Surface Reconstruction and Mesh Generation

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Surface Reconstruction and Mesh Generation

• Nina Amenta
• University of California at Davis

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Singer/Songwriters

• Joni Mitchell

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Singer/Songwriters and Funk Bands

• Joni Mitchell

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Surface Reconstruction
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Mesh Generation
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Other secondary sources

• Jonathan Shewchuk lecture notes on mesh
  generation.
• Surface reconstruction survey by Cazals and
  Giesen.
• Chapter on meshing surfaces by Boissonnat,
  Cohen-Steiner, Mourrain, Rote, and Vegter.

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Surface Reconstruction
Input Samples from object surface.
Output Polygonal model.
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Laser Range Scanners
Minolta NextEngine Use triangulation on a
stripe of laser light.
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Structured Light
Breuckmann white-light scanner. Projects
patterns on object, correlates images seen by
several cameras.
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Other ways to get points

• Stereo/photogrammetry
• LiDAR
• Tend to be messier, CG methods not as
  appropriate.

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Commercial Applications
Reverse engineering, metrology
Customization
Delcam scanner and software
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Academic Applications
Levoy et al, Stanford
Amenta/Delson, UC/CUNY
Allen, Curless, Popovic, U Wash.
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Mesh Generation
Fill in object with well-shaped triangles or
tetrahedra (or other elements). Goal minimum
angles bounded away from zero.
aCute, Alper Ungor
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Application
Simulate physical properties on or around complex
objects.
heat, strain
Mike Hohmeyer
Christof Garth, UCD
fluid flow
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Finite Element/Volume Methods

• Numerically solve PDE for physical quantity
  over space, on triangle/tet mesh.
• Finite Element Linearly interpolate vertex
  data over elements.
• Finite Volume Edges represent fluxes across
  dual Voronoi faces.

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Attack of the Computational Geometers

• Define problems
• Voronoi/Delaunay constructions
• Provably correct algorithms, constants, running
  times
• Plenty of structural geometric theory

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Alpha-shapes
Edelsbrunner, Kirkpatrick, Seidel, 83 Union of
balls -gt restricted weighed Voronoi diagram -gt
weighted Delaunay faces (skeleton)
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Alpha-shape reconstruction
Edelsbrunner Muecke, 94 3D surface
reconstruction

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Difficulty
Usually no ideal choice of radius.
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Ball-pivoting
Bernardini et al, IBM
Fixed-radius ball rolling over points selects
subset of alpha-shape.
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Voronoi Diagram Approximates Medial Axis
For dense surface samples in 2D, all Voronoi
vertices lie near medial axis. Figure out which
are inside and which are outside
Ogniewicz, 92
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2D Medial Reconstruction
Pink Voronoi edges approximate medial axis.
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2D Curve Reconstruction
Blue Delaunay edges reconstruct the curve, pink
triangulate interior/exterior. Many algorithms,
with proofs.
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Sliver tetrahedra
In 3D, some Voronoi vertices are not near medial
axis
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Sliver tetrahedra
. even when samples are arbitrarily dense.
Interior Voronoi balls
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Poles
Subset of Voronoi vertices, the poles,
approximate medial axis. Amenta Bern, 98
Crust papers
Interior polar balls
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Sampling Requirement
e-sample distance from any surface point to
nearest sample is at most small constant e times
distance to medial axis. Zero at sharp corners
uh-oh.
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Sampling Requirement
Intuition dense sampling where curvature is high
or near features.
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Kinds of Results

• Assuming input sampling is dense enough, then
  output triangulation will be homeomorphic to, and
  close to, the original surface.
• Usually also demonstrate robustness by
  implementation.

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Algorithms and Software

• Examine Delaunay triangles
• Amenta and Bern, Crust
• Amenta, Choi, Dey and Leekha, Cocone
• Dey Goswami, (water)-Tight Cocone
• Dey Giesen, undersampling errors
• Inside/Outside
• Boissonnat, sculpting
• Boissonnat and Cazals, Natural neighbor
• Amenta, Choi and Kolluri, Power crust
• Kolluri, Shewchuk, OBrien, Spectral

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Distance function
Giesen and John, 01,02
Distance from nearest sample.
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Distance function flow
Consdier uphill flow . Idea interior is part
that flows to interior maxima.
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Distance function
Compute flow combinatorially using
Delaunay/Voronoi
Max and (some) saddle points.
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Distance Functions are Pretty Stable

• Distance functions of similar (Hausdorff) sets
  are similar
• Maxima lie near near-maxima (points with small
  generalized gradient)

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Gradient Flow Algorithms

• Giesen and John
• Edelsbrunner Wrap.

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Geomagic
Founded by Herbert Edelsbrunner. Leading system
on the market.
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Other Companies

• Dessault Catia Andrei Liutier as resident
  genius, includes Nearest-neighbor
  reconstruction?
• Imageware, RapidForm, ScanTo3D.
• Bottom line - They all know were out here, but
  we are not integral to their business.

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Whats really used in graphics

• Poisson algorithm - Kazhdan, Bolitho, Hoppe
  06. Define gradient at boundaries, solve PDE on
  octree to fill space, take level-set of implicit
  function.

