Designing Quadrangulations with Discrete Harmonic Forms Speaker: Zhang Bo 2007.3.8

References Designing Quadrangulations with Discrete Harmonic Forms Y.Tong P.Alliez D.Cohen-Steiner M.Desbrun Caltech INRIA Sophia-Antipolis, France Eurographics Symposium on Geometry Processing (2006)

About the Author: Yiying Tong 2005-present: Post doctoral Scholar in Computer Science Department, 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 University Eurographics Young Researcher Award 2005 INRIA: 法国国家信息与自动化研究所 CGAL developer Siggraph Significant New Researcher Award 2003

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 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

About Harmonic Form 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

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. because 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

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))/2 sc() 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: wij = cot aij + cot bij Can generate smooth fields even on irregular meshes!

Handling Singularities Vertex with regular continuity as simple as jump in potential: N - N +

Handling Singularities Vertex with reverse continuity

Handling Singularities Vertex with switch continuity

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!

As a special line in the singularity graph Handle Boundaries As a special line in the singularity graph Force the boundary values to be linearly interpolating the two corner values

Mesh Extraction A contouring of the u and v potentials will stitch automatically into a pure quad mesh

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

Mesh 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 REGULAR REVERSE SWITCH

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!