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 ).
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. 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: 1. Have I missed something in existing filters that would handle this? 2. Can the existing Stream Tracer be modified? 3. Does the eigenvector routine in ParaView yield the same sign ambiguity among uncorrelated computations? Thanks, Paul W
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