Moment convex optimization of the inverse I.F.S. affine fractal problem in one dimension, 1991

The study of chaotic dynamical systems has given rise to a new geometry for classifying non-integrally dimensioned, zero measure sets known as fractals. Fractal geometry provides an efficient means for modelling complex, naturally occuring objects and processes (i.e landscapes, trees, diffusion limited aggregates, etc.) as pioneered through the works of Mandelbrot. A more recent development is that of Barnsley and co-workers involving the iteration of contractive, affine linear maps. Such Iterative Function Systems (I.F.S.) are capable of producing a rich fractal theory impacting all areas of nonlinear dynamics, including computer graphics-simulations. When compared to polynomial splinefitting approaches, I.F.S fractals have been shown to yield superior information-compression ratios. This is in part due to the strong dependence of fractal theory on self-similarity features of the system. Selfsimilarity is an important aspect of many chaotic processes and complex formations. The application of I.F.S. fractal analysis, in generating computer graphic displays of such intricate patterns, provides an important numerical laboratory for decoding complexity in nature. Given a target image, one seeks to determine the underlying I.F.S. fractal parameters. This is the inverse fractal problem (IFP). An important theorem in this regard is that of Barnsley and Demko: the Collage Theorem. Unfortunately, its full implementation requires significant human intervention. The objective of this thesis is to define alternate, fully automated, IFP methods enabling the incorporation of fractal-based technologies in satellites and missiles, and enhancing the development of computer generated terrain simulations. An important factor hindering the full automation of the IFP, through conventional variational theories, is the presence of multi-minima ambiguities. These difficulties are present both in one and two space dimensions. This thesis explores the benefits of a moment problem formulation of the IFP which utilizes important convexity properties and circumvents the aforementioned multi-minima pitfalls