Surface Reconstruction and
Mesh Generation
Nina Amenta
University of California at Davis
Singer/Songwriters
Joni Mitchell
Singer/Songwriters and
Funk Bands
Joni Mitchell
Surface Reconstruction
Mesh Generation
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.
Surface Reconstruction
Input: Samples
from object
surface.
Output: Polygonal
model.
Laser Range Scanners
Minolta;
NextEngine
Use triangulation on a
stripe of laser light.
Structured Light
Breuckmann
white-light
scanner. Projects
patterns on
object, correlates
images seen by
several cameras.
Other ways to get points
• Stereo/photogrammetry
• LiDAR
Tend to be messier, CG methods not as
appropriate.
Commercial Applications
Reverse
engineering,
metrology
Customization
Delcam scanner and software
Academic Applications
Levoy et al, Stanford
Amenta/Delson, UC/CUNY
Allen, Curless, Popovic, U Wash.
Mesh Generation
Fill in object with
well-shaped
triangles or
tetrahedra (or
other elements).
aCute, Alper Ungor
Goal: minimum
angles bounded
away from zero.
Application
Simulate physical
properties on or
around complex
objects.
heat, strain
Mike Hohmeyer
Christof Garth, UCD
fluid flow
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.
Attack of the Computational
Geometers
• Define problems
• Voronoi/Delaunay constructions
• Provably correct algorithms,
constants, running times…
• Plenty of structural geometric
theory
Alpha-shapes
Edelsbrunner, Kirkpatrick, Seidel, 83
Union of balls -> restricted weighed
Voronoi diagram -> weighted Delaunay
faces (skeleton)
Alpha-shape reconstruction
Edelsbrunner
& Muecke, 94:
3D surface
reconstruction
Difficulty
Usually no
ideal choice
of radius.
Ball-pivoting
Bernardini et al, IBM
Fixed-radius ball “rolling”
over points selects subset
of alpha-shape.
Voronoi Diagram
Approximates Medial Axis
For dense surface
samples in 2D, all
Voronoi vertices
lie near medial
axis.
Ogniewicz, 92
Figure out which
are inside and
which are
outside…
2D Medial Reconstruction
Pink Voronoi edges
approximate medial
axis.
2D Curve Reconstruction
Blue Delaunay
edges reconstruct
the curve, pink
triangulate
interior/exterior.
Many algorithms,
with proofs.
Sliver tetrahedra
In 3D, some Voronoi vertices are not
near medial axis …
Sliver tetrahedra
…. even when
samples are
arbitrarily
dense.
Interior Voronoi balls
Poles
Subset of Voronoi
vertices, the poles,
approximate medial
axis.
Interior polar balls
Amenta & Bern, 98
“Crust” papers
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.
Sampling Requirement
Intuition: dense
sampling where
curvature is high
or near features.
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.
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, O’Brien, Spectral
Distance function
Giesen and John,
01,02
Distance from
nearest sample.
Distance function flow
Consdier uphill
flow …. Idea:
interior is part
that flows to
interior maxima.
Distance function
Compute flow
combinatorially
using
Delaunay/Voronoi
Max and (some) saddle points.
Distance Functions are
Pretty Stable
• Distance functions of similar
(Hausdorff) sets are similar
– Maxima lie near near-maxima (points
with small generalized gradient)
Gradient Flow Algorithms
• Giesen and
John
• Edelsbrunner
Wrap….
Geomagic
Founded by Herbert Edelsbrunner.
Leading system on the market.
Other Companies
• Dessault – Catia – Andrei Liutier as
“resident genius”, includes Nearestneighbor reconstruction?
• Imageware, RapidForm, ScanTo3D.
• Bottom line - They all know we’re out
here, but we are not integral to their
business.
What’s 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.
• “This CGAL component implements a
state-of-the-art surface
reconstruction method: Poisson
Surface Reconstruction.”
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)?
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).
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
What to work on?
• Fast octree-based algorithms with
proofs -> 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)
Medial axis approximation
Amenta, Choi, Kolluri, 01
Dey & Zhao, 02
Attali & Montanvert, 97
Amenta & Kolluri, 01
Medial Axis Simplification
Miklos, Giesen, Pauly, SIGGRAPH 2010
Look out for…Chambers, Letscher & Ju,
2D-soon-to-be-3D line-skeleton
algorithm.
Mesh generation
….like I know….
Quad/Octree algorithms
Shewchuk notes
Bern, Eppstein, Gilbert ‘90 – first guaranteed
quality mesh generator!
Delaunay refinement
Equivalent to upper
bound on
circumcircle/shortes
t edge.
All triangle
angles > k
(here 25o).
Forces
grading
from small
to larger.
Delaunay refinement
Insert circumcenters of badly-shaped triangles
Handling boundaries
If circumcenter lies across a boundary
edge, divide edge instead.
2D Meshing Software
•
•
•
•
Triangle, Shewchuk.
aCute, Ungor (advancing front).
CGAL.
Very widely used.
Surface meshing
Chew
Adapt planar techniques to surfaces.
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.
Kind of results
• Surface can be covered with wellshaped triangles, and the number of
triangles is O(minimal).
• Requires the input surface boundary
to have no sharp angle; otherwise
algorithm may not terminate!
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!)
Edge Protection
Dey&Levine
Place strings of
barelyintersecting balls
along edges; mesh
faces by Delaunay
refinement.
Comment
• Local feature size is overkill for just
surface meshing.
Volume meshing
Shewchuk notes
Shewchuk; alg generalizes Bajaj, Dey and
Sugihara.
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!
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.
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.
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 day…Pyramid, Shewchuk.
Industry/Government
• Ansys – Sells simulation capability,
not meshes.
• Many CAD systems, eg. SolidWorks.
• Sandia organizes International
Meshing Roundtable.
• This is very incomplete.
What to work on?
…you’re asking me?...
• Stuff I didn’t talk about
– Anisotropic meshing (Canas & Gortler,
this conference)
– Quad/hex meshing
• Digital differential geometry?
• Get out and meet people.
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.
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