Registration Notes

This note is for CS5240

Last time we discussed Rigid & Nonrigid and their methods:

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.