Moving Frames & Computer Vision

该项目致力于将运动框架的几何和微分不变量的数学理论创新地应用于计算机视觉中出现的问题，重点是医学成像。由于对称是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对象识别的数学方法中。要考虑的图像处理应用将包括去噪和平滑，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式，即基于对所考虑的对称群的合适微分不变量的构造。将比较区分不同物体特征曲线的指标，目标是从二维和三维图像中识别和提取物体。图像处理系统的实际实现将依赖于构建保持对称的微分不变量数值近似的新方法，该方法将作为本项目的一部分开发。这些应用将基于一种新的、实用的运动框架理论，可以应用于一般的群体行动，无论是有限维还是无限维。项目的一部分将用于进一步发展这一一般理论。在物理、几何、不变量理论和变分学中出现的各种附加应用也将被研究。

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.