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• This CGAL component implements a
  state-of-the-art surface reconstruction method
  Poisson Surface Reconstruction.

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Why? Noise

• Noisy data sources are increasingly important.
• Computing DT of whole point cloud is overkill.
• Persistence is really not the answer.
• Averaging in 3D is faster and better.
• Distance-like functions (Chazal talk)?

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Why? Delaunay bottleneck

• 3D Delaunay triangulation O(n2), O(n) in
  practice, but still slow.
• Attali, Boissonnat, Lieutier 03 O(n lg n) DT
  complxity
• Funke Ramos, 02, Funke Milosavljevic 07,
  O(n lg n) thinning and then reconstructing.
• Cheng, Jin, Lau, this conference. More practical
  O(n lg n).

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For comparison

• Delaunay of 1 million 3D points 1 minute.
• GPU octree 18 milliseconds
• GPU k-NN answer 1 million 50-NN queries/second
  (based on Bern, Chan reduction to sorting)
• A., Li, Simons, Parkaravor, Abbasinejad, Owens

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What to work on?

• Fast octree-based algorithms with proofs -gt
  surface meshing algorithms.
• Prove results about what people already do in
  practice.
• Work on other problems related to building
  objects from data!
• Eg, alignment ( matching)

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Medial axis approximation
Amenta, Choi, Kolluri, 01
Dey Zhao, 02
Attali Montanvert, 97 Amenta Kolluri, 01
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Medial Axis Simplification

• Miklos, Giesen, Pauly, SIGGRAPH 2010

Look out forChambers, Letscher Ju,
2D-soon-to-be-3D line-skeleton algorithm.
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Mesh generation

• .like I know.

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Quad/Octree algorithms
Shewchuk notes
Bern, Eppstein, Gilbert 90 first guaranteed
quality mesh generator!
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Delaunay refinement
All triangle angles gt k (here 25o). Forces
grading from small to larger.
Equivalent to upper bound on circumcircle/shortest
edge.
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Delaunay refinement
Insert circumcenters of badly-shaped triangles
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Handling boundaries
If circumcenter lies across a boundary edge,
divide edge instead.
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2D Meshing Software

• Triangle, Shewchuk.
• aCute, Ungor (advancing front).
• CGAL.
• Very widely used.

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Surface meshing
Chew
Adapt planar techniques to surfaces.
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Restricted Delaunay Triangulation
3D Voronoi diagram restricted to 2D surface.
Delaunay is dual.

• Edelsbrunner and Shah, 96, showed closed-ball
  property if every rVor cell is a disk, rVoD is
  homeomorphic to surface.

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Kind of results

• Surface can be covered with well-shaped
  triangles, and the number of triangles is
  O(minimal).
• Requires the input surface boundary to have no
  sharp angle otherwise algorithm may not
  terminate!

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Delaunay refinement

• Smooth
• - Chew
• Boissonnat and Oudot
• Cheng, Dey, Ramos and Ray
• Piecewise-smooth
• Rineau and Yvinec
• Cheng, Dey and Ramos
• Cheng, Dey and Levine (software!)

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Edge Protection
Place strings of barely-intersecting balls along
edges mesh faces by Delaunay refinement.
DeyLevine
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Comment

• Local feature size is overkill for just surface
  meshing.

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Volume meshing
Shewchuk notes
Shewchuk alg generalizes Bajaj, Dey and
Sugihara.
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Sliver tetrahedra
Are NOT eliminated by optimizing
circumradius/shortest edge.
This is OK for finite volume methods (Miller,
Talmor, Teng and Walkington, STOC 95, mesh a
Poisson-disk point set). But not OK for finite
element methods!
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Sliver removal

• Sliver exudation, 00, Cheng, Dey, Edelsbrunner,
  Facello and Teng. Adjust weights of mesh vertices
  to squeeze out slivers. Dihedral guaranteed to
  be bounded away from zero.
• Randomized perturbation, Chew 97 and Li and Teng
  01.

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Isosurface Stuffing

• Octree-based method, Labelle and Shewchuk 07.
• Dihedral angles bounded between 10.7o and 164.8o
• Requires smooth manifold boundary, uniform
  sizing on boundary. NOT DELAUNAY.

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Free Tet Meshing Software

• Several algorithms implemented in CGAL - Stéphane
  Tayeb, Yvinec, L. Rineau, Alliez and Tournois.
• TetGen, Hang Si, Weierstrass Institute for
  Applied Analysis and Stochastics (WIAS)
• Some dayPyramid, Shewchuk.

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Industry/Government

• Ansys Sells simulation capability, not meshes.
• Many CAD systems, eg. SolidWorks.
• Sandia organizes International Meshing
  Roundtable.
• This is very incomplete.

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What to work on?

• youre asking me?...
• Stuff I didnt talk about
• Anisotropic meshing (Canas Gortler, this
  conference)
• Quad/hex meshing
• Digital differential geometry?
• Get out and meet people.

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Conclusions

• Real problems, real science/industry, real
  impact.
• Theoretical structures and results, and software.
• Bridging the gap to practice is an ongoing
  challenge, not necessarily our top priority.