T. J. Peters, University of Connecticut K. Abe, J. Bisceglio, A. C. Russell Computational Topology on Approximated Manifolds (with Applications)

Outline: Topology & Approximation Theory Algorithms Applications

Outline: Animation & Approximation Animation for 3D Approximation of 1-manifolds Transition to molecules Molecular dynamics and knots Extensions to 2-manifolds Supportive theorems Spline intersection approximation (static)

Role for Animation Towards ROTATING IMMORTALITY – –Möbius Band in the form of a Trefoil Knot Animation makes 3D more obvious Simple surface here Spline surfaces joined along boundaries Mathematical Discovery

Problem in Approximation Input: Set of unorganized sample points Approximation of underlying manifold Want –Error bounds –Topological fidelity

Typical Point Cloud Data

Subproblem in Sampling Sampling density is important For error bounds and topology

Recent Overviews on Point Clouds Notices AMS,11/04, Discretizing Manifolds via Minimum Energy Points, ‘bagels with red seeds’ –Energy as a global criterion for shape (minimum separation of points, see examples later) –Leading to efficient numerical algorithms SIAM News: Point Clouds in Imaging, 9/04, report of symposium at Salt Lake City summarizing recent work of 4 primary speakers of ….

Recent Overviews on Point Clouds F. Menoti (UMn), compare with Gromov- Hausdorff metric, probabalistic D. Ringach (UCLA), neuroscience applications G. Carlsson (Stanford), algebraic topology for analysis in high dimensions for tractable algorithms D. Niyogi (UChi), pattern recognition

Seminal Paper Surface reconstruction from unorganized points, H. Hoppe, T. DeRose, et al., 26 (2), Siggraph, `92 Modified least squares method. Initial claim of topological correctness.

Modified Claim The output of our reconstruction method produced the correct topology in all the examples. We are trying to develop formal guarantees on the correctness of the reconstruction, given constraints on the sample and the original surface

Sampling Via Medial Axis Delauney Triangulation Use of Medial Axis to control sampling for every point x on F the distance from x to the nearest sampling point is at most 0.08 times the distance from x to MA(F) Approximation is homeomorphic to original. (Amenta & Bern)

Medial Axis Defined by H. Blum Biological Classification, skeleton of object Grassfire method

KnotPlot!!

Unknot

Bad Approximation Why? Curvature? Separation?

Why Bad? No Intersections! Changes Knot Type Now has 4 Crossings

Good Approximation All Vertices on Curve Respects Embedding Via Curvature (local) Separation (global)

Summary – Key Ideas Curves –Don’t be deceived by images (3D !) –Crossings versus self-intersections Local and global arguments Knot equivalence via isotopy

Initial Assumptions on a 2-manifold, M Without boundary 2 nd derivatives are continuous (curvature) Improved to ambient isotopy (Amenta, Peters, Russell)

T

Proof: Similar to flow on normal field. Comment: Points need not be on surface. (noise!) Theorem: Any approximation of F in T such that each normal hits one point of W is ambient isotopic to F.

Tubular Neighborhoods Its radius defined by ½ minimum –all radii of curvature on 2-manifold –global separation distance. Estimates, but more stable than medial axis. and Ambient Isotopy

Medial Axis H. Blum, biology, classification by skeleton Closure of the set of points that have at least 2 nearest neighbors on M

X

Large Data Set ! Partitioned Stanford Bunny

Acknowledgements, NSF I-TANGO: Intersections --- Topology, Accuracy and Numerics for Geometric Objects (in Computer Aided Design), May 1, 2002, #DMS I-TANGO: Intersections --- Topology, Accuracy and Numerics for Geometric Objects (in Computer Aided Design), May 1, 2002, #DMS SGER: Computational Topology for Surface Reconstruction, NSF, October 1, 2002, #CCR SGER: Computational Topology for Surface Reconstruction, NSF, October 1, 2002, #CCR Computational Topology for Surface Approximation, September 15, 2004,Computational Topology for Surface Approximation, September 15, 2004, #FMM