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

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 σκ 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.