection between theoretical and natural fractals. A similar approach is followed in the last section of this chapter, which deals briefly with the increasingly important multifractal measures. A second objective of this chapter is to point out that some uses of fractals amount to little more than curvefitting exercises, and that any attempt to relate the resulting fractal dimensions to geometrical features of natural systems should be approached with great caution. 

In some cases, the limit as r -> 0 is not taken in Equation (2.23) and the ratio r lx By x))/ ar is termed in this case the Holder [36], coarse Holder [37] or Lipschitz-HClder exponent [34]. It is traditionally denoted by a and it may be evaluated for any measure, defined or not by Equation (2.22). This exponent is useful to characterize singular measures, which have no local densities (i.e. for which the limit in Equation (2.23) does not exist), and it plays an important role in the definition of multifractal measures (Section 2.6). 

The introduction to fractal geometry in this chapter would not be complete without a short mention of an area that is conceptually challenging, yet is the object of considerable interest in the literature, i.e. multifractal measures. 

As with fractals, many authors try to get by without having to provide a precise definition of multifractal measures. Consequently, the term multifractal measure ... 

Kwvin [38] humorously comments that multifractal measures are not for the squeamish ... 

For multifractal measures to be useful in practice, we need to find ways to characterize and parameterize their geometrical properties. This is achieved in particular by the... 

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