Spectral Surface Quadrangulation
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Transcript of Spectral Surface Quadrangulation
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Spectral Surface Quadrangulation
Shen Dong, Peer-Timo Bremer, Michael Garland, Valerio Pascucci and John Hart
Reporter: Hong guang Zhou
Math Dept. ZJU
October 26
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Quadrangulating Surfaces
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Why Quad Meshes?
Applications PDEs for fluid, cloth, … Catmull-Clark subdivision NURBS patches in CAD/CAM
Demands Few extraordinary points High quality elements
Stam 2004
DAZ Productions
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Related Work – Semi-Regular Triangle Remeshing
Multiresolution Adaptive Parameterization of Surfaces [Lee et al. 98]
Multiresolution Analysis of Arbitrary Meshes [Eck et al. 95]
Globally Smooth Parameterization [Khodakovsky et al. 03]
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Related Work – Quad Remeshing
Parameterization of Triangle Meshes over Quadrilateral Domains
[Boier-Martin et al. 04]
Periodic Global Parameterization [Ray et al. 05]
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Our Approach
Start with a triangulated 2-manifold
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Our Approach
Start with a triangulated 2-manifold
Construct a “good” scalar function
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Our Approach
Start with a triangulated 2-manifold
Construct a “good” scalar function
Quadrangulate the surface using its Morse-Smale complex
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Our Approach Start with a triangulated 2-
manifold
Construct a “good” scalar function
Quadrangulate the surface using its Morse-Smale complex
Optimize the complex geometry
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Our Approach Start with a triangulated 2-
manifold
Construct a “good” scalar function
Quadrangulate the surface using its Morse-Smale complex
Optimize the complex geometry
Generate semi-regular quad mesh
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Key Features of SSQ
Few extraordinary points
Pure quad, fully conforming mesh
Topological robustness
High element quality
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Computing the Morse-Smale Complex
Given any scalar function
Contrust Morse –Smale function over a manifol
d
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Discrete Laplacian Eigenfunctions
Discretization Smooth surface polygon mesh of n vertices Scalar field real vector of size n
Laplace operator Vertex i : fi = wij ( fj – fi )
Whole mesh : f = L · f
Eigenfunction of : F = F eigenvector of L : L · f =
Morse –Smale function F: v → f
i
j
wij = (cot+cot) / 2
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Our Choice – Laplacian Eigenfunctions
Equivalence of Fourier basis functions in Euclidean space
Capture progressively higher surface undulation modes
1 2 3 4
5 6 7 8
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Computing the Morse-Smale Complex
Given any scalar function Identify all critical points
maximum minimum saddle
Other points :regular
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Computing the Morse-Smale Complex
Each saddle has four lines of steepest
ascent /descent
Trace ascending lines from saddle to maxima
Trace descending lines from saddle to minima
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Shape Dependence
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Properties of the Morse-Smale Complex
Guaranteed fully conforming, purely quadrangular decomposition for Any surface topology Any function
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Noise Removal
• Cancel pairs of connected critical
points
persistence
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Noise Removal
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Quasi-Dual Complexes
Morse-Smale complexMorse-Smale complex Quasi-dual complexQuasi-dual complex
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Quasi-Dual Complexes
In each cell, calculate the easiest path that connect the minimum to the maximum.
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Quasi-Dual Complexes
Primal Quasi-dual
Doubles the number of available base domains Capture different symmetry patterns of the surface
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Bunny Harmonics
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Complex Improvement Patches may be poorly shaped Paths can merge
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Globally Smooth Parameterization
Build 2n2n linear system
0ij j ij
w u u
[0,0]
[1,1]
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Globally Smooth Parameterization
0ij j ij
w u u
Build 2n2n linear system
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Globally Smooth Parameterization
0ij j ij
w u u
Bake transition function into system
Parameterization
[Tong et al. 06] use more general formulation
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Iterative Relaxation
1. For any vertex i, find a patch such that [0,1][0,1]
iu
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Iterative Relaxation
1. For any vertex i, find a patch such that [0,1][0,1]
2. Conform patch boundaries to the in-range charts
iu
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Iterative Relaxation
1. For any vertex i, find a patch such that [0,1][0,1]
2. Conform patch boundaries to the in-range charts
3. Relocate nodes to adjacent paths branching points
4. Resolve parameterization and repeat relaxation
iu
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Complex Refinement
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Mesh Generation Lay down kk grid in each patch
Extraordinary points can only exist at complex extrema
Fully conforming
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Picking Eigenfunctions
Two phases
1. Pick range of spectrum by target number of critical points
2. Pick best eigenfunction within range with lowest parametric distortion
Spectrum
0 k
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Results – Torus
Primal
Quasi-dual
8th 16th 32nd
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Results – Dancer
Input MS-complex
Optimized complex Remesh
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Results – Heptoroid
Input Quadrangulation
Output
|EV|=175
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Results – Bunny
SSQ|EV| = 26
[Ray et al.]|EV| = 314
[Boier-Martin et al.]|EV| = 175
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Results – Bunny
SSQ
[Ray et al.]
[Boier-Martin et al.]
=6.87
=9.63
=12.71
Angle Edge Length
=7e-4
=7.4e-4
=9.3e-4
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Performance
Model |V| |Ev|Time (s)
Eigen
Complex
Relax
Torus 1,600 0 1.36 3.28 0.33
Moai 10,092
12 6.97 4.88 8.67
Kitten 10,000
15 1.76 5.05 28.08
Dancer 24,998
33 3.24 9.68441.4
3
Bunny 72,023
26 10.79 25.551259.
15
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Conclusion Surface quadrangulation
using Morse-Smale complex of Laplacian eigenfunction
Key features Few extraordinary points Pure quad, fully
conforming mesh Topologically robust High element quality
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Future WorkFuture Work
Deeper understanding of the Laplacian spectrum
Full feature and boundary support More efficient complex optimization Select the good eigenfunction whose gradient field most closely follows any such user- specified orientation
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Thank you
Questions ?