1. We have a working implementation of soft transition functions in libigl, for the moment still using soft alignment constraints. In this formulation, every triangle is considered separately (triangle soup) and (u,v) parameters of adjacent triangles are bound together by per-edge transition functions, introducing two new integer variables for translational DoFs.
  2. For the soft alignment constraints, we have introduced auxiliary integer variables. To get rid of such a variable in the new setup, we can fix it to a pre-computed integer value. The pre-computation will be done per triangle, by solving the local parameterization problem for this triangle only.
  3. The per-triangle computation needs to handle the special cases when one, two or even all three vertices of the triangle are constrained (black pixels). I will see if we can enumerate all of the possible cases in order to find a closed-form solution.
  4. The newly introduced integer variables (translational DoFs) can be used to contain the inter-triangle constraints that we want to impose on corners corresponding to the same vertex in the original mesh.
  5. If we just consider a single triangle, which is unconstrained (all three vertices == white pixels), the parameterization problem is under-determined. For this reason, we should consider modifying the greedy solver so that the black pixels are processed first (seeds); we then proceed by iteratively adding adjacent pixels based on some scalar distance field.
  6. The same scalar distance field can be used for weighting the importance of soft transition functions.
  7. David suggested an experiment with the following weighting scheme for soft transition constraints: high weights for black pixels and their k-nearest neighbours; low (zero) weights for all other vertices. This should provide an interesting comparison with Misha and Justin’s paper.
  8. For distance field computation, Adrien pointed out connection to line integral convolution, as used e.g. in [Kang et al. 2007] “Coherent line drawing”
  9. Adrien has some code to compute the distance transform, there is also the bwdist function in Matlab.

As a side note, we don’t need combing the cross field anymore. It seems like we could use this fact to constrain even the vertices with low confidence from the structure tensor.