Post on 08-Aug-2020
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
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