Registration Notes
This note is for CS5240
Last time we discussed Rigid & Nonrigid and their methods:
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Rigid | Nonrigid |
similarity transformation | affine transformation |
ICP | nonrigid ICP |
[/su_table]
Methods below are approximation. Now we discuss interpolation.
Thin Plate Spline
How to get TPS?
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Minimizing bending energy!TPS maps [latex]p_i[/latex] to [latex]q_i[/latex] exactly.
Consider jth component [latex]v_{ij}[/latex] of [latex]q_i[/latex], TPS maps [latex]p_i[/latex] to [latex]v_{ij}[/latex] by [latex]f(p_i)=v_{ij}[/latex] which minimize bending energy denoted as [latex]E_d(f)[/latex].
Bending energy function takes two parameters the first is d(the dimension of the point), the second is m, which denotes order-m derivatives.
Finally the function f that minimize the Bending energy takes the form
[latex]f(x’) = a^Tx’+\sum_{i=1}^{n} w_iU(||x-p_i||)[/latex]
a are affine parameters. w are weights. U(r) is increasing function of distance r.