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?

[su_custom_gallery source=”media: 243″ limit=”19″ target=”blank” width=”800″ height=”480″]

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.

 

 

Categories: 未分类