Mathematics and Implementation of Computer Graphics Techniques 2015 Boundary Aligned Smooth 3D Cross-Frame Field Jin Huang, Yiying Tong, Hongyu Wei, Hujun.

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Presentation transcript:

Mathematics and Implementation of Computer Graphics Techniques 2015 Boundary Aligned Smooth 3D Cross-Frame Field Jin Huang, Yiying Tong, Hongyu Wei, Hujun Bao SIGGRAPH Asia 2011 Kenshi Takayama Assistant National Insitute of Informatics 1 st 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 FieldUV 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 ・・・ 4

Distance between 3D Frames Integral over an entire sphere  Spherical Harmonics! 2 5

Basics of Spherical Harmonics Something like Fourier series on sphere Orthonormality:

Frame represented by SH simplify 7

Rotation about Z axis by θ 8

Frame that aligns with boundary surface 9 (Proof in Appendix)

Discretization & Objective 10

Optimization 11

Results & Performance 12

Questions 13