Curvature and Bending Energy in Digitized 2D and 3D Images

Abstract

Existing curvature estimators of planar curves are applied to a binary representation of the object.The parametric curve description is 1D-smoothed to overcome quantization errors.In this paper we estimate object curvature directly from a properly sampled grayscale image using 2D isotropic derivative-of-Gaussian filters.Three times oversampling or a Gaussain of 2.7 yields sampling-error free results.The algorithm was extended to find the principal curvatures of iso-grey surface patches in 3D.The Gaussian and mean curvatures can easily be computed from the principal curvatures.Integrated curvature and bending energy of a closed object in 2D or 3D is frequently used as shape discriminator.The binary methods sum the estimated curvature/energy values over a chain-code description of the contour (as in binary length estimators).We estimate bending energy through grey-volume measurement.Volume is measured without thresholding and does not introduce a sampling error.Edges are transformed into volumes by giving them a constant height after which they are shifted perpendicular to the edge over a small distance.Subtraction of the two images shifted in opposite direction produces a volume that is proportional to the edge length [1,2].This volume is weighted using the obtained energy values.This method produces very precise (CV < 0.01 %) and accurate measures.