Measurement, orientation determination and recognition of surface shapes in range images
Ping Liang · 1987
Theory and matching algorithms are developed for accurate shape measurement, orientation determination and recognition of three dimensional surface shapes in range image, especially smooth curved surface shapes. Two theorems are proved which give the conditions of congruence of surfaces directly applicable to surface shape description and recognition. A set of geometric descriptors for hyperbolic, elliptic and developable surfaces, which uniquely describes a surface and can be defined geometrically so that it is invariant under parameterization, are identified based on the theorems proved and some known results. Surface shapes are segmented into regions with the Gaussian curvature positive, negative and zero, and along discontinuities of orientation and depth. A relational graph is used in representing the connectivity of the segmented surface patches. The relational graphs are first matched when matching two surface shapes. Criteria for two general surface shapes, which are segmented as described, to be congruent or similar are presented. Unit normal and shape descriptors list array (UNSDLA) representations, and matching algorithms for hyperbolic, elliptic and developable surfaces are developed. The matching algorithms match the UNSDLA representations of surfaces via the Gaussian map. Matching of surfaces is formulated as an optimization problem. The UNSDLA of the sensed surface is rotated to minimize a matching error. The minimum of the matching error is searched over the space of all possible rotations by a gradient descent or quadratic minimization algorithm. This allows more accurate orientation determination and matching. Matching of hyperbolic and elliptic surfaces is performed at hierarchial multiresolution levels of the UNSDLA to accelerate the convergence. The matching algorithms are capable of matching surfaces with the same shape but different sizes, matching surfaces with occlusion and matching part of a surface to the whole surface. The orientation and scaling factor of a sensed surface relative to the object model surface are determined by the matching algorithms. The representation and matching algorithms can accommodate the matching of hyperbolic and elliptic surfaces with Gaussian maps not one to one, and matching developable surfaces with Gaussian maps of lines of curvature with nonzero principal curvature not one to one. Four most commonly encountered developable surfaces are considered. Method for determining the type of a developable surface is provided. A theorem on developable surfaces is proved.