Based Olver Moving Frame Theory Methods And Applications Of Target Recognition

Lie group is one of the most important achievements of modern mathematics in the end of nineteenth century, which is a bond of connecting of analysis, algebra with geometry. On the one hand, Olver,under the foundation of Lie's research work,Olver successfully developed of a set of equivalent theory relevant to moving frame in the nineties of the last century,which combined organiclly with the theory of differential forms developed by Eile Cartan,He obtain a recursive algorithm for computing a set of invariants with regard to pseudo-group and general transformation group and also achieve a new algorithm of Maurer-Cartan structure equation under the Lie pseudo group. On the other hand, under the guidance of Klein's Erlangen program,the modern geometry and group theory were combined closely.The mathematics pay a close attention to the issues of invariant geometric objects in a given transformation group and symmetrical properties of geometric objects have attract human attention with regard to the research of geometry. The theory of moving frame have the superiority in studying this problem of this fields. In addition, more and more physicists and mathematicians are ware of a fact that the essence of modern physics establish the variational problem based on a hypothetical transformation group essentially and a given physical scene,which can obtain a partial differential equations called Euler-Lagrange during building up the model of variational problem.However the Euler-Lagrange equation of most of variational problem is very complicated. The construction of equivalent to the Euler-Lagrange equation in given transformatio group is a very meaningful work, especially on computer image processing is of great significance. The studies of variational problem was first proposed byGriffiths.The recursive formula was applied to this problem successfully by Olver. The complexity of the solution of this problem are greatly simplified. In addition, the theory of invariants has also been a good application of signature curves, especially Kogan and Boutin both study the signature curves independently, their works will be discussed in this paper.In this paper, it based on the theory of Olver's moving frame,we studied three subjects,including Lee pseudo-group, invariance variational problems and signature curves. With symbolic computation, the theory of Olver's rescusive moving frame was studied in the application of invariant, while the summary of each chapter is introduced. The main work of this paper is as follows:(1) The mathematical foundation of the moving frame theory is discussed. The conception of moving frame is introduced in the chapter of the theory of Lee pseudo-group, and further this chapter discusses the recursive algorithm of the moving frame(2) This paper explores the recursive algorithm of Euler-Lagrange equation under the general linear group. The recursive operator is obtained under the transformation group by means of the recursive algorithm. Further the variable problem can be transformed into the invariant variable differential problem.(3) This paper obtains the recursive operator via the invariant variable differential problem, and then studies the evolution of signature curve space by means of the method of level set, which applies the approach of evolution of invariants in the image segmentation and can effectively detects the edge profile of target object.(4) The numerical procedure of various kind of signature curves are studied, which symmetrically detect the geometric object through mathematical modeling. The simulation experiments show that the algorithm can detect the symmetry axis of target object, and the accuracy of detection depends on the local properties of geometric object. In addition, this method produces some certain errors with regard to the geometrical objects which possess the high curvature transformation, since we need some other method such probability to further improve the detection algorithm.