T. J. Peters, University of Connecticut

Computer ScienceMathematics

www.cse.uconn.edu/~tpeterswith

K. Abe, J. Bisceglio, A. C. Russell, T. Sakkalis, D. R.

Ferguson

Computational Topology for Reconstruction ofManifolds With Boundary

(Potential Applications to Prosthetic Design)

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 guaranteeson 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)

Medial Axis

• Defined by H. Blum

• Biological Classification, skeleton of object

• Grassfire method

X

Formal Definition: Medial Axis

The medial axis of F, MA(F), is the closure of the set of all points that have at least two distinct nearest points on S.

Sampling Via Medial Axis

• Nice: Adaptive

• 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)

• Bad– Small change to surface can give large change to MA– Distance from surface to MA can be zero

Need for Positive Separation

• Differentiable surfaces,continuous 2nd derivatives

• Shift from MA to– Curvature (local)– Separation (global)

Topological Equivalence Criterion?

• Alternative from knot theory

• KnotPlot

• Homeomorphism not strong enough

Unknot

BadApproximation

Why?

Curvature?

Separation?

Good Approximation

All Vertices on Curve

Respects Embedding

Via

Curvature (local)

Separation (global)

Boundary or Not

• Surface theory – no boundary

• Curve theory – OK for both boundary & no boundary

Related Work

• D. Manocha (UNC), MA algorithms, exact arithmetic

• T. Dey, (OhSU), reconstruction with MA

• J. Damon (UNC, Math), skeletal alternatives

• K. Abe, J. Bisceglio, D. R. Ferguson, T. J. Peters, A. C. Russell, T. Sakkalis, for no boundary ….

Computational Topology Generalization• D. Blackmore, sweeps, next week

• Different from H. Edelsbrunner emphasis on PL-approximations, some Morse theory.

• A. Zamorodian, Topology for Computing

• Computation Topology Workshop, Summer Topology Conference, July 14, ‘05, Denison.– Digital topology, domain theory– Generalizations, unifications?

Acknowledgements, NSF

• I-TANGO: Intersections --- Topology, Accuracy and Numerics for Geometric Objects (in Computer Aided Design), May 1, 2002, #DMS-0138098.

• SGER: Computational Topology for Surface Reconstruction, NSF, October 1, 2002, #CCR - 0226504.

• Computational Topology for Surface Approximation, September 15, 2004,

#FMM -0429477.