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Chapter 2: Literature Review 21extracted. Shape functions D 1 , D 2 , D 3 , D 4 , and A 3 are computed at each of theserandom points.•D 1 : Distance between a fixed point (centroid) and a random point.•D 2 : Distance between two random points.•D 3 : Square root of the area of triangle formed by three random points.•D 4 : Cube root volume of the tetrahedron of four random points.•A 3 : Angle between three random points.They suggest the use of D 2 shape function for computing Shape Distributions due toits robustness and efficiency along with invariance to rotation and translation. The D 2distances between random points are normalized using the mean distance. The shapedistribution is the histogram that measures the frequency of occurrence of distanceswithin a specified range of distance values. Once the Shape Distributions aregenerated the distance between the two solid shapes is computed using L N norm.Usually L 2 norm is used for comparison, though other distances such as Earth Mover’sdistance or match distances can also be used.This technique is robust and efficient for simple objects and gross shape similarity. Asthe resolution of the 3D model increases the comparison becomes more robust, but thecomputational time increases. Furthermore as objects become more and morecomplex, the Shape Distributions tend to assume similar shape resulting in inaccuratecomparison of solid models. Shape Distributions have been experimented with limitedsuccess on mechanical parts and real laser scanned data. Ohbuchi et al. in [Ohbuchi etal., 2003] improve the performance of Shape Distributions with a 2D histogram ofangle-distance and absolute angle distance that can be computed from the D 2 shapedistribution. Page et al. [Page et al., 2003b] define shape information as the entropy ofthe curvature density. They use it to describe the complexity of the 3D shape. Hetzel etal. [Hetzel et al., 2001] present an occlusion robust algorithm for 3D objectrecognition that makes use of local features such as the shape index, pixel depth andthe surface normal characteristics in a multidimensional histogram. Histograms of twoobjects are matched and verified using the chi–squared hypothesis test to achieveshape recognition.2.3.5 Topology DescriptionThe topology of a 3D model is an important property for measuring similarity betweendifferent models. Topology of models is typically represented in the form of arelational data structure such as trees or directed acyclic graphs. Subsequently, thesimilarity estimation problem is reduced to a graph or tree comparison problem.