<title>Differential geometry for characterizing 3D shape change</title>

Abstract

In this paper, we present certain computational procedures for measurement of differential geometric quantities of surfaces. A condensed overview of differential geometry is presented and methods for measurement of various quantities are given. Our measurements are made from a stack of pre-segmented image slices, although no such explicit requirements besides knowledge of a set of 3-D points in space is needed by our algorithms. We assume that sufficiently small surface patches can be approximated by a biquadric polynomial. As differential characteristics are local properties, local surface fits are all that will be needed. Once the tangent plane (normal to the surface) is estimated from covariance matrix of the actual coordinate values of a surface patch, any given surface patch may be treated as a height map. This allows for invariant surface fitting where the tangent plane is transformed to algin with the x - y plane. Area, curvature, and principal directions can then be computed from surface fits. Knowledge of differential geometric quantities will allow for matching surface features in applications involving registration of 3-D medical images assuming rigid transformations or for arriving at point correspondences where non-rigid transformations are necessary. In non-rigid motion computation, once initial match vectors are obtained from a bending and stretching model, membrane smoothing with confidences optimizes the flow estimates. To this end, a linear vector equation in terms of components of flow vectors is derived which must be satisfied at all nodes of a finite element grid.© (1992) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.