pHCM_VariationalShapeApproximation.txt

First, why is Variational shape approximation (VSA) a good idea?
- we get an exact error function for a given proxy set.  We can exactly represent the error function (as a height field or color field) over each proxy.
- we have a good chance of finding the globally optimal proxy configuration for a given error bound.  If the globally optimal configuration is not found, then a locally optimal configuration is still found.

pHCM for variational shape approximation
	Here we focus on a fast method for computing the optimal proxies for a given input mesh and error bound.  The constrained Delaunay Triangulation (CDT) is not considered (but its implementation ought to be massively parallel.)

Background:
pHCM is a parallel version of HCM, which is a fast method for solving Eikonal equations.  HCM is a domain decomposition method that combines Fast Marching Method (FMM) on the coarse level nad Fast Sweeping Method on the fine level.


first, FMM for VSA is already the algorithm stated in the paper.  Few modifications:
- triangles are ACCEPTED iff popped off Q
- when T is popped, recompute neighbors' error, taking the min error available (and setting the label as the one with the min error)
- add FAR neighbors to the queue and update the position within Q for the CONSIDERED neighbors if their label changed.

A triangle doesn't need to appear multiple times on Q with a different label.  Only the smallest error label matters.

FSM for VSA:
Each "sweep" is an initial proxy.  Each proxy computes the error over the entire triangle set, updating the label only if the error is smaller.
Unlike FSM for Eikonal in 2D and 3D where there are 4 and 8 sweep directions, here there could be thousands --> massively parallelize, probably according to Zhao's original parallelization.  Or maybe this is clearly a bad idea as a standalone method.

HCM:
does domain decomposition alone give any advantage?
	Yes!  Each cell C still has a list of tagged sweep directions.  Neighbor cells N(c) can activate directions of C as in HCM.  When it comes time to process a C, it deactivates all its tagged directions and performs those sweeps.
	
Domain decomposition:
Region growing on input mesh using a Fast-access neighborhood to segment the input into cells	
	Since there is no simple geometric relationship between cells, each triangle must hold onto its cell label.
Alternatively, we can divide the bounding box into boxes.  A triangle belongs to a box if its barycenter is inside the box.  Each cell then holds onto a list of its triangles.

Initialization: start with only the sweeping directions that are defined within the initial cell(s).  These come from the seed triangles.

The first implementation makes more mathematical sense-- a wavefront spreads over a manifold represented by the mesh.  The cells are then 2D regions over the parameterization domain


Other interesting consideration:
Since it was so easy to translate an algorithm developed for an Eikonal equation to variational shape approximation, is the VSA error function actually an Eikonal equation?