User:Jadrian Miles/Triangle toy problem

Problem statement:

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.

The proposed solution follows these steps:

1. Forward-modeling initialization
   1. Fit tensors to DWIs
   2. Cluster tensors
   3. Fit triangles to the clusters
2. Multi-scale iterative fitting

There is a place for probabilistic thinking in 1.2 and 2. Things to consider for both:

• 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?
• What is the prior on a given model configuration?
• What is the likelihood of the data given a model configuration?
• Does chi-square-style reasoning fit in anywhere?
• What's the relationship between model feasibility (no gaps between triangles) and the objective function (which comes from probabilities)?

Potential Extensions

• 3-D instead of 2-D
• Adjoin adjacent triangles into groups, with interior orientation determined by the aggregate shape, and work with 2-D streamlines
• Non-triangular shapes (quadrilaterals?)
• Crossing regions, either by overlapping shapes or having one triangle belong to multiple groups
• Realistic image artifacts: blurring, distortion
• Detailed microstructure rather than just an orientation