<div dir="ltr">Oh boy, the math does slip away too fast :)<br></div><div class="gmail_extra"><br><br><div class="gmail_quote">On Fri, Aug 23, 2013 at 4:31 PM, Burlen Loring <span dir="ltr"><<a href="mailto:bloring@lbl.gov" target="_blank">bloring@lbl.gov</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
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<div>Eigenvectors are unique up to a
constant<span style="text-indent:0px;letter-spacing:normal;font-variant:normal;text-align:start;font-style:normal;display:inline!important;font-weight:normal;float:none;line-height:normal;text-transform:none;font-size:13px;white-space:normal;font-family:Verdana,Geneva,Helvetica,Arial,sans-serif;word-spacing:0px"></span>
so if you took any eigenvector and multiplied it by -1 it's still
an eigenvector. You could see it in the definition,<br>
<br>
M x=\lambda x<br>
<br>
eigenvector x appears in both sides of the eqn.<br>
<br>
I had a similar problem with tensor glyphs in ParaView. In that
case I was able to solve by looking at the sign of the determinant
of the transformation matrix (see bug report, patch and mail list
posts below). I wonder if you could adapt/build on this solution
there to solve your issue here?<br>
<br>
<a href="http://vtk.org/Bug/view.php?id=12179" target="_blank">http://vtk.org/Bug/view.php?id=12179</a><br>
<a href="http://vtk.1045678.n5.nabble.com/tensor-glyph-inward-pointing-surface-normals-td4388361.html" target="_blank">http://vtk.1045678.n5.nabble.com/tensor-glyph-inward-pointing-surface-normals-td4388361.html</a><br>
<br>
Burlen<br>
<br>
<br>
On 08/23/2013 01:10 PM, Andy Bauer wrote:<br>
</div>
<blockquote type="cite">
<div dir="ltr">
<div>
<div>
<div>Hi Paul,<br>
<br>
</div>
Apologies as my math is a bit rusty but isn't the sign of
the eigenvector related to the sign of its corresponding
eigenvalue? In that case if you make sure that all of the
eigenvalues are positive then all of their corresponding
eigenvectors should be aligned properly. If that's the case
and you have access to the eigenvalues of the eigenvectors
you could use the calculator or python calculator to
properly orient the eigenvectors.<br>
<br>
</div>
In any case, if you can come up with an algorithm that
properly orients the eigenvectors you should be able to do
that in the python calculator or calculator filters. If not,
then things could get a bit hairy as far as computationally
figuring out which is the "proper" direction your eigenvalues
should have.<br>
<br>
</div>
Regards,<br>
Andy<br>
</div>
<div class="gmail_extra"><br>
<br>
<div class="gmail_quote">On Fri, Aug 23, 2013 at 3:38 PM,
pwhiteho <span dir="ltr"><<a href="mailto:pwhiteho@masonlive.gmu.edu" target="_blank">pwhiteho@masonlive.gmu.edu</a>></span>
wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div>
<div style="direction:ltr;font-size:10pt;font-family:Tahoma">The
term "eigenvector", used to describe the principal
directions of a tensor, is a bit of a misnomer since
it's not a "vector" as interpreted by the Stream Tracer
filter - it's more accurately bi-directional like
tension/compression and could be termed
"eigenaxis/eigenaxes". When interpreted as a vector,
there is an inherent sign ambiguity in each eigenvector
- the sign is indeterminate and one is free to choose +
or -, and that is exactly what Mathematica does ( likely
true for other routines also ).<br>
<br>
I've been using Mathematica to prototype computations
for the investigation of tensor topology which I then
visualize in ParaView. Eigen-decomposition of a tensor
field at each grid point returns an orthonormal set of
eigenvectors, uncorrelated with any neighbors. Taken
separately, each eigenvector field exhibits large
regions of smoothly varying orientation, but there are
systematic and random reversals of orientation that
confound the Stream Tracer filter, sending streamlines
wandering around the field. What is needed is a true
tangent curve ( tensor line ) integrator that would
avoid "doubling back" as the "streamline" propagates,
similar to the scheme of Weinstein, et. al., ( IEEE
VIS'99 ) which computes the dot product of the incoming
propagation vector with the eigenvector; and if near -1,
negate the outgoing propagation vector. This can also be
fancied-up to accommodate noisy initial tensor data as
in Weinstein.<br>
<br>
I think I would be taking on too much at this point in
learning to write my own filter so have been exploring
ways to pre-process the eigenvector fields before
visualizing in Paraview, but I ask:<br>
1. Have I missed something in existing filters that
would handle this?<br>
2. Can the existing Stream Tracer be modified?<br>
3. Does the eigenvector routine in ParaView yield the
same sign ambiguity among uncorrelated computations?<br>
<br>
Thanks,<br>
Paul W<br>
<br>
</div>
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