Designing Quadrangulations with Discrete Harmonic Forms

Speaker: Zhang Bo2007.3.8

References

Designing Quadrangulations with Discrete Harmonic Forms

Y.Tong P.Alliez D.Cohen-Steiner M.DesbrunCaltech INRIA Sophia-Antipolis, France

Eurographics Symposium on Geometry Processing (2006)

About the Author: Yiying Tong 2005-present: Post doctoral Scholar in Computer Science Depar

tment, Calteth. 2000-2004: Ph.D. in Computer Science at the Unversity

of Southern California (USC). Thesis title: “Towards Applied Geometry in Graphics” Advisor: Professor Mathieu Desbrun. 1997-2000: M.S. in Computer Science at Zhejiang University Thesis Title: “Topics on Image-based Rendering” 1993-1997: B. Engineering in Computer Science at Zhejiang Univ

ersity

Siggraph Significant New Researcher Award 2003

Eurographics YoungResearcher Award 2005

INRIA: 法国国家信息与自动化研究所

CGAL developer

Methods for Quadrangulations Among many:

clustering/Morse [Boier-Martin et al 03, Dong et al. 06] global conformal param [Gu/Yau 03] curvature lines [Alliez et al. 03, Marinov/Kobbelt 05] isocontours [Dong et al. 04]

two potentials (much) more robust than streamlines

periodic global param (PGP) [Ray et al. 06] PGP : nonlinear + no real control

This paper: one linear system only This paper: discrete forms & tweaked Laplacian

About Discrete Forms

Discrete k-form A real number to every oriented k-simplex

0-forms are discrete versions of continuous scalar fields

1-forms are discrete versions of vector fields

About Exterior Derivative

)()(: 1 MMd kk

Associates to each k-form ω a particular (k+1)-form dω

If ω is a 0-form (valued at each node), i.e., a function on the vertices, then dω evaluated on any oriented edge v1v2 is equal to ω(v1) -ω(v2)

Potential: 0-form u is said to be the potential of w if w = du

Hodge star: maps a k-form to a complimentary (n-k)-form

On 1-forms, it is the discrete analog of applying a rotation of PI/2 to a

vector field

)()(: MM knk

About Harmonic Form

)()(: 1 MM kk dkn 1)1()1(

)(),(,, 1 MMd kk

Codifferential operator:

Laplacian:

满足 的微分形式称为调和形式 , 特别 的函数 称为调和函数

)()(: MMdd kk

0 0f

)(0 Mf

One Example

Why Harmonic Forms ? Suppose a small surface patch composed of locally “nice” quadrangles Can set a local coordinate system (u, v) du and dv are harmonic, so u and v are also harmonic. bec

ause the exterior derivative of a scalar field is harmonic iff this field is harmonic

This property explain the popularity of harmonic functions in Euclidean space

Discrete Laplace Operator

u = harmonic 0-form

0)()(

iNj

jiij uu

0)()(

)cot(cot

iNj

jiijij uu

Necessity of discontinuities

Harmonic function on closed genus-

0 mesh? Only constants! Globally continuous harmonic scalar potentials are too restrictive for quad meshing

Adding singularities

Poles, line singularity

du dv contouring

With more poles…

Crate saddles

Why ?

Poincaré–Hopf index theorem!

ind(v)=(2-sc(v))/2 ind(f)=(2-sc(f))/2sc() is the number of sign changes as traverses in order

Discrete 1-forms on meshes and applications to 3D mesh parameterization

StevenJ.Gortler ,Craig Gotsman ,Dylan Thurston, CAGD 23 (2006) 83–112

Line Singularity -> T-junctions

Singularity graph

reverse

regular

Singularity lines between “patches”

Special continuity of 1-forms du and dv

i.e., special continuity of the gradient fields

only three different cases

in order to guarantee quads

Vertex with no singularities ?

Discrete Laplace Equation:

wwij = cot aij + cot bijij = cot aij + cot bij

Can Can generate generate smooth fieldssmooth fields even on irregular mesheseven on irregular meshes!!

Handling Singularities

Vertex with regular continuity

NN

--NN

++

as simple as jump in potential:as simple as jump in potential:

Handling Singularities

Vertex with reverse continuity dvdvdudu ,

Handling Singularities

Vertex with switch continuity dudvdvdu ,

Building a Singularity Graph

Meta-mesh consists of Meta-vertices, meta-edges, meta-faces

Placing meta-vertices Umbilic points of curvature tensor (for alignment) User-input otherwise

Tagging type of meta-edges can be done automatically or manually

Geodesic curvature along the boundary will define types of singularities

Small linear system to solve for corner’s (Us,Vs) “Gauss elimination”: row echelon matrix

Assisted Singularity Graph Generation

Two orthogonal principal curvature directions emin & emax everywhere, except at the so-called umbilics

Final Solve

Get a global linear system for the 0-forms u and v of the original mesh as discussion above

The system is created by assembling two linear

equations per vertex, but none for the vertices on

corners of meta-faces This system is sparse and symmetric, Can use the

supernodal multifrontal Cholesky factorization option of TAUCS, Efficient!

Handle Boundaries

As a special line in the singularity graph Force the boundary values to be linearly int

erpolating the two corner values

Mesh Extraction

A contouring of the u and v potentials

will stitch automatically into a pure quad

mesh

Mesh alignments controlMesh alignments control

provide (soft) control over provide (soft) control over the the final mesh alignmentsfinal mesh alignments

Mesh size controlMesh size control

Results

Singularity graph

Harmonic Functions u,v

du, dv

Final Remesh

B-Spline Fitting

More result

Summary

Extended Laplace operator along singularity lines Only three types:

regular, reverse, switch Provide control over

singularity: type locations

sizing

REGULARREGULAR REVERSEREVERSE SWITCHSWITCH

Summary

Sparse and symmetric linear system, average 7 non-zero elements per line, can be compute fast!

Not a fully automatic mesher

Singularity graph

Thank you!