Parametrization of Closed Surfaces for 3-D Shape Description

摘要

This paper presents procedures for the explicit parametric representation and global description of surfaces of simply connected 3-D objects. The novel techniques overcome severe limitations of earlier methods (restriction to star-shaped objects (D. H. Ballard and Ch. M. Brown, Computer Vision, Prentice-Hall, Englewood Cliffs, NJ, 1981), constraints on positioning and shape of cross-sections (F. Solina and R. Bajcsy, IEEE Trans. Pattern Anal. Much. Intell. 12(2), 1990, 131-147; L. H. Staib and J. S. Duncan, in Visualization in Biomedical Computing 1992 (R. A. Robb, Ed.), Vol. Proc. SPIE 108, pp. 90-104, 1992), and nonhomogeneous distribution of parameter space). We parametrize the surface by defining a continuous, one-to-one mapping from the surface of the original object to the surface of a unit sphere. The parametrization is formulated as a constrained optimization problem. Practicable starting values are obtained by an initial mapping based on a heat conduction model. The parametrization enables us to expand the object surface into a series of spherical harmonic functions, extending to 3-D the concept of elliptical Fourier descriptors for 2-D closed curves (E. Persoon and K. S. Fu, IEEE Trans. Syst. Man Cybernetics 7(3), 1977, 388-397; F. P. Kuhl and Ch. R. Giardina, Comput. Graphics Image Process. 18(3), 1982, 236-258). Invariant, object-centered descriptors are obtained by rotating the parameter net and the object into standard positions. The new methods are illustrated with 3-D test objects. Potential applications are recognition, classification, and comparison of convoluted surfaces or parts of surfaces of 3-D shapes.