Direct Computation of 3d Shape and Motion Invariants

Structure from motion often refers to the computation of 3D structure from a matched sequence of images. However, a (relative) depth map of a surface may not be a good representation for storage and recognition; a more concise representation seems necessary. The sign of the Gaussian curvature of a surface is one candidate to be a part of a useful representation of the surface. I will show that in order to compute the sign of the Gaussian curvature it is not necessary rst to go through the computationally expensive and error sensitive process of recovering the exact function of the surface and the motion parameters. I will rst show that the sign of the normal curvature in a given direction at a given point in the image can be computed from a simple diierence of slopes of line-segments in one image. Using this result, local surface patches can be classiied as convex, concave, parabolic (cylindrical), hyperbolic (saddle point) or planar. At the same time the translational component of the optical ow is obtained, from which the focus of expansion can be computed. In addition, the axes of principal curvature and the axes of zero curvature are obtained.