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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 !!