Mathematics and Implementation of Computer Graphics Techniques 2015

Boundary Aligned Smooth 3D Cross-Frame Field

Jin Huang, Yiying Tong, Hongyu Wei, Hujun BaoSIGGRAPH Asia 2011

Kenshi Takayama

Assistant Prof @ National Insitute of Informatics

takayama@nii.ac.jp

1st round

25 July 2015

Background: 2D Frame Field & Quad Meshing

• 2D Frame Field Auto-computed

• UV Parameterization Auto-computed

Mixed-Integer Quadrangulation [Bommes,Zimmer,Kobbelt,SIGGRAPH09]

2D Frame Field UV Parameterization= Quad Mesh

2

Background: 3D Frame Field & Hex Meshing

CubeCover - Parameterization of 3D Volumes [Nieser,Reitebuch,Polthier,SGP11]

“Meta-Mesh” to define 3D Frame Field

UVW Parameterization= Hex Mesh

• 3D Frame Field Heuristic

• UVW Parameterization Auto-computed

3

Definition of 3D Frame

• Don’t care about orientation / ordering of axes

• Question: How distant is from ?

• Key insight:

• Invariant under sign flip / axis reordering!

・・・𝑅1

𝑅2𝑅3

𝑅4

4

Distance between 3D Frames

• Integral over an entire sphere Spherical Harmonics!

𝑑 (𝑅𝑎 ,𝑅𝑏)≔ ∫𝑠∈ 𝑆2

❑

❑

h (𝑅𝑎⊤𝑠 ) h (𝑅𝑏

⊤𝑠 )

𝑑𝑠

2

h (𝑠)≔𝑠𝑥2 𝑠𝑦

2 +𝑠𝑦2 𝑠𝑧

2+𝑠𝑧2 𝑠𝑥

2

5

Basics of Spherical Harmonics

• Something like Fourier series on sphere

• Orthonormality:

6

𝑓 (𝜃 ,𝜙 )=∑𝑙=0

∞

∑𝑚=−𝑙

𝑙

𝑓 𝑙𝑚𝑌 𝑙

𝑚 (𝜃 ,𝜙 )

∫𝑠 ∈𝑆2

❑

𝑌 𝑙𝑚1 (𝑠)𝑌 𝑙

𝑚2 (𝑠 )𝑑𝑠={1 if𝑚1=𝑚2

0 otherwise

𝑚

𝑙

3210-1-2-3

3

2

1

0

Frame represented by SH

• “Frequency” unaffectedby rotation:

• Frame represented as SH coeffs for band (i.e. 9-vector) :

• Coeffs mapped by some 9x9 matrix :

• Distance between & :

h (𝑠 )≔√7𝑌 40 (𝑠 )+√5𝑌 4

4 (𝑠 )h (𝑠)=− 2√𝜋15 (𝑌 4

0 (𝑠 )+√ 57 𝑌 44 (𝑠 )+16√𝜋𝑌 0

0 (𝑠 ))h (𝑅⊤𝑠 )=: 𝑓 [𝑅 ] (𝑠)= ∑

𝑚=− 4

4

𝜆𝑚𝑌 4𝑚 (𝑠)

�� [𝑅 ]=�� h

𝑑 (𝑅𝑎 ,𝑅𝑏)=‖��𝑎 h− ��𝑏 h‖2

simplify

𝑌 40𝑌 4

−1𝑌 4−2𝑌 4

−3𝑌 4−4 𝑌 4

1 𝑌 42 𝑌 4

3 𝑌 44

7

Computing 9x9 from 3D rotation

• Not immediately obvious

• Insight: Obvious for certain cases: & • (Not sure...)

Represent rotation by ZYZ Euler angle

Rotation about Z axis by θ

8

Frame that aligns with boundary surface

• Frame aligns with:• Z axis iff

• Surface normal iff

• : Rotation that brings n to Z axis

Coeff for

9

(Proof in Appendix)

Discretization & Objective

• Tetrahedral mesh over domain

• Frame var at center of every (interior/exterior) triangle

• Piecewise-linear frame field • Gradient constant within each tetrahedron

• Objective to be minimized:

𝐸 smooth≔∑TET 𝑗

volume (TET 𝑗 ) ∑𝑚=− 4

4

‖𝛁 �� TET 𝑗(𝑚)‖2

𝐸align≔ ∑TR I 𝑖∈𝜕Ω

area (TR I 𝑖 )‖( ��𝑛𝑖→Z𝑓 𝑝𝑖

) (0 )−√7‖2𝐸 full≔

𝐸 smooth

volume (Ω )1/3+𝑤align

𝐸align

area (𝜕Ω )

10

Optimization

• Energy quadratic in Simple Laplace-like least squares <Step 1>

• Problem: Arbitrary doesn’t represent rotation!

<Step 2> Project to its closest rotation • (Not sure how to do it...)

• <Step 3> Using as initial guess, run nonlinear optimization over • L-BFGS (solver: ALGLIB, dlib, etc)• (Not sure about analytic form of derivative...)

11

Results & Performance

12

Questions

• Regarding implementation:• Expressions for & • Projection of to its closest rotation • Analytic derivative of w.r.t.

• Possible idea for improvement:• Can we sidestep nonlinear optimization

by alternating Laplace smoothing and “normalization”?

13