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# Moreover when the number of sampled point tends to be

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Unformatted text preview: , z | M , Ω ,V (D; Ω )) p (1) i i i i 3 For any set of points in ℜ , it is possible to build the seven PDFs representing the classes of symmetries. In order to identify which PDF provides the best description of the measured points it is necessary to compare their performances by computing their likelihood by means of the following expression: ˆ L(M i ) = ∏ ~ (x j , y j , z j | M i , Ω ij , Vi (D j ; Ω ij )) p n j =1 (2) where Dj=D\{(xj,yj,zj)} ∀j∈{1,...,n} and Ωij is the maximum likelihood estimate of Minimum Reference Geometric Elements Ωi based on Dj, namely: Ω ij = arg max ~ (D j | M i , Ω i , Vi (D j ; Ωi )) p (3) Ω i An exhaustive formulation of the seven PDFs is illustrated in [Chiabert et al., 2003], with a detailed description of the semi-parametric model Mi, the set of reference parameters Ωi and the projecting function Vi(·;Ωi). 4. THE EXPERIMENTAL TESTS The strength of the proposed methodology relies on its ability to identify a surface according to the seven classes of symmetries available in the Euclidean space. The experimental test involves seven simple surfaces and demonstrates the ability of algorithms in classifying each surface by identifying its symmetries. Some considerations on the sampling methodology have to be made before analyzing experimental results. Algorithm...
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