Representation and recognition of surface shapes in range images: A differential geometry approach

Theory and matching algorithms are developed for accurate orientation determination and recognition of 3D surface shapes in range images. Two corollaries to the fundamental theory of surface theory are proved. The first corollary proves the invariance of the fundamental coefficients when lines of curvature are used as the intrinsic parameter curves. The second corollary proves that a diffeomorphism which preserves the intrinsic distance along the principal directions, in addition to preserving the eigenvectors and eigenvalues of the shape operator (Weingarten map), is necessarily an isometry. Based on these two corollaries, a set of geometric descriptors which satisfy the uniqueness and invariance requirements are theoretically identified for all classes of surfaces, namely, hyperbolic, elliptic, and developable surfaces. Theunit normal and shape descriptors list array (UNSDLA) representation and the corresponding matching algorithm are developed. The UNSDLA is a generalization of the extended Gaussian image (EGI). The EGI has a fundamental limitation; that is, it can only uniquely represent convex shapes. The new representation overcomes this limitation of the EGI and extends the scope of unique representation to all classes of surfaces. Moreover, it still has all the advantages of the EGI. This is achieved by preserving the connectivity of the original data. Connectivity here should include not only the adjacency relation of points or patches on a surface, but also the direction and order in which the points or patches are traversed in a connected path. The importance of the direction and order of connectivity is emphasized. Surface matching can be performed more accurately using the UNSDLA than the EGI. Based on the UNSDLA representations, surfaces can be matched via the Gaussian map by optimization over all possible rotations of a surface shape. The representation and matching algorithm can deal with hyperbolic and elliptic surfaces whose Gaussian maps are not one-to-one. Developable surfaces whose Gaussian maps of lines of curvature with nonzero principal curvature are not one-to-one can also be accommodated. Two theorems on developable surfaces are proved.