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Unformatted text preview: ⎥ ⎢ 0 0 1⎦ ⎣0 The transformation matrix is characterized by six degrees of freedom so t = [ψ, θ, ϕ, tx, ty, tz] is the set of transformation parameter. The minimization of the distance between the set of measured points and the nominal surface is made by the following objective function: Q(t ) = ∑ (T(t ) ⋅ m i − n i ) ,
N 2 i =1 (3) where ni represents the nearest point of T(t)⋅mi on the reference element, point chosen in according of geometrical definition of distance. 160 W. Polini, U. Prisco and G. Giorleo Equation (3) represents the standard expression of the objective function. The expression itself is sufficient to give us an acceptable solution of the problem, but the resultant algorithm is relatively inefficient. In order to improve the speed of the algorithm it is useful the concept of equivalence class, based on the Lie subgroups. Speaking shortly, the concept of equivalence class implies that the feature is invariant for certain class of translations or rotations. So there are some parameters in the (3) that have no influence on the value of the objective function. This parameters can be simply set to zero, and the objective function can be rewritten without these...
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