Measuring Curvature

Many techniques for computing surface curvature have been developed to match the similarly wide range of input data formats. Polygonal meshes are predominantly used in graphics applications. Taubin uses a least-squares derivation to match the local geometry about a vertex to compute the principal directions and curvatures[5]. Meyer et al. rigorously derive formulas for discrete operators for measures of curvature based on averaging over the area and consistency with several theorem [6]. These two methods only are able to compute curvature on the vertexes, and they are highly affected by the regularity of the triangulation. Curvature can also be estimated by fitting a quadratic surface. Methods for fitting quadric surface patches to conform to the coordinates and properties of neighboring vertexes have been proposed for mesh simplification[7]. Volumetric data is often the input for image analysis as well as computer graphics. Volume data discretized into voxels are the basis of both modern level set methods as well as most volume visualization techniques, while analytical forms are used as continuous volume generating functions to generate the 3D scalar fields within which implicit surfaces are embedded as isophotes. Volume rendering systems and edge finding algorithms usually compute first order derivatives, and sometimes explore second-order differential properties as well. Thirion published the mathematics on computing level set curvature from second derivatives of volume intensities [4]. In his monograph, Sethian also derives curvature approximations from discrete data, and goes on to develop a broad foundation for complex applications involving deformable, discrete implicit surfaces based on level sets [8]. Many of these techniques are subject to sampling errors inherent in the data representation. Aliasing artifacts often perfuse the results with systemic errors that are difficult to ameliorate. We submit that we need to apply the concepts of scale-space to address these issues.