Add updated SPHARM-PDM functionality to SlicerSALT build

[bpaniagua]

https://github.com/Kitware/SlicerSALT/blob/master/CMakeLists.txt#L399

@allemangD could you please link here the SPHARM-PDM fork that includes your WIP on the python implementation of SPHARM-PDM?

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Plan is to include the rework version as an "experimental" alternative for SPHARM-PDM.

[allemangD]

https://github.com/slicersalt/py-spharm-pdm/

Two attempts at optimization with branches torch and torch-optim - neither work very well for one reason or another.

Currently the two core challenges:

1. scipy.optimize uses dense matrices for storing Jacobian and Hessian information, so its constrained optimizers use a prohibitive amount of memory.
   • The branch torch attempts to use pytorch backward autograd to compute these instead. Runtime improves but memory usage still prohibitive.
   • The branch torch-optim attempts to use torch.optim library, however it doesn't support constrained optimizers. I attempted a constraint penalty term, but it doesn't perform well: it optimizes for the constraint or the metric, not both. There are too many degrees of freedom and the constraints are too sparse.
2. There is a bug somewhere in core.initial_parameterization that causes self-intersecting initial parameterization; since the initial conditions are wrong the optimizer fails to produce any meaningful result.

The problem with the python optimizers is we need sparse and constrained optimization. I'm curious to investigate the package pyOptSparse; it claims to support this. It doesn't provide an autograd interface but I can attempt with the same technique as in the torch branch. I'm not super confident since it's not a terribly popular library but I think it's worth looking at before diving into a custom implementation (that's the current situation with EqualAreaParametricMeshNewtonIterator!!)

The first issue to resolve is the discrepency in initial parameterization. My current Python implementation is based on the paper, not the code. I've isolated the relevant sections of EqualAreaParametricMeshNewtonIterator and am comparing them now. I will follow up here with details, or schedule a breakout with you and Jared if the bug is too subtle.

Note for posterity: the relevant methods in EqualAreaParametricMeshNewtonIterator are:

• start_values invokes general workflow.
• generate_matrix populates laplacian matrix for latitude problem.
• set_lati_rhs populates rhs vector for latitude problem.
• Solve Ax = b; extract latitudes from x.
• modify_matrix updates the matrix for the longitude problem.
• set_longi_rhs populates rhs vector for longitude problem.
• Solve Ax = b; extract longitudes from x.
• Compute cartesian coordinates.

[allemangD]

I believe I located and resolved the bug: https://github.com/slicersalt/py-spharm-pdm/commit/e5a47f5a007fe81d4d313a386da8169d2273eb0d Now working on more detailed tests and automated validation to be sure, but I wanted to report this progress.

While investigating this I realize that the legacy TSurfaceNet and new vtkSurfaceNets3D output have different vertex ordering; then the spherical parameterization and resulting harmonics will differ. Therefore I do not think it will be possible to maintain correspondence between models generated with this rework and the legacy implementation.

It may be possible to resolve it in certain cases with ellipsoid alignment but I'm not confident. Assuming not, I plan to use model to model distance as the metric to validate.