Computational Topology on
Approximated Manifolds
(with Applications)
T. J. Peters, University of
Connecticut
www.cse.uconn.edu/~tpeters
K. Abe, J. Bisceglio, A. C. Russell
Outline: Topology
& Approximation
• Theory
• Algorithms
• Applications
Role for Animation
Towards
Mathematical
Discovery
• ROTATING IMMORTALITY
– www.bangor.ac.uk/cpm/sculmath/movimm.htm
– Möbius Band in the form of a Trefoil Knot
• Animation makes 3D more obvious
• Simple surface here
• Spline surfaces joined along boundaries
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 ….
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
Separation?
Curvature?
Why?
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
• 2nd derivatives are continuous
(curvature)
• Improved to ambient isotopy
(Amenta, Peters, Russell)
T
Theorem: Any approximation of F in T
such that each normal hits one point of
W is ambient isotopic to F.
Proof: Similar to flow on normal field.
Comment: Points need not be on surface. (noise!)
Tubular Neighborhoods
and Ambient Isotopy
• Its radius defined by ½ minimum
– all radii of curvature on 2-manifold
– global separation distance.
• Estimates, but more stable than medial axis.
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-0138098.
• SGER: Computational Topology for
Surface Reconstruction, NSF, October 1,
2002, #CCR - 0226504.
• Computational Topology for Surface
Approximation, September 15, 2004,
#FMM -0429477.