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# 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[1,2,4] was introduced by Besl in 1986. He defines Gaussian and Mean curvatures, which are calculated from the two principal curvatures and . The Gaussian curvature equals the product of the principal curvatures. The Mean curvature equals the arithmetic average of the principal curvatures.

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.

Table 1: Classification for the HK segmentation based on the sign H and K

Koenderink defined an alternative curvature representation[3]. 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 (). 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 ( for convex objects. 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 plane surface.

Table 2: Classification for Koenderink's approach based on the shape index (S)

Next: Classification regions Up: Comparison of HK and Previous: Comparison of HK and

Helmut Cantzler
Tue Oct 24 15:53:47 BST 2000