In this thesis we study the multifractal structure of graph directed self-conformal measures. We begin by introducing a number of notions from geometric measure theory. In particular, several notions of dimension, graph directed iterated function schemes, and the thermodynamic formalism. We then give an historical introduction to multifractal analysis. Finally, we develop our own contribution to multifractal analysis. Our own contribution to multifractal analysis can be broken into three parts; the proof of two multifractal density theorems, the calculation of the multifractal spectrum of self-conformal measures coded by graph directed iterated function schemes, and the introduction of a relative multifractal formalism together with an investigation of the relative multifractal structure of one graph directed self-conformal measure with respect to another. Specifically, in Chapter 5 we show that by interpreting the multifractal Hausdorff and packing measures Olsen introduced in [0195] as Henstock-Thomson variation measures we are able to obtain two stronger density theorems than those obtained by Olsen. In Chapter 6 we give full details of the calculation of the multifractal spectrum of graph directed self-conformal measures satisfying the strong open set condition and show that the multifractal Hausdorff and packing measures introduced by Olsen in [0195] take positive and finite values at the critical dimension provided that the self-conformal measures satisfy the strong separation condition. In Chapter 7 we formalise the idea of performing multifractal analysis with respect to an arbitrary reference measure by developing a formalism for the multifractal analysis of one measure with respect to another. This formalism is based on the ideas of the 'multifractal formalism' as first introduced by Halsey et. al. [HJKPS86] and closely parallels Olsen's formal treatment of this formalism in [0195]. In Chapter 8 we illustrate our relative multifractal formalism by investigating the relative multifractal structure of one graph directed self-conformal measure with respect to another where the two measures are based on the same graph directed self-conformal iterated function scheme which satisfies the strong open set condition.

by Wang Xiang-Yang. / "May 2002." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (p. 95-99). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Zhou, Yu. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 152-165). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

Thesis (M.S.)--West Virginia University, 2006. / Title from document title page. Document formatted into pages; contains xii, 102 p. : ill. (some col.). Vita. Includes abstract. Includes bibliographical references (p. 96-99).

The purpose of this dissertation is to introduce a natural and unifying multifractal formalism which contains the above mentioned multifractal parameters, and gives interesting results for a large class of natural measures. In Part 2 we introduce the proposed multifractal formalism and study it properties. We also show that this multifractal formalism gives natural and interesting results when applied to (nonrandom) graph directed self-similar measures in Rd and "cookie-cutter" measures in R. In Part 3 we use the multifractal formalism introduced in Part 2 to give a detailed discussion of the multifractal structure of random (and hence, as a special case, non-random) graph directed self-similar measures in R^d.

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The investigation of the multifractal spectrum of the equilibrium measure for a parabolic rational map with a Lipschitz continuous potential, φ, which satisfies sup φ < P(φ) x∈J(T) is conducted. More specifically, the multifractal spectrum or spectrum of singularities, f(α) is studied.

With the development of Geographical Information System (GIS) and remote sensing techniques, a great deal of data has provided a set of continuous samples of the earth surface from local, regional to global scales. Several multi-scale, multi-resolution, pyramid or hierarchical methods and statistical methods have been developed and used to investigate the scaling property of remotely sensed data: local variance, texture method, scale variance, semivariogram, and fractal analysis. This research introduces the wavelet transform into the realm of scale study in remote sensing and answers three research questions. Three specific objectives corresponding to the three research questions are answered. They include: (1) exploration of wavelets for scale-dependent analysis of remotely sensed imagery; (2) examination of the relationships between wavelet coefficients and classification accuracy for different resolutions and their improvement of classification accuracy; and (3) multiscaling analysis and stochastic down-scaling of an image by using the wavelet transform and multifractals. The significant results obtained are: (1) Haar wavelets can be used to investigate the scale-dependent and spatial structure of an image and provides another method for selection of optimal sampling size; (2) there is a good relationship between classification accuracy and wavelet coefficients. High/low wavelet coefficient reflects low/high classification accuracy in each land cover type. (3) the maximum likelihood classifier with inclusion of wavelet coefficients can improve land cover classification accuracies. (4) the moment-scale analysis of wavelet coefficients can be used to investigate the multifractal properties of an image. Also the stochastic down-scaling model developed based on wavelet and multifractal generates good simulation results of the fine resolution image.

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Thesis (Ph.D.)--York University, 2008. Graduate Programme in Earth and Space Science and Engineering. / Typescript. Includes bibliographical references (leaves123-134). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR46019