http://vrl.cs.brown.edu/wiki/index.php?action=history&feed=atom&title=User%3AJadrian_Miles%2FTriangle_toy_problemUser:Jadrian Miles/Triangle toy problem - Revision history2026-04-20T16:58:53ZRevision history for this page on the wikiMediaWiki 1.43.1http://vrl.cs.brown.edu/wiki/index.php?title=User:Jadrian_Miles/Triangle_toy_problem&diff=4811&oldid=prevJadrian Miles: New page: Problem statement: {{infobox|The ground truth is a set of closed, non-overlapping triangles in the plane whose union is homeomorphic to a disc, with an arbitrary orientation associated wit...2011-04-12T19:16:20Z<p>New page: Problem statement: {{infobox|The ground truth is a set of closed, non-overlapping triangles in the plane whose union is homeomorphic to a disc, with an arbitrary orientation associated wit...</p>
<p><b>New page</b></p><div>Problem statement:<br />
{{infobox|The ground truth is a set of closed, non-overlapping triangles in the plane whose union is homeomorphic to a disc, with an arbitrary orientation associated with the interior of each triangle. The observation is a set of noisy "DWIs" generated from the ground truth. The problem is to reconstruct the ground truth from the observation.}}<br />
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The proposed solution follows these steps:<br />
# Forward-modeling initialization<br />
## Fit tensors to DWIs<br />
## Cluster tensors<br />
## Fit triangles to the clusters<br />
# Multi-scale iterative fitting<br />
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There is a place for probabilistic thinking in 1.2 and 2. Things to consider for both:<br />
<br />
* What are the parameters of the model? What is the space of these parameters? Is it reasonable to talk about probabilities at every point in this space?<br />
* What is the prior on a given model configuration?<br />
* What is the likelihood of the data given a model configuration?<br />
* Does chi-square-style reasoning fit in anywhere?<br />
* What's the relationship between model feasibility (no gaps between triangles) and the objective function (which comes from probabilities)?<br />
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=== Potential Extensions ===<br />
* 3-D instead of 2-D<br />
* Adjoin adjacent triangles into groups, with interior orientation determined by the aggregate shape, and work with 2-D streamlines<br />
* Non-triangular shapes (quadrilaterals?)<br />
* Crossing regions, either by overlapping shapes or having one triangle belong to multiple groups<br />
* Realistic image artifacts: blurring, distortion<br />
* Detailed microstructure rather than just an orientation</div>Jadrian Miles