Decomposing the Tangent of Occluding Boundaries according to Curvatures and Torsions

Huizong Yang, Anthony Yezzi ;

"This paper develops new insight into the local structure of occluding boundaries on 3D surfaces. Prior literature has addressed the relationship between 3D occluding boundaries and their 2D image projections by radial curvature, planar curvature, and Gaussian curvature. Occluding boundaries have also been studied implicitly as intersections of level surfaces, avoiding their explicit description in terms of local surface geometry. In contrast, this work studies and characterizes the local structure of occluding curves explicitly in terms of the local geometry of the surface. We show how the first order structure of the occluding curve (its tangent) can be extracted from the second order structure of the surface purely along the viewing direction, without the need to consider curvatures or torsions in other directions. We derive a theorem to show that the tangent vector of the occluding boundary exhibits a strikingly elegant decomposition along the viewing direction and its orthogonal tangent, where the decomposition weights precisely match the geodesic torsion and the normal curvature of the surface respectively only along the line-of-sight! Though the focus of this paper is an enhanced theoretical understanding of the occluding curve in the continuum, we nevertheless demonstrate its potential numerical utility in a straight-forward marching method to explicitly trace out the occluding curve. We also present mathematical analysis to show the relevance of this theory to computer vision and how it might be leveraged in more accurate future algorithms for 2D/3D registration and/or multiview stereo reconstruction."