# Registration Notes

This note is for CS5240

Last time we discussed Rigid & Nonrigid and their methods:

[su_table]

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.