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We present
algorithms for constructing a hierarchy of increasingly coarse Morse complexes
that decompose a piecewise linear 2-manifold. While Morse complexes are
defined only in the smooth category, we extend the construction to the
piecewise linear category by ensuring structural integrity and simulating
differentiability. We then simplify Morse complexes by cancelling pairs
of critical points in order of increasing persistence. We see the ability
to construct and simplify Morse complexes as a fundamental tool with a
variety of applications, - the analysis of three-dimensional electron
densities in structure determination, - the enhancement of the visualization
and analysis of two-dimensional densities, such as the electro-static
potential on a molecule surface, just to name two.
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