Visually complex objects with infinitesimally fine features, naturally call for mathematical representations. The geometrical property of self-similarity - the whole similar to its parts - when iterated to infinity generates such features. Finite sets of affine contractions called Iterated Function Systems (IFS), with their compact attractors IFS fractals, can be applied to represent detailed self-similar shapes, such as trees or mountains. The fine local features of such attractors prevent their straightforward geometrical handling, and often imply a non-integer Hausdorff dimension. The main goal of the thesis is to develop an alternative approach to the geometry of IFS fractals in the classical sense via bounding sets. The results are obtained with the objective of practical applicability. The thesis thus revolves around the central problem of determining bounding sets to IFS fractals - and the convex hull in particular - emphasizing the fundamental role of such sets in their geometry. This emphasis is supported throughout the thesis, from real-life and theoretical applications to numerical algorithms crucially dependent on bounding.

Determining fracture characteristics at the laboratory scale is a major challenge. It is known that fracture characteristics are scale dependent; as such, the minimum sample size should be deduced in order to scale to reservoir dimensions. The main factor affecting mechanical and hydrological characteristics of natural fractures is aperture distribution, which is a function of scale and confining pressure, rather than roughness of one fracture surface. Scale and pressure dependencies of artificial and natural fractures were investigated in this study using an X-Ray CT Scanner. Fractal dimension, D, and amplitude parameter, A, of fracture aperture approaches a constant value with increased sampling area, similar to the behavior of fracture roughness. In addition, both parameters differ under different confining pressures for a reference sampling area. Mechanical properties of fracture-fracture deformation behavior and fracture normal stiffness were obtained from CT scan data as well. Matrix porosity is relatively easy to measure and estimate compared to fracture porosity. On the other hand, fracture porosity is highly heterogeneous and very difficult to measure and estimate. When matrix porosity of naturally fractured reservoirs (NFR) is negligible, it is very important to know fracture porosity to evaluate reservoir performance. Since fracture porosity is highly uncertain, fractal discrete fractal network (FDFN) generation codes were developed to estimate fracture porosity. To reflect scale dependent characteristics of fracture networks, fractal theories are adopted. FDFN modeling technique enables the systematic use of data obtained from image log and core analysis for estimating fracture porosity. As a result, each fracture has its own fracture aperture distribution, so that generated FDFN are similar to actual fracture systems. The results of this research will contribute to properly evaluating the fracture porosity of NFR where matrix porosity is negligible.

A new multilayer ceramic balun with fractal curve for 802.11b/a is presented in this thesis. The fractal curve is implemented in balun design. By using the fractal geometry, the reduction in size of the proposed balun is evident and only several metal layers are used. Excellent results of amplitude balance and phase difference are obtained. In the mean time, there is a 1.3-GHz bandwidth of return loss and the design results in a good matching. The measured results of the LTCC multilayer ceramic balun with fractal curve agree well with the simulation. Satisfactory performance is also obtained. The dimension of the proposed Koch¡¦s and Minkowski_1st balun can be reduced by about 46.4% and 50%, respectively. Finally, the fractal balun is combined with fractal band pass filter and fractal antenna to become a passive module. Simulation of the module has been carried out and we find satisfactory results.

Digital multimedia can be distributed via the internet efficiently with superior compression technologies. The chance of distributing digital intellectual properties, such as image, music, films, and software, being large-scale unauthorized copied and distributed are much increasing one possible and practical solution for the copyright protection is information hiding technology. Information hiding technology embeds a special data into multimedia data for copyright protection. However, the embedded data may be damaged by malicious attacks or common signal processing. In this thesis, an information hiding technique based on Fractal Orthonormal Basis is proposed. First, the original image is divided into NxN Range blocks, each range block is substituted by several Domain blocks (Fractal Orthonormal Basis), then the watermark information is embedded into the coefficients of the fractal orthonormal basis. Besides, our technique will be compare with the other two watermarking algorithm (using DCT and DWT). After the attacks of cropping, down-scaling, median filter, smoothing, noise, JPEG, SPIHT and EZW compression, the Fractal Orthonormal Basis watermarking technique shows better result of capacity, transparency and robustness. In addition, we only store parts of compression fractal codes and the permutation seed, and these can be the secret key for the security.

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In this thesis, a target detection technique using a rotational invariant wavelet-based scheme is presented. The technique is evaluated on Synthetic Aperture Rader (SAR) imaging and compared with a previously developed fractal-based technique, namely the extended fractal (EF) model. Both techniques attempt to exploit the textural characteristics of SAR imagery. Recently, a wavelet-based fractal feature set, similar to the proposed one, was compared with the EF feature for a general texture classification problem. The wavelet-based technique yielded a lower classification error than EF, which motivated the comparison between the two techniques presented in this paper. Experimental results show that the proposed techniques feature map provides a lower false alarm rate than the previously developed method.

En esta tesis se explora experimentalmente el comportamiento multibanda que le da una característica multifuncional al diseño de la antena. Se busca demostrar que gracias a la autosimilitud o autosimilaridad del fractal, la antena responda con un comportamiento multibanda.

The main portion of my thesis focuses on a 2-dimensional second order heat transmission problem in domains with pre-fractal interfaces. My focus is on the numerical approximation of the solutions. Precisely, I€™m concerned to develop a suitable mesh refinement algorithm that could be adapted to our situation, by taking into account the regularity of the solutions and the geometry of irregular pre-fractal interfaces. I obtain an error estimate between the weak solution and the discrete solution, which indicates an optimal rate of convergence as in the classical case when the solution has H^2-regularity. In addition, numerical simulations are also included, which demonstrates the features of our heat transmission model. Another portion of my thesis focuses on the asymptotic analysis of singular boundary value problems with highly conductive layers of pre-fractal type. My models illustrate the problems of a lower- dimensional highly conductive material intruding into a higher- dimensional material with lower conductivity. I consider a 2D model of Sierpinski pre-fractal layers and 3D models of hierarchical layers. The main results consist in the so-called Mosco-convergence of certain energy functionals, which implies the strong convergence of the solutions and of the spectral resolutions as a byproduct in real applications.