Mirror symmetry most probable shape

Figure 18a shows a configuration of points whose locations are given by a normal distribution function (marked as rectangles having width and length proportional to the standard deviation). In this section we show a method of evaluating the most probable symmetric shape closest to the data. For simplicity we derive the method with respect to rotational symmetry of order n (Cn-symmetry). The solution for mirror symmetry is similar (see Appendix D). 

Several examples are shown in Figure 18 where for a given set of measurements (Figure 18a), the most probable symmetric shapes are shown (Figure 18b-e for C2, C3, C6 and mirror symmetry respectively). 

In Section VIII we described a method for finding the most probable rotationally symmetric shape given measurements of point location. The solution for mirror symmetry is similar. In this case, given m measurements (where m - 2q), the unknown parameters are fyjpj, (0 and 0 where 0 is the angle of the reflection axis. However these parameters are redundant and we reduce the dimensionality of the problem by replacing two-dimensional (0 with the one dimensional x0 representing the x-coordinate at which the reflection axis intersects the x-axis. Additionally we replace Rt, the rotation matrix with ... 

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See also in sourсe #XX -- [ ]

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