Atlas Refinement with Bounded Packing Efficiency Hao-Yu Liu , - - PowerPoint PPT Presentation

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Atlas Refinement with Bounded Packing Efficiency Hao-Yu Liu , - - PowerPoint PPT Presentation

Jan 22, 2023 161 likes β€’614 views

Atlas Refinement with Bounded Packing Efficiency Hao-Yu Liu , Xiao-Ming Fu, Chunyang Ye, Shuangming Chai, Ligang Liu University of Science and Technology of China Atlas Normal Color Texture Packing Efficiency (PE) PE=86.1% PE=45.6% High

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SLIDE 1

Atlas Refinement with Bounded Packing Efficiency

Hao-Yu Liu, Xiao-Ming Fu, Chunyang Ye, Shuangming Chai, Ligang Liu

University of Science and Technology of China

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SLIDE 2

Atlas

Color Normal

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SLIDE 3

Texture

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SLIDE 4

Packing Efficiency (PE)

PE=86.1% PE=86.1% High pixel usage rate PE=45.6% Low pixel usage rate

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SLIDE 5

Packing Efficiency (PE)

Maximizing atlas packing efficiency is NP-hard!

[Garey and Johnson 1979; Milenkovic 1999]

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SLIDE 6

Other Requirements

  • Low distortion

High Distortion Low Distortion

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SLIDE 7

Other Requirements

  • Low distortion
  • [Golla et al. 2018; Liu et al. 2018; Shtengel et al. 2017; Zhu et al. 2018]
  • Consistent orientation
  • [Floater 2003; Tutte 1963; Claici et al. 2017; Hormann and Greiner 2000;

Rabinovich et al. 2017; SchΓΌller et al. 2013]

  • Bijection
  • [Jiang et al. 2017; Smith and Schaefer 2015]
  • Low boundary length
  • [Li et al. 2018; Poranne et al. 2017; Sorkine et al. 2002]

These methods do not consider PE!

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SLIDE 8

Atlas Refinement

Bijective High PE Input

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SLIDE 9

Previous Work

Box Cutter [Limper et al. 2018]

  • Cut and repack

No guarantee for a high PE result!

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SLIDE 10

Motivation

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SLIDE 11

Packing Problems

Irregular shapes Hard to achieve high PE Rectangles Simple to achieve high PE Widely used in practice

?

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SLIDE 12

Axis-Aligned Structure

Axis-aligned structure Rectangle decomposition High PE (87.6%)!

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SLIDE 13

General Cases

Not axis-aligned Axis-aligned Higher distortion

Axis-aligned deformation

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SLIDE 14

Distortion Reduction

Axis-aligned High distortion Bijective & High PE High distortion Bijective & High PE Low distortion Bounded PE Scaffold-based method [Jiang et al. 2017] Distortion reduction

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SLIDE 15

Axis-aligned deformation Distortion reduction Rectangle decomposition and packing

Pipeline

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SLIDE 16

Axis-Aligned Deformation

  • Input

Single chart Not bijective 10 charts Bijective

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SLIDE 17

Axis-Aligned Deformation

  • Targets of boundary edges
  • Smoothing
  • Labeling
  • Deformation
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SLIDE 18

Axis-Aligned Deformation

Direction vector Ambiguous rotating directions Fail!

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SLIDE 19

Axis-Aligned Deformation

Polar angle Clear rotating direction Success!

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SLIDE 20

Polar Angle

πœ„ = atan2(𝑧, 𝑦) + 2π‘™πœŒ π‘’πœ„ = 𝑦𝑒𝑧 βˆ’ 𝑧𝑒𝑦 𝑦2 + 𝑧2

(x,y)

