Comparison of HK and SC Curvature Description Methods

This paper compares two different local surface shape description methods. The general goal of surface shape description methods is to classify different surface shapes from range data. One well-known method to classify patches of various shapes is the HK classification space [2, 1, 10]. Another way to classify patches is the SC method introduced by Koenderink [9]. This paper presents several experiments designed to show the (1) qualitatively different classification, (2) the impact of thresholds and (3) the impact of different noise levels. We conclude that Koenderink’s approach has some advantages at low thresholds, complex scenes and at dealing with noise. 1 Description of the algorithms Gaussian (K) and Mean (H) curvatures are the most widely used indicators for surface shape classification in range image analysis. The HK segmentation [2, 1, 10] was introduced by Besl in 1986. He used Gaussian and Mean curvatures, which are calculated from the two principal curvatures 1 and 2. The Gaussian curvature equals the product of the principal curvatures. K = 1 2 (1) The Mean curvature equals the arithmetic average of the principal curvatures. H = 1 + 2 2 (2) Image points can be labelled as belonging to a viewpoint independent surface shape class type based on the combination of the signs from the Gaussian and Mean curvatures as shown in Table 1. We found that it is not necessary to differentiate between the different kinds of saddles (ie. K < 0). Therefore, we classify all saddle points as hyperboloid (Hy) points. K < 0 K = 0 K > 0 H < 0 Saddle Valley Concave Concave (Sv Hy) Cylinder (-Cy) Ellipsoid (-El) H = 0 Minimal Plane Impossible (M Hy) (Pl) H > 0 Saddle Ridge Convex Convex (Sr Hy) Cylinder (+Cy) Ellipsoid (+El) Table 1: Classification for the HK segmentation based on the sign of the Mean (H) and Gaussian (K) curvatures Koenderink defined an alternative curvature representation [9]. His approach (SC classification) decouples the shape and the magnitude of the curvedness. The surface in terms of relative curvature remains invariant under changes in scale. He defined a shape index S, which is a number in the range [-1,1]. The index covers all shapes except for the planar shape which has an indeterminate shape index ( 1 = 2 = 0). The shape index provides a continuous gradation between shapes, such as concave shapes (-1 < S < -1/2), hyperboloid shapes (-1/2 < S < 1/2) and convex shapes (1/2< S< 1). The image points can be classified as shown in Table 2. We use the positive principal curvatures ( 1;2 > 0) for convex objects. S = 2 ar tan 1 + 2 1 2 1 2 (3) Beside the shape index, Koenderink introduced the positive value C for describing the magnitude of the curvedness at a point. It is a measure of how highly or gently curved a point is. At a point that has no curvedness the value becomes zero. Therefore, this variable can be used to recognise a planar surface.