Fractal analysis, mathematical imaging and stochastic methods in optimization

点击翻译按钮获取中文摘要

Most people have seen these amazing pictures of fractals and heard of fractals.  But what good are they?  The defining characteristic of a fractal is its self-similarity.  That is, small bits of a fractal bear a striking resemblance to the entire fractal.  Many natural phenomena have this same type of scaling behaviour, but usually only as an approximation.  However, it is often a good enough approximation to be descriptively or predictively useful.                                                                                                                                              This research project aims to develop further theoretical tools in fractal analysis and to apply these tools to digital imaging applications.   One such application is to medical imaging where one wants to track the boundary of some region, like the boundary of a cancerous tumor.  These tumors often distinguish themselves from the surrounding tissue by having a more irregular shape.   One outcome of this research will be another class of models and algorithms to track the evolving boundaries of such irregular growths.                                                                                                                                              The standard fractal image algorithms search within an image to find similar image features at different scales and uses these relationships to provide a compressed description of the image.  Optimizing this fit ensures that the approximation will be as good as possible.  Thus, another area of interest is optimization.  The project's second main area of research is geared towards a theoretical and practical examination of stochastic methods in optimization.  Two such methods which are very popular are Simulated Annealing and Genetic Algorithms, and the project will perform a theoretical and computational examination of hybrids of these two methods.