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?

    [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.



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