Toponogov comparison theorem for open triangles

Dedicated to Professor Gromoll: The aim of our article is to generalize the Toponogov comparison theorem to a complete Riemannian manifold with smooth convex boundary. A geodesic triangle will be replaced by an open (geodesic) triangle standing on the boundary of the manifold, and a model surface will be replaced by the universal covering surface of a cylinder of revolution with totally geodesic boundary. Applications of our theorem are found in our article "Applications of Toponogov's comparison theorems for open triangles" (arXiv:1102.4156).Comment: This version 4 (36 pages, no figures) is a version to appear in Tohoku Mathematical Journa

The 3-D image recognition based on fuzzy neural network technology

Three dimensional stereoscopic image recognition system based on fuzzy-neural network technology was developed. The system consists of three parts; preprocessing part, feature extraction part, and matching part. Two CCD color camera image are fed to the preprocessing part, where several operations including RGB-HSV transformation are done. A multi-layer perception is used for the line detection in the feature extraction part. Then fuzzy matching technique is introduced in the matching part. The system is realized on SUN spark station and special image input hardware system. An experimental result on bottle images is also presented