Compact and efficient 3D shape description through radial function approximation

Abstract

A fast and simple method for three-dimensional shape description is described. The method views a 3D object as a radial distance function on the unit sphere, and thus reduces the dimensionality of the description problem by one. The radial distance function is approximated by Fourier methods in the basis of the spherical harmonic polynomials. The necessary integration is carried out on the object boundary, rather than on the unit sphere. Consequently, there is no need of a parameterisation of the object surface. The description makes it possible to compare shapes in a computationally very simple way. Solutions on how to cope with translated and rotated objects are discussed. The method is developed for star-shaped objects, but is stable even if the input image is non-star-shaped. The method is tested in a data set from magnetic resonance imaging (MRI) of the brain. Potential medical applications are discussed.