Practical Foldover-Free Volumetric

Mapping Construction

Jian-Ping SuSJPing@mail.ustc.edu.cn

University of Science and Technology of China

Joint work with Xiao-Ming Fu and Ligang Liu

Introduction

3

Test

Piecewise linear

volumetric mapping

Volumetric Mapping

Source tetrahedral mesh Target tetrahedral mesh

One interior

cross section

4

Foldover-Free: the determinant of the Jacobian matrix is positive

Foldover-Free Volumetric Mapping

𝑣3

𝑣0

𝑣2

𝑣1

ො𝑣1Flipped

Negative signed volume

𝑣0

𝑣2

𝑣3

𝑣1

ො𝑣1

Orientation-preserving

Positive signed volume

5Applications

Shape deformation

Volumetric PolyCube

6Previous Work

TestThese methods are usually slow or rely on extra inputs !

Representation-based methods

—[Pailléet al. 2015; Fu et al. 2016]

Bounded distortion mapping methods

—[Aigerman et al. 2013; Kovalsky et al. 2014; Kovalsky et al. 2015]

The penalty-based methods

—[Liu et al. 2016]

7

Extra input

Motivation

95.58 seconds

Average conformal

distortion: 4.47

[Fu et al. 2016]

Time-consuming

39.25 seconds

Average conformal

distortion: 9.76

[Kovalsky et al. 2015]

5.97 seconds

Average conformal

distortion: 4.98

Our methods

8Contributions

Present a practical method for computing foldover-

free volumetric mappings:

Practically robust

Practically efficient

Don’t require any other extra inputs

Easily extend to meshless mappings

Our Method

10Problem

Source tetrahedral meshFoldover-free

volumetric map

Input Output

Initial

volumetric map

11Preliminaries

Signed singular value decomposition

𝐽𝑖 𝐮 = 𝑈𝑖𝑆𝑖𝑉𝑖𝑇, 𝑆𝑖 = 𝑑𝑖𝑎𝑔(𝜎𝑖,1, 𝜎𝑖,2, 𝜎𝑖,3)

𝜎𝑖,1 ≥ 𝜎𝑖,2 ≥ 𝜎𝑖,3 .

Foldover-free constraints

det 𝐽𝑖 𝐮 > 0, 𝑖 = 1,⋯ ,𝑁 ⟺ 𝜎𝑖,3 > 0

Conformal distortion

𝜏 𝐽𝑖 𝐮 = 𝜎𝑖,1/𝜎𝑖,3

Bounded conformal distortion constraints

1 ≤ 𝜏 𝐽𝑖 𝐮 ≤ 𝐾

12Constraints

Foldover-free

constraints

det 𝐽𝑖 𝐮 > 0

Bounded conformal

distortion constraints

1 ≤ 𝜏 𝐽𝑖 𝐮

𝜏 𝐽𝑖 𝐮 ≤ 𝐾

𝜎𝑖,3 > 0, 𝜏 𝐽𝑖 = Τ𝜎𝑖,1 𝜎𝑖,3

𝜎𝑖,1 ≥ 𝜎𝑖,2 ≥ 𝜎𝑖,3

𝐾 = max𝑖=1,⋯,𝑁

𝜏 𝐽𝑖

𝜏 𝐽𝑖 ≥ 1, 𝜎𝑖,3 > 0, 𝜎𝑖,3 > 0 ?It is difficult to satisfy the constraints!

13Our idea

1 ≤ 𝜏 𝐽𝑖 𝐮 ≤ 𝐾

Alternatively solving 𝐾 and 𝐮

Update K: generate a conformal distortion bound;

Update 𝐮 : project the mapping into the bounded

distortion space;

If there are foldovers, go to Step 1;

Input:initial mapping

Update bound 𝐾

Update vertices 𝐮

Output:Foldover-free

mapping

14

Monotone projection

ℋ𝑖 = 𝐻𝑖|1 ≤ 𝜏(𝐻𝑖) ≤ 𝐾 : bounded conformal distortion space.

Update vertices 𝐮

min𝐮

𝐸𝑑 =

𝑖=1,⋯,𝑁

𝐽𝑖 𝐮 − 𝐻𝑖 𝐹2 ,

𝑠. 𝑡. 𝐻𝑖 ∈ ℋ𝑖 , 𝑖 = 1,⋯ ,𝑁,

𝐴𝐮 = 𝑏.

