simplify: Rework point simplification to use reservoirs #618
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This makes it easier to debug the underlying algorithm and metrics.
Before this, point simplifier encoded non-sensical functions into the quadric which resulted in fairly suboptimal candidate point selection, raising the effective error (when computed separately as a distance function) by ~2x. To make mistakes like this less likely in the future we now annotate each quadric structure with the formula it encodes.
While quadrics are very important for triangle mesh simplification, they are decidedly overkill for point cloud simplification. We encode a simpler function into them compared to triangles, and even that function is too complicated for what we need - minimizing the sum of square distances to all points in the cell ends up yielding approximately the same result as averaging all points in the cell and minimizing the distance to that average. The benefit of a simpler formulation is that there's less memory used for encoding the function and less computation required to aggregate the points / minimize the error. This also sets us up for including color error into the calculation (by computing a weighted average of the colors and minimizing the 6D distance to XYZ+RGB). Even once we include RGB into the reservoirs, we'll be at 7 floats per cell, vs 11 floats per cell that we spend right now for weighted quadrics.
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While quadrics are very important for triangle mesh simplification, they
are decidedly overkill for point cloud simplification. We encode a
simpler function into them compared to triangles, and even that function
is too complicated for what we need - minimizing the sum of square
distances to all points in the cell ends up yielding approximately the
same result as averaging all points in the cell and minimizing the
distance to that average.
The benefit of a simpler formulation is that there's less memory used
for encoding the function and less computation required to aggregate the
points / minimize the error.
This also sets us up for including color error into the calculation (by
computing a weighted average of the colors and minimizing the 6D
distance to XYZ+RGB). Even once we include RGB into the reservoirs,
we'll be at 7 floats per cell, vs 11 floats per cell that we spend right
now for weighted quadrics.
The RGB support is planned for a future PR; for now this improves the
error (due to a bug fix in quadric math, which is approximately equivalent
to reservoir based code) and makes space for extra data and calculation
for color support.