Hi Philipp
the Hessian is estimated at each vertex by doing a quadratic fit to the
local surface as the height function over the tangent plane of all the
vertices in a 2-neighborhood of that vertex. I'm not sure what the gradient
vanishing is about, but the curvatures are just the eigenvalues of the
Hessian, so I don't think the gradient has anything to do with it. Not that
we also have some discrete tools for computing curvature
make sense?
cheers
Bruce
On Thu, 3 Oct 2019, LOSKE, PHILIPP
(PGR) wrote:
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Hi,
I am trying to understand how exactly FreeSurfer estimates the curvature
values from the white surface. From the mailing lists I understood that the
white surface is modeled by fitting a second-order polynomial function and
curvatures are estimated from the Hessian matrix at each vertex (thank you
Bruce). However, I still have trouble to understand how this works in
detail. First, as I understand it, curvature can only be derived from the
Hessian if the gradient vanishes (why is this the case?), and from
differential geometry, shouldn't instead the shape operator be calculated at
each vertex on the surface? Second, are the Gaussian and mean curvatures
then directly calculated from Hessian/Shape operator or first principle
curvatures (and are they saved somewhere?). I tried to find a detailed
explanation in some of the FreeSurfer papers, but couldn't find anything
really.
Thank you very much in advance!
Cheers
Philipp
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