Curvature measure for fractals and application in pattern recognition

分形曲率测量及其在模式识别中的应用

• 批准号: 61396530
• 负责人: Professorin Dr. Martina Zähle
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2008
• 资助国家: 德国
• 起止时间: 2007-12-31 至 2015-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/61396530?language=en
• 关键词: Curvature; measure; fractals; application; pattern

In the previous period of our project we were able to determine fractal curvatures for rather general self-similar and self-conformal sets. The recent combination with ergodic theory of dynamical systems, which leads to simpler methods of proof, is planned to be further developed. A main aim consists now in extension to new classes, in particular, V-variable and graph directed random fractals.Furthermore, relationships to analysis on fractals and their curvature properties will be investigated, like known from the classical case of spectral behavior and heat kernels. Here connections to potential theory arise via Dirichlet forms. We also intend to support the further development of a computer program for approximations of fractal curvatures with applications.

在我们项目的前一阶段，我们能够确定相当一般的自相似和自共形集合的分形曲率。最近与动力系统遍历理论的结合，导致了更简单的证明方法，计划进一步发展。一个主要的目标是扩展到新的类，特别是V-变量和图有向随机分数。此外，将研究关于分数及其曲率性质的分析的关系，就像从经典的谱行为和热核的情况中已知的那样。在这里，与位势理论的联系是通过狄利克莱形式产生的。我们还打算支持进一步开发用于分形曲率近似的计算机程序及其应用。

项目成果

Curvature-direction measures of self-similar sets

自相似集的曲率方向测度

• DOI: 10.1007/s10711-012-9810-5
• 发表时间: 2013
• 期刊: Geometriae Dedicata
• 影响因子: 0.5
• 作者: T. Bohl;M. Zähle
• 通讯作者: M. Zähle

Stability Properties of Fractal Curvatures

分形曲率的稳定性性质

• DOI: 10.1007/978-3-662-43920-3_13
• 发表时间: 2014
• 作者: M. Zähle

Legendrian Cycles and Curvatures

勒让循环和曲率

• DOI: 10.1007/s12220-014-9506-1
• 发表时间: 2015
• 期刊: The Journal of Geometric Analysis
• 作者: J. Rataj;M. Zähle
• 通讯作者: M. Zähle

Multivariate variational principles for topological pressure

• DOI: 10.1142/s0219493715500161
• 发表时间: 2015-05
• 期刊: Stochastics and Dynamics
• 影响因子: 1.1
• 作者: Marc Rauch