Moving Frame Theory And Applied Research In Object Recognition

This thesis first give a profound study on equivariant moving frame theory improved by Fels and Olver, and then presents its application in object recognition.We know that Maurer-Cartan normalization method can be used to construct moving frames for any free and regular group actions, thus obtain the fundamental invariants. However, for one group action which does not satisfy this condition, in the paper, one can prolong the group transformation to let it act on a Jet space, and then using normalization method to construct the moving frames. Similarly, the fundamental invariants can be got, but the key problem is how to get higher order differential invariants, which relates to the core method of this paper—moving frame recursive construction method, so called recursive moving frame method for abbreviation. The traditional method is to compute total differential operator under the moving frames, the corresponding invariant differential operators, then make it act on the obtained fundamental invariants, so that re-using the invariant differential operators can yield the higher order differential invariants. The problem is that, with the increase of the order, the calculation complexity of the seemingly simple and classical method increases rapidly, resulting in the computer algebra system (symbolic computation) cannot accomplish its computation, thereby the method's practical scope is limited. The proposed recursive moving frame method is proposed, not only solved this difficult problem commendably, but also broke through the limitation of recursive algorithm proposed by Irina A. Kogan in that it needs a slice.An important application of moving frame in computer vision is object recog-nition. As is known to all, object recognition is one of the main goals of computer vision, seeking the invariants of the object features plays a key role in object recog-nition. In two-dimensional space, one can obtain the object's contour curve by edge detection method, thereby the problem of object recognition reduces to curve matching. Based on the Fels-Olver equivariant moving frame theory, by means of the recursive algorithm to get invariants and differential invariants of curves in the plane, that is, the curvature and its derivatives with respect to arc length (includ- ing all higher order derivatives with respect to arc length)respectively. Therefore, signature curves of the curves are constructed by differential invariant, differential invariants are not changed under the rigid motion and affine transformation. Then, in computer vision field, signature curves are generally applied to the problem of object recognition, visual tracking and symmetry detection. Besides, E. Cartan's equivalence theorem is fundamental theorems of signature curves, using joint invari-ants caused by noise resistant numerically approximate signature curves, and gives differential invariant signature curves of several Euclidean curves and affine curves. The instances in this thesis show that differential invariant method is efficient in the computer vision.