Computational topology for image understanding (topologie computationnelle pour la compréhension d'image)

用于图像理解的计算拓扑（topologieComputationnelle pour la compréhension dimage）

• 批准号: 46201-2011
• 负责人: Kaczynski, Tomasz
• 金额: $ 0.8万
• 依托单位: Université de Sherbrooke
• 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=568935
• 关键词: Computational; topology; image; understanding; topologie; Computational; topology; image; understanding; topologie

My general goal is to develop an emerging research field called computational topology in the context of analysis and processing of multidimensional data. Rooted in both Computer Science and Mathematics, computational topology overlaps with computational geometry and combinatorial topology. It has applications in several domains of science and engineering, in particular, in imaging and multimedia. The specific goals of my proposal are to: - Develop the multi-parameter persistent homology introduced by Carlsson at al., also advanced by Frosini at al. In particular, investigate connections between reductions of the persistence diagram of a vector-valued measuring function and those of the cellular complex of a discrete Morse function. This study will offer new mathematical tools for realizing a system of observer-based 3D media contents retrieval. This is a joint project with collaborators from the universities of Bologna, Modena - Reggio Emilia, and Bishop's. - Apply the above concepts to complement my earlier work on detection and classification of singular zones in multidimensional data. The multi-parameter persistence will permit to extract image characteristics which persist in small changes in both spatial and dynamical resolution. - Extend homology reduction and coreduction algorithms to the cohomology, including the computation of its ring structure. This is a joint project with collaborators from the Jagiellonian University of Krakòw. The next step is to promote the use of these algorithms in computational electromagnetism. - Study modern extensions of the Morse theory to functions with non-isolated singularities in the context of imaging science.

我的一般目标是在分析和处理多维数据的背景下，开发一个称为计算拓扑的新兴研究领域。植根于计算机科学和数学，计算拓扑与计算几何和组合拓扑重叠。它在科学和工程的几个领域中都有应用，特别是在成像和多媒体中。我的建议的具体目标是： - 开发Carlsson在Al。引入的多参数持久同源性，也是Al的Forsini提出的。特别是，研究矢量值测量功能的持久图的减少与离散摩尔斯函数的细胞复合物的持久图之间的联系。这项研究将为实现基于观察者的3D媒体内容检索系统提供新的数学工具。这是一个联合项目，与博洛尼亚大学，摩德纳 - 雷吉奥·埃米利亚和主教的合作者。 - 应用上述概念来补充我在多维数据中对单数区域检测和分类进行分类的早期工作。多参数持久性将允许提取图像特征，这些图像特征在空间和动态分辨率的小变化中持续存在。 - 将同源性还原和排放算法扩展到共同体，包括其环结构的计算。这是一个与贾吉洛尼亚大学Krakòw大学的合作者的联合项目。下一步是促进这些算法在计算电子磁性中的使用。 - 在成像科学的背景下，研究摩尔斯理论的现代扩展对具有非异形奇异性的功能。