Invariants Theory in Computer Vision and Omnidirectional Vision

Abstract

This chapter will demonstrate that geometric algebra provides a simple mechanism for unifying current approaches in the computation and application of projective invariants using n-uncalibrated cameras. First, we describe Pascal’s theorem as a type of projective invariant, and then the theorem is applied for computing camera-intrinsic parameters. The fundamental projective invariant cross-ratio is studied in one, two, and three dimensions, using a single view and then n views. Next, by using the observations of two and three cameras, we apply projective invariants to the tasks of computing the view-center of a moving camera and to simplified visually guided grasping. The chapter also presents a geometric approach for the computation of shape and motion using projective invariants within a purely geometric algebra framework (Hestenes and Sobczyk (1984). Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, Hestenes and Ziegler (1991). Acta Applicandae Mathematicae 23: 25–63.) [138, 139]. Different approaches for projective reconstruction have utilized projective depth (Shashua (1994). IEEE Transations on Pattern Analysis and Machine Intelligence 16 (8): 778–790 and Sparr (1994). Proceedings of the European Conference on Computer Vision II.) [269, 284], projective invariants (Csurka and Faugeras (1998). Journal of Image and Vision Computing 16: 3–12.) [61], and factorization methods (Poelman and Kanade (1994). European Conference on Computer Vision, Tomasi and Kanade (1992). International Journal of Computer Vision 9 (2): 137–154 and Triggs (1995). IEEE Proceedings of the International Conference on Computer Vision (ICCV’95).) [237, 296, 300] (factorization methods incorporate projective depth calculations). We compute projective depth using projective invariants, which depend on the use of the fundamental matrix or trifocal tensor. Using these projective depths, we are then able to initiate a projective reconstruction procedure to compute shape and motion. We also apply the algebra of incidence in the development of geometric inference rules to extend 3D reconstruction.