Spectral Surface Quadrangulation

Shen Dong, Peer-Timo Bremer, Michael Garland, Valerio Pascucci and John Hart

Reporter: Hong guang Zhou

Math Dept. ZJU

October 26

Quadrangulating Surfaces

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

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]

Related Work – Quad Remeshing

Parameterization of Triangle Meshes over Quadrilateral Domains

[Boier-Martin et al. 04]

Periodic Global Parameterization [Ray et al. 05]

Our Approach

Start with a triangulated 2-manifold

Our Approach

Start with a triangulated 2-manifold

Construct a “good” scalar function

Our Approach

Start with a triangulated 2-manifold

Construct a “good” scalar function

Quadrangulate the surface using its Morse-Smale complex

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

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

Key Features of SSQ

Few extraordinary points

Pure quad, fully conforming mesh

Topological robustness

High element quality

Computing the Morse-Smale Complex

Given any scalar function

Contrust Morse –Smale function over a manifol

d

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

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

Computing the Morse-Smale Complex

Given any scalar function Identify all critical points

maximum minimum saddle

Other points :regular

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

Shape Dependence

Properties of the Morse-Smale Complex

Guaranteed fully conforming, purely quadrangular decomposition for Any surface topology Any function

Noise Removal

• Cancel pairs of connected critical

points

persistence

Noise Removal

Quasi-Dual Complexes

Morse-Smale complexMorse-Smale complex Quasi-dual complexQuasi-dual complex

Quasi-Dual Complexes

In each cell, calculate the easiest path that connect the minimum to the maximum.

Quasi-Dual Complexes

Primal Quasi-dual

Doubles the number of available base domains Capture different symmetry patterns of the surface

Bunny Harmonics

Complex Improvement Patches may be poorly shaped Paths can merge

Globally Smooth Parameterization

Build 2n2n linear system

0ij j ij

w u u

[0,0]

[1,1]

Globally Smooth Parameterization

0ij j ij

w u u

Build 2n2n linear system

Globally Smooth Parameterization

0ij j ij

w u u

Bake transition function into system

Parameterization

[Tong et al. 06] use more general formulation

Iterative Relaxation

1. For any vertex i, find a patch such that [0,1][0,1]

iu

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

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

Complex Refinement

Mesh Generation Lay down kk grid in each patch

Extraordinary points can only exist at complex extrema

Fully conforming

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

Results – Torus

Primal

Quasi-dual

8th 16th 32nd

Results – Dancer

Input MS-complex

Optimized complex Remesh

Results – Heptoroid

Input Quadrangulation

Output

|EV|=175

Results – Bunny

SSQ|EV| = 26

[Ray et al.]|EV| = 314

[Boier-Martin et al.]|EV| = 175

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

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

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

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

Thank you

Questions ?