Geometric Computation Of Object Recognition

Now, we want to use another new geometric representation and calculation system, conformal geometric algebra. First, it provides a natural and elegant method which can represent the geometric objects and the transformations of the geometric objects. In the CGA, the points, circles, spheres, lines, planes and so on, they all can be represented by the vectors, and the geometric transformations(rigid body motion, Reflection, intersection and so on) can be represented by the geometric product; and CGA combine with the Quaternion, dual quaternion, Lie group and Lie Algebra, and embed them in the geometric algebra, used for the Geometric Modeling of the Euclidean geometry, affine geometry, projective geometry, and conformal geometric. It unifies mathematical systems used computer graphics in a simple and intuitive way. This paper discusses the conformal geometric algebra in the application of object recognition.The main contents and contributions of this thesis are summarized as follows:(1)On the basis of analyzing Conformal Geometric Algebra as the new mathematical framework of the computer graphics, aiming at weakness of the recent algorithms for detecting geometric object, combining with their merits, an efficient recognized algorithm for line and circle detection in the Conformal Geometric Algebra framework is designed and implemented. This arithmetic performs well when an image has strong background noise. Furthermore, certain blending and covered line and circle can be detected precisely. At first, by the geometric algebra theory, circle and line in Euclidean space can be represented as multivectors in conformal geometric algebra, then the distance was computed from every point to each multivector which presents the line or the circle by this detecting algorithm. Based on this distance, the points on the line or circle and near the line or circle belonged to the same cluster, so several points could be obtained whose number was equal to number of the line or circle. The point sets in the vicinity of each line or circle were extracted and the disturbance or noise far from the line or circle was deleted, at last this algorithm can detected lines and circles in images effectively. Experimental result shows that this algorithm is robust and the result is accurate.(2) Object recognition is a challenging problem in the field of pattern recognition and computer vision. On the basis of analyzing Conformal Geometric Algebra as the new mathematical framework of the computer graphics, a series of projective and point-permutation invariants are constructed based on the corner points. Some properties of these invariants include the invariance on shift, rotation and scale transforms. So, an algorithm for object shape recognition is designed based on the geometric invariant. Experiments give a high recognition rates.(3)As a new algebra frame of the computer graphics an the computer vision, the CGA provide a new method for the circle and sphere blending. CGA Is suitable for solving this kind of problem, because it has concise express of the geometric objects and the transformations of the geometric objects, and has a simpler method to define the geometric model. The method of forming the curves and the surfaces is not only simple, but also highly reduce the computation.