Fractal analysis, mathematical imaging and stochastic methods in optimization

大多数人都看过这些令人惊叹的分形图，也听说过分形图。但它们有什么用呢？分形图的定义特征是它的自相似性。也就是说，分形图的小部分与整个分形图有着惊人的相似之处。许多自然现象都有同样类型的标度行为，但通常只是作为近似值。然而，它往往是足够好的近似值，可以用来描述或预测有用的东西。*，*他说，这个研究项目旨在进一步发展分形分析的理论工具，并将这些工具应用于数字成像应用。其中一个这样的应用是医学成像，人们想要跟踪某个区域的边界，就像癌症肿瘤的边界。但这些肿瘤往往通过具有更不规则的形状而与周围组织区分开来。这项研究的一个结果将是另一类模型和算法，以跟踪这种不规则生长的演变边界。*，*该算法利用标准的分形图像算法在一幅图像中搜索不同尺度的相似图像特征，并利用这些关系来提供对图像的压缩描述。通过优化这种拟合，可以确保获得尽可能好的逼近。因此，另一个感兴趣的领域是优化。该项目的第二个主要研究领域是针对优化中的随机方法的理论和实践检验。非常流行的两种这样的方法是模拟退火法和遗传算法，该项目将对这两种方法的混合进行理论和计算检验。

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.