We conclude the discussion of the implementation by showing some more examples, produced by this algorithm. As rotational alignment and closure constraints are not considered here, these results do not yet constitute ideal shape matchings. To take these features into account, one would need to combine the dynamic programming with an exhaustive search over SO(2). Also one would have to allow for different initial points of the parametrization, in case of closed curves, effectively resulting in another exhaustive search over SO(2). The grid size used here was n = 100. The neighbourhood Ni,j was
taken to be of size 5 in x and y direction. Each group of pictures features the optimal γ (a), the initial parametrization of q1 and q2 (b)-(c), the final parametrization of q2 (d)
and the connecting geodesic (e). (b)-(d) are in low resolution to emphasize the changes in parameter. Gaps are to be interpreted as particularly stretched segments and do not occur in a higher resolution. Occurrence of discontinuities in the optimal γ are particularly notable in figure 15(a) at t = 0.14, t = 0.71 and t = 0.98. Figure 16(a) features less pronounced discontinuities at t = 0.42 and t = 1. No singularities at all are seen in figure 17(a). Apparently, in this case the shapes are close enough not to require any ‘over-stretching’. The last shape matching, again, features a small discontinuity of the optimal γ (figure 18(a)) at t = 0.93. Reparametrization was always performed on both shapes, using the decomposition γ = γ2 ◦ γ1− (as in lemma 4.19). However, only the
changes of parameter in the second shape are notable, as γ1 simply introduces intervals
of constancy to the first shape. These examples further emphasize that discontinuous optimal matching prescriptions do arise for generic shape constellations and not only in examples that are specifically designed to exhibit such behaviour.
0 0.5 1 0 0.2 0.4 0.6 0.8 1 (a) (b) (c) (d) (e)
0 0.5 1 0 0.2 0.4 0.6 0.8 1 (a) (b) (c) (d) (e)
Figure 16: Geodesic distance d = 0.747
0 0.5 1 0 0.2 0.4 0.6 0.8 1 (a) (b) (c) (d) (e)
Figure 17: Geodesic distance d = 0.345
0 0.5 1 0 0.2 0.4 0.6 0.8 1 (a) (b) (c) (d) (e)
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