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SLIDE 21

Polar Angle

πœ„π‘—+1 = πœ„π‘— + 𝜌 βˆ’ 𝛽𝑗 Gauss–Bonnet formula ෍

𝑗

𝜌 βˆ’ 𝛽𝑗 = 2𝜌

Discrete boundary curvature

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SLIDE 22

Target Calculation

  • Boundary smoothing
  • Gaussian smooth

𝐻𝜏 s𝑗

𝑙 = ෍

bπ‘˜ π‘šπ‘˜ exp βˆ’ dist 𝑐𝑗, 𝑐

π‘˜ 2

2𝜏2 𝑑

π‘˜ 𝑙

ΖΈ 𝑑𝑗

𝑙 = 𝐻𝜏

ΰ΅— 𝑑𝑗

𝑙

β€–π»πœ 𝑑𝑗

𝑙 β€–

  • Accept ΖΈ

𝑑𝑗

𝑙 if ΖΈ

𝑑𝑗

𝑙 β‹… 𝑑𝑗 𝑙 β‰₯ 0

  • Update interior angles

ො 𝛽𝑗

𝑙+1 = ො

𝛽𝑗

𝑙 + ∠ 𝑑𝑗 𝑙, 𝑑𝑗 𝑙+1 βˆ’ ∠ 𝑑𝑗+1 𝑙 , 𝑑𝑗+1 𝑙+1

  • Global rotation
  • Polar angle axis-alignment
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SLIDE 23

Axis-Aligned Deformation

Target polar angle π›ͺ𝑗 Corners

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SLIDE 24
  • Energy of boundary alignment

𝐹edge πœπ‘— = 1 2 (1 βˆ’ 𝛿) πœ„π‘— βˆ’ 𝜌 2 π›ͺ𝑗

2

+ 1 2 𝛿 π‘šπ‘— π‘šπ‘—

0 βˆ’ 1 2

𝐹align(𝐝) = ෎

𝑗=1 𝑂𝑐

π‘šπ‘— π‘š0 𝐹edge πœπ‘—

Axis-Aligned Deformation

Rotate polar angle Keep length

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SLIDE 25

Axis-Aligned Deformation

  • Energy of isometric distortion(symmetric Dirichlet)

Keep low distortion and orientation consistency. 𝐹d(c) = 1 4 ෎ fi∈Fc Area f𝑗 Area Mc ‖𝐾𝑗‖𝐺

2 + ‖𝐾𝑗 βˆ’1‖𝐺 2

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SLIDE 26

Axis-Aligned Deformation

0.2X Playback

min

c

𝐹d(c) + πœ‡πΉalign(c) s.t. det 𝐾𝑗 > 0, βˆ€π‘—

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SLIDE 27

Rectangle Decomposition and Packing

The faces are all rectangles. But the number is too many.

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SLIDE 28

Rectangle Decomposition and Packing

  • Motorcycle graph algorithm

PE Score 87.0% 0.688 83.6% 0.659 84.4% 0.658

Score = PE βˆ’ πœ•BL1/BL0

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SLIDE 29

min

C

𝐹reduction = 𝐹d(C) + 𝐹PE(C)

Distortion Reduction

Scaffold-based method [Jiang et al. 2017] Isometric energy Barrier function

  • f PE bound

s.t. 𝛸 is bijective

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SLIDE 30

Distortion reduction

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SLIDE 31

Experiments

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SLIDE 32

PE Bound

Input Ed=1.039 PE=80% Ed=1.037 PE=85% Ed=1.041 PE=90% Ed=1.049

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SLIDE 33

Collection of Models

PE=80% Ed=1.026 Input Ed=1.022

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SLIDE 34

Comparison to Box Cutter [Limper et al. 2018]

Box Cutter PE=81.1% Ed=1.149 179.8s Ours PE=88.9% Ed=1.087 1.69s Input #F=4,656

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SLIDE 35

Comparison to Box Cutter [Limper et al. 2018]

Input #F=100,000 Box Cutter PE=75.8% Ed=1.114 247.8s Ours PE=91.3% Ed=1.066 43.84s

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SLIDE 36

Benchmark (5,588)

PE=86.2% Ed=1.020 PE=86.7% Ed=1.024

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SLIDE 37

Benchmark (5,588)

PE=90.5% Ed=1.011 PE=91.0% Ed=1.001

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SLIDE 38

Texture

PE=92.6% Ed=1.018 PE=80.4% Ed=1.119

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SLIDE 39

Single-source Geodesics [Prada et al. 2018]

PE=89.1% Ed=1.041

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SLIDE 40

Conclusion

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SLIDE 41

Conclusions

  • Our method provides a novel technique to refine input atlases with

bounded packing efficiency.

  • Key idea: converting polygon packing problems to a rectangle packing

problems

  • High and bounded packing efficiency
  • Good performance and quality
  • Practical robustness
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SLIDE 42

Limitation & Future Work

  • Modification of the input atlas may not meet the original intention.
  • Boundary length elongation is not explicitly bounded.
  • There is no theoretical guarantee, especially for the axis-aligned

deformation process.

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SLIDE 43

Thank you!