[Insight-users] Cubic BSpline kernel Theory?

[motes motes]

I recognize the expression from the wiki page. But how does the itk kernel:


 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;
 }
}

relate to the expression on wiki? When I write the itk kernel out I get:


4/6 - u^2 + 1/2*u^3          : u<1
4/3 - 2*u + u^2 - u^3/6     : u<2

I can't really see the connection between this expression and the
basis matrix from the wiki page.






On Tue, Oct 13, 2009 at 12:39 AM, Luis Ibanez <luis.ibanez at kitware.com> wrote:
> 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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>