Volume Parameterization

Reporter : Lei Zhang

10\24\2007

What is volume

Surface model 2D manifold

Solid\Volume model 3D manifold

Discrete Representation

triangle tetrahedron

quadrangle hexahedron

Surface Parameterization

Parameterization can be viewed as a one-to-one mapping from a suitable domain to the surface.

1997~ 2007Huge amount of papers

Surface Parameterization Application

Re-meshing

Morphing

Texture mapping

Volume Parameterization

Parameterization can be viewed as a one-to-one mapping from a suitable domain to the volume.

Not proposed formally!

Texture Mapping

Tetrahedral\Hexhedral Meshing

(a) (b) (c) (d)

Rendering Acceleration

(a)(d)

(c)(b)

Volume Parameterization Difficulties Math ground

High dimensional geometry 3 manifold is special

Data structure Enormous datas Complex connectivity

Tiny amount of papers ? ? ?

Problems

Surface: conformal mapping harmonic mapping … …

Volume: ? ? … …

Harmonic Volume Mapping

( )f p q

0 1 2, , 0, 0, 0f f f f

Harmonic equation

1M 2Mp q

3R 3R

• Yalin Wang, Xianfeng Gu, and Shing-Tung Yau. Volumetric harmonic map. Communications in Information and Systems, 2004.

• Yalin Wang, Xianfeng Gu, Tony F. Chan, Paul M. Thompson, and Shing-Tung Yau. Volumetric harmonic brain mapping. IEEE International Symposium on Biomedical Imaging, 2004.

Harmonic mapping from 3 manifold to 3D solid sphere

Surface conformal mapping

0 1 2, ,f f f f

min f

30 1 2 1, , :f f f f M R

0 1 2, ,PL PL PL PLf f f f

2 2 2

0 1 2

min PL

PL PL PL

E f

f f f

Steepest descent method

Xin Li, Xiaohu Guo, Hongyu Wang, Ying He, Xianfeng Gu, and Hong Qin. Harmonic volumetric mapping for solid modeling application. SPM, 2007.

Problem Formulation

f

1M 2M

3R 3R

'f

1M 2M

1

'2

( ) ,

( ) ,

f M

f M

p 0 p

p q q

00000000000000

Solving Equation

self-adjoint

10,1,2 0,1,2i if g

' ' '0,1,2 0,1,2

'

'

,

1 1,

4

i if K g d

K

x x x x

x xx x

Green function

Fundamental Solution Method (MFS)G. Fairweather and A. Karageorghis. The method of fundamental solutions for elliptic boundary value problems. Advances in Computational Mathematics, 1998.

MFS

n 11

( , ; ) , ,sN

i nn

f w w k M

Q P P Q P

f

1M 2MP

nQ

Electric field

Intuitive Explanation

f

1M 2MP

e

Algorithm

n1

( , ; ) ,sN

i nn

f w w k

Q P P Q

nQ

1M2M

1 2, ( )i

n

M f M

Aw b w

P P

1int( ) ( )iM f P P

Discussion

Continuous Volume Mapping

• M. S. Floater. Mean Value Coordinates. CAGD, 2003,20, 19-27.• T. Ju, S. Schaefer and J. Warren. Mean Value Coordinates for Closed Triangular Meshes. Siggraph2005.

Mean Value Interpolation

[ , ] [ ]ˆ[ ]

[ , ]

vx

vx

w x v f x dSf v

w x v dS

f̂1M 2M

3R 3R

f

1M 2M

[ ]f x

v

Summary Research Status

Not hot Parameterization

2 manifold 3 manifold Application

Maybe not much desirous ?

Thanks for your attention!