[Insight-users] Upsample bspline grid in deformableregistration6

[Torsten Rohlfing]

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
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   torsten at synapse.sri.com        http://www.stanford.edu/~rohlfing/

     "Though this be madness, yet there is a method in't"