Fractal-based methods in analysis

基于分形的分析方法

• 批准号: 238898-2011
• 负责人: Kunze, Herbert
• 金额: $ 0.8万
• 依托单位: University of Guelph
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570519
• 关键词: Fractal; based; methods; analysis

Fractals are mathematical entities that have useful multiscale properties; no matter the level to which we zoom in on a fractal, we always reveal more detail, structure, and self-similarity. "Fractal-based methods" refers not just to the theory and tools of fractals themselves, but also to the driving philosophy behind this mathematics. Recently, this philosophy has been used to treat problems far removed from classical fractals. In many application areas, phenomena of interest can have important features at multiple scales (spatial or temporal). For physical models, one might consider continuum or macroscale models (which capture no lower level information), mesoscale or nanoscale models (which include information about groups of atoms and molecules), molecular dynamical models (which include information about individual atoms), or quantum mechanical models (which includes information about electrons). In recent years, scientists have become able to gather data at multiple scales, for pathological tissues or tumors, for example. These scientists now seek to formulate and analyze compatible multiscale models. When measurements are available, scientists often seek to solve the "inverse problem" of estimating the values (or functional form) of the parameters in these models. The first goal of fractal imaging was to represent a given target image by a fractal. Subsequent research sought to perform image processing tasks within this framework. The program seeks to develop (1) fractal-based frameworks for treating problems in biomedical science (such as tumor detection, modeling, and analysis), ecological science (such as ecosystem modeling and analysis), environmental science (such as pollution-driven population dynamics), and physics/engineering (such as heat flow and vibrations); and (2) new imaging frameworks based on some very recent generalizations of the original fractal imaging notions. The work is unified by a common philosophical approach and by the fact that measurements (of a tumor, say) may well come in the form of an image, in which case the related analysis can involve elements of (1) and (2).

分形是具有有用的多尺度特性的数学实体;无论我们放大到分形的哪个层次，我们总是能揭示更多的细节、结构和自相似性。“基于分形的方法”不仅指分形本身的理论和工具，还指这种数学背后的驱动哲学。最近，这种哲学已经被用来处理远离经典分形的问题。 在许多应用领域，感兴趣的现象可以在多个尺度（空间或时间）上具有重要特征。对于物理模型，可以考虑连续或宏观模型（不捕获较低级别的信息），中尺度或纳米尺度模型（包括原子和分子组的信息），分子动力学模型（包括单个原子的信息）或量子力学模型（包括电子的信息）。近年来，科学家们已经能够在多个尺度上收集数据，例如病理组织或肿瘤。这些科学家现在寻求制定和分析兼容的多尺度模型。当测量可用时，科学家通常寻求解决估计这些模型中参数值（或函数形式）的“逆问题”。 分形成像的第一个目标是用分形来表示给定的目标图像。随后的研究试图在这个框架内执行图像处理任务。 该计划旨在开发（1）基于分形的框架来处理生物医学科学中的问题（如肿瘤检测、建模和分析）、生态科学（如生态系统建模和分析），环境科学（如污染驱动的人口动态）和物理/工程（如热流和振动）;和（2）新的成像框架的基础上，一些非常最近的概括原来的分形成像概念。 这项工作是统一的一个共同的哲学方法和事实，即测量（肿瘤，说）很可能以图像的形式出现，在这种情况下，相关的分析可以涉及（1）和（2）的元素。