Download Morse-Smale Complexes for Piecewise Linear 3

Critical points and flow structure
for density data
Vijay Natarajan
Duke University
Joint work with
H. Edelsbrunner, J. Harer, and V. Pascucci
Problem
Given smooth generic function f
:  R
3
Define Morse-Smale complexes for 3-manifolds and give
an algorithm to construct them for piecewise linear data
Motivation
Density function
f:R  R
3
X-ray crystallography
electron density
Medical imaging (MRI) proton density
Lattice dislocation
atom energy
Electron microscopy
atom density
Critical Points
regular
p
• Have zero gradient
upper link
(continent)
• Characterized by lower link
lower link
(ocean)
minimum
maximum
1-saddle
2-saddle
Ascending/Descending Manifolds
Ascending manifold:
Points with common origin.
Descending manifold: Points with common destination.
Ascending 3-cell
of a minimum
1-saddle
Desc mfld
Asc mfld
Min
0-cell
3-cell
1-saddle
1-cell
2-cell
2-saddle
2-cell
1-cell
Max
3-cell
0-cell
2-saddle
Morse-Smale Complex
Overlay of Asc and
Desc manifolds
Morse-Smale Complex
A cell is a connected component of points
with common origin and destination
Node
Arc
Quadrangle
Crystal
Continuous to Piecewise Linear
• Input: tetrahedral mesh, density at vertices
• Critical points characterized by lower link
• Quasi Morse-Smale complexes
– same combinatorial property
– cells monotonic and non-crossing
1-saddle + 2-saddle
Morse-Smale Complex Construction
Asc arc
0. Sort Vertices
1. Downward sweep
descending 1- and 2-manifolds
(arcs and disks)
2. Upward sweep
ascending 1- and 2-manifolds
(arcs and disks)
Desc arc
High Level Operations
• Starting (at 1-saddles)
• Extending (at all vertices)
• Gluing (at minima)
Desc arc construction
START DISK
i. Construct short cycle passing
thru q around annulus
ii. Add triangles from p to fill
disk
Simultaneous Construction
At p :
1.1. Start 1 desc disks
1.2. Extend desc disks touching p
1.3. Start (0 –1) desc arcs
1.4. Extend desc arcs touching p
p
0 = 2 oceans
1+1 = 2 continents
Substructures
• Descending / Ascending arcs
– filter using density threshold
– display corresponding isosurface
• Descending / Ascending disks
– overlay on isosurfaces
• Crystals
Future work
• Case studies
– dislocation study
– x-ray crystallography
• Hierarchy
– cancel pairs of critical points
– order by persistence
References
•
•
•
H. Edelsbrunner, J. Harer, V. Natarajan and V. Pascucci.
MorseSmale complexes for piecewise linear 3-manifolds.
ACM
Symposium on Computational Geometry, 2003.
P. T. Bremer, H. Edelsbrunner, B. Hamann and V. Pascucci.
A
Multi-resolution Data Structure for Two-dimensional Morse Functions.
IEEE Visualization, 2003.
H. Edelsbrunner, J. Harer and A. Zomorodian.
Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds.
Discrete and Computational Geometry, 2003.
No time for another slide !!