Elastic Shape Matching: Theoretical Models and their Algorithmic Complexity

几何形状匹配问题是计算几何领域的核心研究课题之一。在这样一个问题中，一个是给定的两个几何对象（一个“图案”和一个“模型”），一个变换类，以及一个相似性测度，并寻求从给定的类的变换，这样的相似性测度的模式转换由这个变换到模型是maximized.In这个项目中，我们引入了弹性（或非均匀）几何形状匹配问题的概念。在这样的问题中，模式不是通过单个变换来变换，而是通过所谓的变换集合来变换。转换集成允许模式的非均匀变形-模式的不同部分可以通过不同的映射进行转换。它们还允许结合模式的时间依赖性以及其形状随时间的变化。因此，它是可能的计算匹配是有效的在一定的时间框架内，即使参考对象的变化在此框架。详细的项目考虑了以下问题：-结构和代数复杂性研究的弹性形状匹配问题-设计和分析的有效算法策略（精确，近似，参数化）和检索数据结构-相似性措施的转换合奏-空间以及时间内插的弹性模式。

Geometric shape matching problems are one of the core research topics in the field of computational geometry. In such a problem one is given two geometric objects (a "pattern" and a "model"), a transformation class, as well as a similarity measure, and one seeks a transformation from the given class such that the similarity measure of the pattern transformed by this transformation to the model is maximized.In this project we introduce the concept of elastic (or non-uniform) geometric shape matching problems. In such a problem the pattern is not transformed by a single transformation, but by a so-called transformation ensemble. Transformation ensembles allow for non-uniform deformations of the pattern - different parts of the pattern can be transformed by different mappings. They also allow to incorporate temporal dependencies of the pattern as well as changes of its shape over time. Consequently it is possible to compute matchings that are valid within a certain time frame even if the reference objects change during this frame.In detail the project consides the following questions:- structural and algebraic complexity studies of elastic shape matching problems- design and analysis of efficient algorithmic strategies (exact, approximate, parametrized) and retrieval data structures - similarity measures for transformation ensembles- spacial as well as temporal interpolation of elastic patterns.