Moving Frames & Computer Vision

这个项目致力于运动框架几何和微分不变量数学理论在计算机视觉中出现的问题的创新应用，重点是医学成像。由于对称性构成了人类视觉系统的一个基本组成部分，它应该自然地融入到图像处理和物体认知的数学方法中。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例，即基于为所考虑的对称群构造适当的微分不变量。将比较区分不同对象的特征曲线的度量，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构造对微分不变量的对称保持数值近似的新方法，该方法将作为本项目的一部分而开发。这些应用程序将基于一种新的、实用的移动框架理论，该理论可以应用于一般的有限维和无限维群行为。该项目的一部分将致力于这一一般理论的进一步发展。此外，还将研究物理、几何、不变理论和变分中出现的问题的各种应用。

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.