The Line of Curvature Constraint and the Interpretation of 3-D Shape from Parallel Surface Contours

Abstract

Human vision is adept at inferr ing surface shape from a projection of curves lying across a surface, part icular ly from paral lel undulating curves. Since the image projection loses 3-D information, the interpretation must be constrained by certain a p r io r i assumptions. A theory of constraint on this problem [Stevens 1981a] proposes three assumptions, namely that neither the viewpoint nor the placement of the physical curves on the surface is misleading (two forms of general posit ion), and that the physical curves are lines of curvature across the surface. It follows that parallel contours in an image correspond to paral lel curves across an approximately cy l indr ical surface, and since the lines of curvature on a cylinder are geodesic and planar, one may reconstruct the image projection of an orthogonal net covering the surface. There is suff ic ient information available to estimate surface orientation at certain strongly constrained points, then to determine, given the geodesic and planarity properties, the surface orientation elsewhere by a simple extrapolation method. An implementation of this computation predicts local surface orientation that closely matches the orientation we perceive, thereby supporting the theory that our visual interpretation of surface shape from paral le l contours embodies the l ine of curvature constraint.