Local-global solver

15

Local-global solver

Local step

Fix 𝐮 and 𝐽𝑖, solve 𝐻𝑖

min𝐮

𝐸𝑑 =

𝑖=1,⋯,𝑁

𝐽𝑖 𝐮 − 𝐻𝑖 𝐹2 ,

𝑠. 𝑡. 𝐻𝑖 ∈ ℋ𝑖 , 𝑖 = 1,⋯ ,𝑁,

Global step

Fix 𝐻𝑖, solve 𝐮

min𝐮

𝐸𝑑 =

𝑖=1,⋯,𝑁

𝐽𝑖 𝐮 − 𝐻𝑖 𝐹2 ,

𝑠. 𝑡. 𝐴𝐮 = 𝑏

Very slow convergence…

Update vertices 𝐮

16

Anderson acceleration method [Peng et al. 2018]

Update vertices 𝐮

17Why update bound 𝐾?

Projection cannot

eliminate all foldovers

18

Bound generation

𝐾𝑛𝑒𝑤 = 𝛽𝐾

Update bound 𝐾

𝛽 = 2

initialize 𝐾 = 4

19

Apply a maintenance-based method

Post-optimization

Average / maximum

conformal distortion:

2.72 / 107.10

Average / maximum

conformal distortion:

2.08 / 22.61

Before post-optimization After post-optimization

Source tetrahedral mesh

20Recap of our algorithm

Experiments

22Efficiency

(443K, 62.10s) (1441K, 108.16s)

23Different initializations

24Meshless mappings

25Data set: 719 examples

26Comparisons

Large-scale bounded distortion

map [Kovalsky et al, 2015]𝜃 = Τ𝑡𝑙 𝑡𝑜

LBDM

Simplex assembly method

[Fu et al, 2016]𝜂 = Τ𝑡𝑠 𝑡𝑜

SA

𝜃𝑎𝑣𝑔 = 39.25

𝜃𝑠𝑡𝑑 = 31.41𝜃𝑚𝑎𝑥 = 158.13𝜃𝑚𝑖𝑛 = 2.71

𝜂𝑎𝑣𝑔 = 9.19

𝜂𝑠𝑡𝑑 = 9.54𝜂𝑚𝑎𝑥 = 120.78𝜂𝑚𝑖𝑛 = 1.02

27Comparisons

28Bound generation on [Kovalsky et al, 2015]

292D Mappings

Conclusion

31Conclusion

A practically robust and efficient method for computing

foldover-free volumetric mappings.

Key idea: monotonically project the mapping to the bounded

conformal distortion mapping space

No extra inputs

Easily extend to meshless mappings

32Limitation & future work

Limitation

No theoretical guarantee of success for any model.

Future work

Connectivity modifications

More applications

all-hex mesh optimization

33All-hex mesh optimization

Code and data are available at:

http://staff.ustc.edu.cn/~fuxm/

Code and data are available at:

http://staff.ustc.edu.cn/~fuxm/

36Bound generation on [Kovalsky et al, 2015]

37Recap of our algorithm

38Comparisons

Source tetrahedral mesh Initial mapping

38.99 seconds

Maximal conformal

distortion: 17.32

[Fu et al. 2016]

2.92 seconds

Maximal conformal

distortion: 17.29

Our methods

39Comparisons

Large-scale bounded distortion

map [Kovalsky et al, 2015]𝜃 = Τ𝑡𝑙 𝑡𝑜

LBDM

Simplex assembly method

[Fu et al, 2016]𝜂 = Τ𝑡𝑠 𝑡𝑜

SA

𝜃𝑎𝑣𝑔 = 39.25

𝜃𝑠𝑡𝑑 = 31.41𝜃𝑚𝑎𝑥 = 158.13𝜃𝑚𝑖𝑛 = 2.71

𝜂𝑎𝑣𝑔 = 9.19

𝜂𝑠𝑡𝑑 = 9.54𝜂𝑚𝑎𝑥 = 120.78𝜂𝑚𝑖𝑛 = 1.02