[Insight-users] LevelSetFunction / CurvesLevelSetFunction and minimal curvature

[Luca Antiga]

Hi Michael,
  I'm not sure I agree with you, but to be sure one should reproduce  
the cases in the paper and highlight the differences. Actually, it  
would be nice to have them as a test.

If you consider Figure 3 and 6, taking the minimum signed curvature  
as you suggest would result in a different treatment of the two sides  
of the wiggles (with positive and negative curvatures respectively),  
and the line would dramatically change its cross section during the  
evolution, which I don't think it's what happens (and I don't think  
it would be desirable either). Similarly, at bifurcations, you would  
keep moving the bifurcation apex away from the bifurcation trying to  
lower its negative curvature.
In general, you would invariably move the inner side of a wiggle, and  
never the outer side.

You indeed get some change in the cross section in the extreme case  
in Figure 3, but I don't think this results as the evolution of the  
surface along signed minimal curvature, but rather as the progressive  
flattening of the low curvature region at the corners within the  
plane where two tubes meet. I must say, however, that what happens to  
curvatures during the evolution is often tricky to figure out by just  
looking at screenshots.

I might be wrong, so it would be great if you took the effort of  
reproducing the cases. If it turns out that the cases in the paper  
work as you suggest, we could add a flag to control which kind of  
minimal curvature to choose (signed or absolute) and default it to  
absolute, to preserve backwards compatibility.

Last, minimal principal curvature is defined with sign in  
differential geometry (the way you suggest). The paper never mentions  
"minimal principal curvature", but "smaller principal  
curvature" (which is admittedly not very informative).

By for now


Luca



On Jul 6, 2008, at 3:17 PM, Michael Schmidt wrote:

> 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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