# Registration Notes

This note is for CS5240

Last time we discussed Rigid & Nonrigid and their methods:

 Rigid Nonrigid similarity transformation affine transformation ICP nonrigid ICP

Methods below are approximation. Now we discuss interpolation.

## Thin Plate Spline

How to get TPS?

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.