Discrete and continuous curvature computation for real data

Abstract

This paper describes two methods for estimating the minimum and maximum curvatures for a 3D surface and compares the computational efficiency of these approaches on 3D sensor data. The classical method of Least Square Fitting (LSF) finds an approximation of a cubic polynomial fit for the local surface around the point of interest P and uses the coefficients to compute curvatures. The Discrete Differential Geometry (DDG) algorithm approximates a triangulation of the surface around P and calculates the angle deficit at P as an estimate of the curvatures. The accuracy and speed of both algorithms are compared by applying them to synthetic and real data sets with sampling neighborhoods of varying sizes. Our results indicate that the LSF and DDG methods produce comparable results for curvature estimations but the DDG method performs two orders of magnitude faster, on average. However, the DDG algorithm is more susceptible to noise because it does not smooth the data as well as the LSF method. In applications where it is not necessary for the curvatures to be precise (such as estimating anchor point locations for face recognition) the DDG method yields similar results to the LSF method while performing much more efficiently.