Learning theoretic approach to differential and perceptual geometry: I. Curvature and torsion are the independent components of space curves

In standard differential geometry, the Fundamental Theorem of Space Curves states the that two differential invariants of a curve, namely curvature and torsion, determine its geometry, or equivalently, the 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 emphasized a learning theoretic approach to construct a perceptual geometry of the surfaces in the environment. In particular, we have described a computational method for mathematical representation of objects in the perceptual geometry 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 learning 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 environment. The Fundamental Theorem of Space Curves inspires an analogous result in perceptual geometry 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 geometry. That is, in an appropriate resolution, an observer can distinguish two perceptually identical physical models in different locations. If these pairs of functions are approximately the same for two given space curves, then after possibly some changes of viewing planes, the observer confirms the two are the same.