[Insight-users] LevelSetFunction / CurvesLevelSetFunction and minimal curvature

[Michael Schmidt]

Hi there,

I think there is a small mistake in the implementation of the CurvesLevelSetFunction or the LevelSetFunction respectively, regarding computation of minimal curvature. Although I currently don't use this feature, I think I might have found a litte mistake in the implementation, and want to share it with you so it can be fixed or somebody can correct me.

As far as I understand the implementation of ComputeMinimalCurvature, this function returns the principal curvature term that has minimal *absolute* value.

I don't know, if this function is used for other LevelSet-Filters as well, but at least in the context of CurvesLevelSetFunction this doesn't make sense in my opinion. In this filter the regularization force is defined on the minimal curvature and supposed to "smoothen" small tubelike shapes only along their centerline, not it's crossection. Now, given ITK's definitions of inside and outside, the principal curvature along the cross section of the tube will *always* be positive. Accordingly, negative principal curvature can only result from curvature of the centerline and should always be used for regularization. 

Now, in most cases this might not be a problem, but suppose a small tube with a very sharp corner, where the negative curvature on the "inner" side of the corner has a higher amount than the curvature of the crosssection. Then the regularization force is computed using the curvature of the cross section and driving the surface in the wrong direction, resulting possibly in a cut in the tube. Especially look at Figure 6 in the work (in the version I got at http://certis.enpc.fr/publications/papers/01mia.pdf). There the regularization quickly smoothens the sharp corners with negative curvature. Has anybody tested the ITK-Filter on similar data? In their work the authors only refer to taking "the smaller nonzero eigenvalue", which probably can be interpreted in both ways, juding by absolute values or not.

Regards, Michael
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