designing quadrangulations with discrete harmonic forms
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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!