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• LiDARTend to be messier, CG methods not

as appropriate.

Commercial ApplicationsReverse engineering,metrology

Customization

Delcam scanner and software

Academic Applications

Levoy et al, Stanford

Allen, Curless, Popovic, U Wash.

Amenta/Delson, UC/CUNY

Mesh GenerationFill in object with well-shaped triangles or tetrahedra (or other elements). Goal: minimum angles bounded away from zero.

aCute, Alper Ungor

Application

heat, strain

fluid flow

Simulate physical properties on or around complex objects.

Mike Hohmeyer

Christof Garth, UCD

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-pivotingBernardini 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.

Figure out which are inside and which are outside…

Ogniewicz, 92

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

Interior polar balls

Subset of Voronoi vertices, the poles, approximate medial axis.

Amenta & Bern, 98 “Crust” papers

Sampling Requirement

-sample: distance from any surface point to nearest sample is at most small constant 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 SoftwareExamine 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

Distance from nearest sample.

Giesen and John, 01,02

Distance function flow

Consdier uphill flow …. Idea: interior is part that flows to interior maxima.

Distance function

Max and (some) saddle points.

Compute flow combinatorially using Delaunay/Voronoi

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….

GeomagicFounded by Herbert Edelsbrunner.

Leading system on the market.

Other Companies

• Dessault – Catia – Andrei Liutier as “resident genius”, includes Nearest-neighbor 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

Dey & Zhao, 02

Amenta, Choi, Kolluri, 01

Attali & Montanvert, 97Amenta & 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

Bern, Eppstein, Gilbert ‘90 – first guaranteed quality mesh generator!

Shewchuk notes

Delaunay refinement

All triangle angles > k (here 25o). Forces grading from small to larger. Equivalent to upper

bound on circumcircle/shortest edge.

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

Adapt planar techniques to surfaces.

Chew

Restricted Delaunay Triangulation

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

3D Voronoi diagram restricted to 2D surface.

Delaunay is dual.

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!

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

Place strings of barely-intersecting balls along edges; mesh faces by Delaunay refinement.

Dey&Levine

Comment

• Local feature size is overkill for just surface meshing.

Volume meshing

Shewchuk; alg generalizes Bajaj, Dey and Sugihara.

Shewchuk notes

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.