# 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

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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 $p_i$ to $q_i$ exactly.

Consider jth component $v_{ij}$ of $q_i$, TPS maps $p_i$ to $v_{ij}$ by $f(p_i)=v_{ij}$ which minimize bending energy denoted as $E_d(f)$.

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

$f(x’) = a^Tx’+\sum_{i=1}^{n} w_iU(||x-p_i||)$

a are affine parameters. w are weights. U(r) is increasing function of distance r.

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