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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