A Study Of Computer Vision Based On Geometric Algebra

In this dissertation, we introduce a new mathematical tool into the research of computer vision, namely geometric algebra, which is a mathematical framework possesses the characters of both geometry and algebra. It has distinct advantages in multi-dimensional signal processing, in representing geometric entities and geometric relationships, and in geometric computing. The current thesis usefully explores the issue of applications of geometric algebra in 3D reconstruction based on images in computer vision, including the correspondences of color images, the epipolar geometry in two-view system, and the calibration of intrinsic camera parameters. The main work of this dissertation is as follows:(1) The projective geometry expressed in geometric algebra is derived. Based on the introduction of the vector model and homogeneous model of geometric algebra, we analyze the representations of points, lines, planes in projective geometry, and the representation of join, meet, and dualization of geometry entities. Then the invariants of cross-ratio in projective geometry with geometric meanings are derived in geometric algebra. It can be seen from the analysis that the geometric algebra provides a coordinate-free representation for projective geometry in a concise and intuitive manner, and provides a robust and holistic method for geometric problem solving.(2) A method of color images corresponding based on the phase correlation is proposed. In this dissertation, a color pixel is expressed by a pure quaternion, and the color images are corresponded based on the quaternion phase correlation. Firstly, the definitions and properties of quaternion Fourier transformations are analyzed. And then, the corresponding method is proposed based on the quaternion Fourier transformation and log-polar coordinate transformation. This method which is in a holistic manner is different from the methods based on grayscale images and the component-wise method, since the vector nature of color is preserved. The experiments on different color images demonstrate that the proposed method is not only overwhelming the others in precision of corresponding, but also much faster.(3) The feature description in color images is discussed. In the corresponding of images based on feature points, color information has been embedded in the local descriptors in order to improve the photometric invariance and discriminative power. However, the dimensionalities of such descriptors are quite large, which limits the performance of feature matching in terms of speed. In the current thesis, a compact local color descriptor is introduced based on the quaternion principle component analysis. The experimental results show that our descriptor is more efficient in matching computing than the standard RGB-SIFT without sacrificing the robustness and discriminative power.(4) A geometric approach for analyzing and computing the epipolar geometry and the calibration of intrinsic camera parameters are studied. From the pure geometric viewpoint, the expression of the camera model is derived in geometric algebra. And then, we analyze the epipolar geometry in two view system, and obtain the fundamental matrix in geometric algebra. Based on the properties of the fundamental matrix, the constraints on points in two camera planes are derived. In the study of the camera calibration, we firstly analyze the projective definition of quadric conics in geometric algebra. An invariant representation of quadric conics is obtained based on this definition, which is satisfied by the image of absolute conic. Utilizing this invariant representation of absolute conic, a method of computing the image of absolute conic is proposed, and the calibration of intrinsic parameters of camera could be obtained by making a further development.Based on the studies above, this dissertation studies the realization of 3D reconstruction and explores the feasibility and availability of the application of geometric algebra in computer vision.