This note is for CS5240

Last time we discussed Rigid & Nonrigid and their methods:

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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.