Robust Estimation of Surface Curvature from Deformation of Apparent Contours

Abstract

Surface curvature along extremal boundaries is potentially useful information for navigation, grasping and object identification tasks. Previous theories have shown that qualitative information about curvature can be obtained from a static view. Furthermore it is known that, for orthographic projection, under planar viewer-motion, quantitative curvature information is available from spatio-temporal derivatives of flow. This theory is extended here to arbitrary curvilinear viewer-motion and perspective projection. We show that curvatures can actually be computed this way in practice, but that they are highly sensitive to errors in viewer-motion estimates. Intuitively, relative or differential measurements of curvature might be far more robust. Rather than measuring the absolute deformation of an apparent contour, differential quantities depend on the rate at which surface features are swept over an extremal boundary as the viewer moves. It is shown that, theoretically, such differential quantities are indeed far less sensitive to uncertainty in viewer-motion. Ratios of differential measurements are less sensitive still. In practice sensitivity is reduced by about two orders of magnitude. We believe this represents a significant step in the development of practical techniques for robust, qualitative 3D vision.