等价活动标架理论及其在微分方程和计算机视觉与模式识别中的应用研究

项目名称： 等价活动标架理论及其在微分方程和计算机视觉与模式识别中的应用研究

项目编号： No.11471004

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 姚若侠

作者单位： 陕西师范大学

项目金额： 68万元

中文摘要： 本项目寻求Fels和Olver关于李群的等价活动标架理论和传统李群理论新的结合点，并结合计算机代数理论给出比较一般和系统的等价活动标架理论和方法。新理论和方法的应用因此可从有限维李群李代数拓展到无穷维李伪群，而李伪群为计算机视觉中图像跟踪提供方法。此外，因经典理论的局限性和困难性，本项目借助符号计算研究等价活动标架的递推构造方法，该方法可构造微分不变量及其完备系统、微分不变算子、不变量间的全局关系syzygy,不变变分问题、规范形和签名曲线等。模式识别中，表明子流形等价的等价映射对子流形坐标的依赖性所导致的形式的完全不同而无法判断其等价性，而syzygy恰恰是等价子流形的根本和内在属性，形式完全相同，此属性是模式识别的关键。此外，等价子流形的签名曲线相等，但即便是对一维子流形，签名曲线的计算也比较繁难，况且多数曲线的表示并不知道。为此，引入符号数值计算方法，以解决签名曲线的计算和比较问题。

中文关键词： 等价活动标架；微分不变量；可积系统；签名曲线；计算机视觉

英文摘要： This proposal hopes to look for the bonding point between the Fels and Olver's equivariant moving frame theory for Lie groups and the classical Lie group theory, and combined with computer algebra theory to develop more general and systematic theory and methods. The applications could be extended from finite-dimensional Lie groups and Lie algebra to infinite-dimensional Lie pseudo-groups, which can provides new method for motion tracking in computer vision. Duo to the dificulities and limitations of the classical theory, this proposal will study equivariant moving frame method to construct differential invariants,invariant differential operators, the complete systems of differential invariants,the universal relations between various differential invariants,i.e. syzygies, invariant variational problem,normal form and signature cuves. In objects recognition, the dependence of the equivariant map of submanifolds on the coordinates results in complete differentces in forms and then one could not determine whether or not they are equivalent accordingly. However, syzygy is one of their essential and intrinsic properties and possesses the same form such that this key property could be used in computer objects recognition. Furthermore, the signatures of equivariant submanifolds are the same, but even for 1-dimensional submanifold, such as a curve in Euclidean space, the computation of which is tedious and difficult. Another problem is that for most curves, we do not know their equations. Hence, this proposal will also propose an algorithm of symbolic-numeric computation to solve the computation of signatures combined with the advantages of the high efficiency of numerical computation and the accuracy of symbolic computation.

英文关键词： equivariant moving frame;differential invariant;integrable system;signature curve;computer vision