Comparing curvature estimation techniques.

This article presents a careful comparative evaluation of two techniques for numerical curvature estimation of 2D closed contours (more specifically closed, regular and simple parametric curves). The considered methods are: (a) a 1-D Fourier-based approach; and (b) a 2-D Fourier-based approach involving the embedding of the contour into a 2-D regular surface (presented for the first time in this article). Both these techniques employ Gaussian smoothing as a regularizing condition in order to estimate the first and second derivatives needed for curvature estimation. These methods are considered according to a multiresolution approach, where the standard deviation of the Gaussians are used as scale parameters. The methods are applied to a standard set of curves whose analytical curvatures are known in order to estimate and compare the errors of the numerical approaches. Three kinds of parametric curves are considered: (i) curves with analytical description; (ii) curves synthesized in terms of Fourier components of curvature; and (iii) curves obtained by splines. A precise comparison methodology is devised which includes the adoption of a common spatial quantization approach (namely square box quantization) and the explicit consideration of the influence of the related smoothing parameters. The obtained results indicate that the 1D approach is not only faster, but also more accurate. However, the 2-D approach is still interesting and reasonably accurate for applications in situations where the curvature along the whole 2-D domains is needed. Key-words: Differential geometry, numerical techniques, cuvature estimation, performance evaluation, Fourier transform.