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Computational Peeling Art Design
Hao Liu, Xiao-Teng Zhang, Xiao-Ming Fu, Zhi-Chao Dong, Ligang Liu
University of Science and Technology of China
Computational Peeling Art Design Hao Liu, Xiao-Teng Zhang, Xiao-Ming - - PowerPoint PPT Presentation
Computational Peeling Art Design Hao Liu, Xiao-Teng Zhang, Xiao-Ming - - PowerPoint PPT Presentation
Computational Peeling Art Design Hao Liu, Xiao-Teng Zhang, Xiao-Ming Fu, Zhi-Chao Dong, Ligang Liu University of Science and Technology of China Peeling art design Popular art form Peeling art examples Yoshihiro Okadas method Peeling art
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Popular art form
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Peeling art examples
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Yoshihiro Okada’s method
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Peeling art design problem
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Challenges of the computational method
- Non-trivial to optimize the similarity
- Unsuitable input shape
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Existing work: cut generation
- Minimum spanning tree method [Chai et al. 2018; Sheffer 2002;
Sheffer and Hart 2002]
- Mesh segmentation approaches [Julius et al. 2005; Lévy et al.
2002; Sander et al. 2002, 2003; Zhang et al. 2005; Zhou et al. 2004]
- Simultaneous optimization [Li et al. 2018; Poranne et al.2017]
- Variational method [Sharp and Crane 2018]
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Existing work: cut generation
- Minimum spanning tree method [Chai et al. 2018; Sheffer 2002;
Sheffer and Hart 2002]
- Mesh segmentation approaches [Julius et al. 2005; Lévy et al.
2002; Sander et al. 2002, 2003; Zhang et al. 2005; Zhou et al. 2004]
- Simultaneous optimization [Li et al. 2018; Poranne et al.2017]
- Variational method [Sharp and Crane 2018]
unfolded shapes ≠ input shapes
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Our approach
Cut generation
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Key idea
Cut generation Difficult Mapping computation Easy
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𝑆 𝑇𝑛 = Φ(𝑇) Input 𝑇
min 𝐹𝑗𝑡𝑝 𝑇𝑛, 𝑇 + 𝑥𝐹𝑡ℎ𝑠(𝑆)
Mapping computation
Φ
Two goals:
- 1. Low isometric distortion
- 2. Area of 𝑆 approaches zero
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- ARAP distortion metric [Liu et al. 2008]
Isometric energy
𝐹𝑗𝑡𝑝 𝑇𝑛, 𝑇 =
𝑗=1 𝑂𝑔
𝐵𝑠𝑓𝑏 𝑔
𝑗 ||𝐾𝑗 − 𝑆𝑗||𝐺 2
𝑆𝑗 is an orthogonal matrix
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- Our novel rank-one energy
𝐹𝑡ℎ𝑠 𝑆 = 𝑗=1
𝑂𝑆𝑔 𝐵𝑠𝑓𝑏 𝑢𝑗 ||𝐾𝑗 − 𝐶𝑗||𝐺 2
𝐶𝑗 is a rank one matrix
Shrink energy
||𝐾𝑗||𝐺
2
det 𝐾𝑗 rank-one Input
- Other choices
- Frobenius energy
||𝐾𝑗||𝐺
2
- Determinant energy
det 𝐾𝑗
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Local-global solver
min 𝐹𝑗𝑡𝑝 𝑇𝑛, 𝑇 + 𝑥𝐹𝑡ℎ𝑠(𝑆)
𝑡𝑢. 𝜖𝑆 = 𝜖𝑇𝑛 and 𝑤𝑛, 𝑤𝑆 ∈ 𝑁
𝐹𝑗𝑡𝑝 𝑇𝑛, 𝑇 =
𝑗=1 𝑂𝑔
𝐵𝑠𝑓𝑏 𝑔
𝑗 ||𝐾𝑗 − 𝑆𝑗||𝐺 2
𝑆𝑗 = 𝑉𝑗𝑊
𝑗 𝑈
𝐶𝑗 = 𝑉𝑗𝑒𝑗𝑏 𝜏𝑗, 0 𝑊
𝑗 𝑈
𝐹𝑡ℎ𝑠 𝑆 =
𝑗=1 𝑂𝑆𝑔
𝐵𝑠𝑓𝑏 𝑢𝑗 ||𝐾𝑗 − 𝐶𝑗||𝐺
2
𝑤𝑏𝑠𝑗𝑏𝑐𝑚𝑓𝑡 𝜀𝑤𝑙 𝑗𝑜 tangent space 𝑤𝑙
𝑜𝑓𝑥 = 𝑄(𝑤𝑙 + 𝜀𝑤𝑙)
Local step： Global step：
𝜀𝑤𝑙 𝑗𝑜 𝐺
𝑘 𝑗𝑡 𝜀𝑤𝑙 𝑘 = 𝐺 𝑙 𝑘 𝑈𝐺 𝑙 𝑤𝜀𝑤𝑙
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Some details
stalk locations Representations of M
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Suitable input
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Unsuitable input
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Almost cover Mapping Process Interaction Process
Iterative interaction
Cut Generation Final resulting cut
Iterative design
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Interaction place
Prune and Decompose Unfold 𝑇𝑛 and 𝑆
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Interaction 1: shape augmentation
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Interaction 2: part deletion
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Interaction 3: angle augmentation
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Interaction 4: curvature reduction
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Interaction 5: pre-processing
Unprocessed high distortion Processed low distortion Input Input with specify area
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Interaction 5: pre-processing
Input with specify area Unprocessed: high distortion Processed: low distortion Align to initialize
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Cut generation
Resulting cut Mapped shape Simplify boundary
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Real peeling
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Real design
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Real peeling
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Experiments
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Shapes designed by Yoshihiro Okada
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Comparison to Yoshihiro Okada
Okada’s Ours Dove Eagle Shrimp
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Our results
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More results
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Conclusion
- A computational tool for peeling art design and construction.
- Unsuitable input 2D shapes are rectified by an iterative process.
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Limitations: conservation principle
User input
…
Interaction many times also cannot keep posture
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