[Insight-users] Cubic BSpline kernel Theory?

[Luis Ibanez]

Hi Mottes:


You may find useful to look at the following tutorial:
http://www.na-mic.org/Wiki/images/0/06/Insight-DeformableRegistration-BSplines.ppt

It will explain you how to derive the equations by performing
successive convolutions with basic kernels.


You will also find the explicit expression
for cubic B-Splines in the Wikipedia:
http://en.wikipedia.org/wiki/B-spline#Uniform_cubic_B-splines


   Regards,


     Luis


----------------------------------------------------------------------------------
On Mon, Oct 12, 2009 at 4:11 PM, motes motes <mort.motes at gmail.com> wrote:
> In the BSplineDeformableTransform a BSpline kernel is used which in
> the Cubic case is:
>
> ::BSplineKernel( const double & u) const
> {
>  double absValue = vnl_math_abs( u );
>  double sqrValue = vnl_math_sqr( u );
>
>  if ( absValue  < 1.0 ) {
>    return ( 4.0 - 6.0 * sqrValue + 3.0 * sqrValue * absValue ) / 6.0;
>  } else if ( absValue < 2.0 ) {
>    return ( 8.0 - 12 * absValue + 6.0 * sqrValue - sqrValue * absValue ) / 6.0;
>  } else {
>    return 0.0;
>  }
> }
>
> But where does this expression come from? In the BSpline litterature I
> have only found the basis functions expressed using the Cox De Boor
> recursion formula. Where can I read up on the above explicit
> expression?
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