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Designing Quadrangulations with Discrete Harmonic Forms
Speaker: Zhang Bo2007.3.8
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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)
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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
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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
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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
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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
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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
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One Example
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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
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Discrete Laplace Operator
u = harmonic 0-form
0)()(
iNj
jiij uu
0)()(
)cot(cot
iNj
jiijij uu
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Necessity of discontinuities
Harmonic function on closed genus-
0 mesh? Only constants! Globally continuous harmonic scalar potentials are too restrictive for quad meshing
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Adding singularities
Poles, line singularity
du dv contouring
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With more poles…
Crate saddles
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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
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Line Singularity -> T-junctions
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Singularity graph
reverse
regular
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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
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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!!
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Handling Singularities
Vertex with regular continuity
NN
--NN
++
as simple as jump in potential:as simple as jump in potential:
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Handling Singularities
Vertex with reverse continuity dvdvdudu ,
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Handling Singularities
Vertex with switch continuity dudvdvdu ,
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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
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Assisted Singularity Graph Generation
Two orthogonal principal curvature directions emin & emax everywhere, except at the so-called umbilics
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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!
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Handle Boundaries
As a special line in the singularity graph Force the boundary values to be linearly int
erpolating the two corner values
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Mesh Extraction
A contouring of the u and v potentials
will stitch automatically into a pure quad
mesh
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Mesh alignments controlMesh alignments control
provide (soft) control over provide (soft) control over the the final mesh alignmentsfinal mesh alignments
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Mesh size controlMesh size control
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Results
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Singularity graph
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Harmonic Functions u,v
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du, dv
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Final Remesh
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B-Spline Fitting
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More result
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Summary
Extended Laplace operator along singularity lines Only three types:
regular, reverse, switch Provide control over
singularity: type locations
sizing
REGULARREGULAR REVERSEREVERSE SWITCHSWITCH
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Summary
Sparse and symmetric linear system, average 7 non-zero elements per line, can be compute fast!
Not a fully automatic mesher
Singularity graph
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Thank you!