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Topic: sub-voxel-precise estimation of the extremum of a tabulated function
Replies: 5   Last Post: Mar 14, 2012 9:29 AM

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Hannes Fassold

Posts: 3
Registered: 3/6/12
sub-voxel-precise estimation of the extremum of a tabulated function
Posted: Mar 6, 2012 3:40 AM
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I have a regular 3D voxel grid (spacing 1) and a arbitrary (real)
function evaluated at a 3x3x3 window in ('voxel grid') around a
'center' position (xc, yc, zc). So I have 27 function values at the
postions (xc +{-1|0|1}, yc +(-1|0|1}, zc + (-1|0|1})).

Now i want to interpolate a quadratical 'taylor' polynom into the
function and find the position (xc_0, yc_0, zc_0) where the
quadratical polynom has an extremum. How to do ?
For the 1D case, i know how to do - see fn. 'P3Interpolate_Extremum_1'
from http://www.ebyte.it/library/codesnippets/P3Interpolation.html.
But how to 'generalize' this function to 3 dimensions ??
Furthermore, I'd like to know whether these 27 grid points and
function values define my an unique quadratical polynom, or whether I
have to fit a 'best-fitting' quadratical polynom into these points.

By the way the problem comes from implementing section 2 in paper
http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=A9B821E909E3269BFFDCB11CC4C07DA9?doi=10.1.1.1.8475&rep=rep1&type=pdf

Many thx for any suggestions.



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