Application Of Geometric Algebra For 3D Reconstruction In Computer Vision

Generally, standard linear algebra is used to deal with the problems of computer vision. In this paper, we will introduce a new mathematical framework of Geometric (Clifford) algebra. Geometric algebra is a practical and powerful framework which unifies the linear algebra and geometry for the representation and computation of geometric problems. It can represent elements in an intuitive manner. The use of geometric invariants can lead to important simplifications in geometric computing. The reconstruction of there dimensional scene structure from images has been an active research topic in computer vision. Geometric algebra is a very good analysis for the research of 3D reconstruction.The introduction is separated into two parts. The first part concentrates on the geometric interpretation of elements and the second part on algebraic properties of Clifford algebra elements. These are just two names for the same thing. The only difference is that we would like to emphasize the geometric interpretation of the elements when we use the name of Geometric algebra. In terms of the geometric product, a number of important algebraic products can be simply defined. This provides a powerful approach to a unified system of algebraic and geometric structures, because it reduces similarities in different algebraic systems to a common body. Despite of its importance and influence in actual applications, projective geometry has not been fully integrated into mathematics. The coordinate-based methods and the ordinary geometric analysis do not meld well with the popular mathematics of today. Geometric algebra provides a very natural language for projective geometry.There are many algorithms about solving the problems of reconstruction. Invariance has been widely used in the field of computer vision. The cross ratio is invariant under projective transformations. We discuss the projective invariant in terms of different formulations using Geometric algebra. Using the dual bracket and dual operation, the geometry structure of two cameras is described in a new way. Geometric constraints in sterol-vision are also discussed in this new framework. Here, we propose a new approach for three dimension scene reconstruction based on the use of invariant without camera calibration. Our method uses four reference points to build two projective reference planes by exploiting some geometrical shapes in the scene. The experiment shows that the method is validated.