A Learning Theoretic Approach to Differential and Perceptual Geometry Part I: Curvature and Torsion are the Independent Componets of Space Crves.

摘要

In standard differential geometry, the Fundamental Theorem of Space Curves states the that two differnetial invariants of a curve, namely curvature and torsion, determien its geometry, or equivalently, he isometry class of the curve up to rigid motions in the Euclidean three-dimensional space. Consider a physical model of a space curve made from a sufficiently thin, yet visible rigid wire, and the problem of perceptual identification (by a human observer or a robot) of two given physical model curves. In a previous paper (perceptual geometry) we have emphaized a learning theoretic approach to construct a perceptual geoemtry of the surfaces in the environment. In paritcular, we have described a computational method for mathematical representation of objects in the perceptual geoemtry inspired by the ecological theory of Gibson, and adhering to the principles of Gestalt in perceptual organization of vision. In this paper, we continue our learnign theoretic treatment of perceptual geometry of objects, focusing on the case of physical models of space curves. In particular, we address the question of perceptually distinguishing two possibly novel space curves based on observer's prior visual experience of physical models of curves in the environemtn. The Fundamental Theorem of Space Curves inspires an analogous result in perceptual geoemtry as follows. We apply learning theory to the statistics of a sufficiently rich collection of physical models of curves, to derive two statistically independent local functions, that we call by analogy, the curvature and torsion. This pair of invariants distinguish physical models of curves in the sense of perceptual geoemtry. That is, in an appropriate resolution, an observer can distinguish two perceptually identical physical models in differnet locations. If these pairs of functions are approximately the same for two givne space curves, then after possibly some changes of viewing planes, the observer cofirms the two are the same.