[Insight-users] Upsample bspline grid in deformableregistration6

[Nick Tustison]

To add to Torsten's comments --- I generalized Lee's original
algorithm to N dimensions and arbitrary order within ITK (including
the BSplineKernelFunction) and submitted it to the Insight Journal.
You can find the submission at

http://hdl.handle.net/1926/140

The submission also includes a discussion of the upsampling process
for arbitrary order b-splines.

Good luck,
Nick Tustison

On 3/6/07, Torsten Rohlfing <torsten at synapse.sri.com> wrote:
> Hi there:
>
> I am not an expert in ITK's implementation of the B-spline
> transformation model, so I may be completely redundant here. And I may
> be missing the point completely. Anyway, here are my thoughts:
>
> The B-spline grid should be upsampled (let's say: refined) using
> appropriate relationships that guarantee identical transformations
> before and after refinement.
>
> For third-order 2D transformations, the explicit formulas are given in
>
> S. Lee, G. Wolberg, and S. Y. Shin, "Scattered data interpolation with
> multilevel B-splines," IEEE Transactions on Visualization and Computer
> Graphics, vol. 3, no. 3, pp. 228-244, 1997.
>
> Their 2D expressions, however, are simple to generalize using the tensor
> product of 1D refinement formulas. These can, for example, be found
> (apologies for self citation) in
>
> T. Rohlfing, C. R. Maurer, Jr., W. G. O'Dell, and J. Zhong, "Modeling
> liver motion and deformation during the respiratory cycle using
> intensity-based free-form registration of gated MR images," in /Medical
> Imaging: Visualization, Display, and Image-Guided Procedures/, S. K.
> Mun, Ed., 2001, vol. 4319 of /Proceedings of the SPIE/, pp. 337-348.
>
> Again, this is for third-order splines. For different orders I am sure
> there is a paper out there somewhere that generalizes things further.
> Anyway, the correct upsampling depends on the spline order, which should
> therefore be considered in the upsampling process.
>
> Not sure if that happens in the proposed procedure of getting the
> parameter image, upsampling that and then transforming back into a
> spline transformation.
>
> Best,
> Torsten
>
> --
> Torsten Rohlfing, PhD          SRI International, Neuroscience Program
>  Research Scientist             333 Ravenswood Ave, Menlo Park, CA 94025
>   Phone: ++1 (650) 859-3379      Fax: ++1 (650) 859-2743
>    torsten at synapse.sri.com        http://www.stanford.edu/~rohlfing/
>
>      "Though this be madness, yet there is a method in't"
>
>
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