Moving Frames & Computer Vision

这个项目致力于创新应用的几何ofmoving框架和数学理论的微分不变量的问题出现在计算机视觉，重点是医学成像。由于对称性构成了人类视觉系统的基本组成部分，因此它应该自然地纳入图像处理和目标识别的数学方法中。要考虑的图像处理应用程序将包括去噪和平滑，分割和边缘检测，以及对象识别，包括对称检测和部分遮挡。实际实现将基于签名曲线范例，即基于所考虑的对称群的合适的微分不变量的构造。 将比较区分不同对象的签名曲线的方法，目标是从二维和三维图像中识别和提取对象。 图像处理系统的实际实现将依赖于一种新的方法，该方法将作为本项目的一部分开发，用于构建保精度的微分不变量的数值近似。 应用程序将基于一个新的，实用的理论，移动框架，可以适用于一般的群体行动，有限和无限维。 该项目的一部分将致力于进一步发展这一一般理论。各种额外的应用程序，在物理，几何，不变理论和变分法中出现的问题，也将进行调查。

This project is devoted to innovative applications of the geometry ofmoving frames and the mathematical theory of differential invariants toproblems arising in computer vision, with emphasis on medical imaging. Since symmetry forms a basic component of the human visual system, it should benaturally incorporated into mathematical methods for image processing and objectrecognition. The image processing applications to be considered will includedenoising and smoothing, segmentation and edge detection, as well as objectrecognition, including symmetry detection and partial occlusions. Practicalimplementations will be based on the signature curve paradigm, that is based on the construction of suitable differential invariants for the symmetry group under consideration. Metrics that distinguish signature curves of different objects will be compared, with the goal being to recognize and extract objects from both two and three-dimensional images. Practical implementations of the image processing systems will rely on a new method of constructing symmetry-preserving numerical approximations to differential invariants that is to be developed as a part of this project. The applications will be based on a new, practical theory of moving frames, that can be applied to general group actions, both finite and infinite-dimensional. Part of the project will be devoted to the further development of this general theory. Various additional applications, to problems arising in physics, geometry, invariant theory, and the calculus of variations, will also be investigated.