Vis 2008: Topology and Tensors

Good morning from the Topology and Tensors session at Vis 2008. I overslept and walked in about halfway through Attila Gyullasy’s Morse-Smale computation paper, so this will, again, be seat-of-my-pants. Their group’s series of topology papers at Vis 2003 with Peer-Timo Bremer et al’s 2D Morse complex computation paper, and they have been publishing at least a paper a year at Vis on the subject. These papers are a remarkable mix between theory and application. From redefining the notion of critical points so that it makes sense in a discrete fashion to designing out-of-core algorithms for processing the data, there’s something for everyone. Coming from the big data needs of the national labs, their examples are typically gigantic, which is always great to see. It looks like this year they’re using a different flavor of Morse theory, but I haven’t read the paper yet.

Xavier Tricoche is now talking about their work in diffusion tensor imaging, which I’m far less familiar with. They use Kindlmann’s diffusion tensor eigendecomposition approach to compute the tensor mode. They start showing that creases and valleys of mode are exactly the topological singularity lines that Zheng and Pang presented in 2004. However (as prsented by Schultz in 2007), these are not anatomically relevant for brain imaging. The solution is to take this general ridge-valley framework and use another tensor invariant. They show that ridge and crease lines of fractional anisotropy actually delineate meaningful structures, automatically capturing (for example) the cingular bundles (which look to be the bundles of neurons going along the corpus callosum, but I don’t know any real brain anatomy). The implementation seemed kind of messy, requiring an adaptive subdivision that didn’t seem to guarantee the detection of the crease lines. But I’m sure that can be figured out in a later study – the important idea here is the use of these extremal lines over tensor invariants.

Thomas Schultz is now presenting some work that tries to overcome the basic limitations of DTI imaging. He describes this as the “partial voluming imaging”: the resolution of DTI imaging is much coarser than the structure they’re trying to determine (I heard it described as trying to measure the direction of people walking around in building corridors by taking the average direction field over the entire block where the building is). In general, at each point it is possible to get an orientation distribution function, which traditional DTI imaging approximates by an ellipsoid. The approaches that use the ODF up to now according to Schultz only use the maxima, and that’s not particularly great. Schultz proposes taking the ODFs and decomposing them in a set of rank-1 tensors. His results for tractography seem to show more anatomical structure, particular in places where multiple fibers cross one another in a single cell. His last slide sentence is “ODFs should not simply be reduced to their maxima”.

Since I know next to nothing about surface matching and parametrization, I’ll simply mention that the final paper, “Geodesic Distance-weighted Shape Vector Image Diffusion”, is doing something with conformal mappings and surface matching. Claudio tells me the work is very good, so the lack of comments is only my ignorance about the subject showing.

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