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 Okada’s method

Peeling art design problem

Challenges of the computational method • Non-trivial to optimize the similarity • Unsuitable input shape

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]

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

Our approach Cut generation

Key idea Cut generation Difficult Mapping computation Easy

Mapping computation 𝑆 Φ Input 𝑇 𝑇 𝑛 = Φ(𝑇) Two goals: min 𝐹 𝑗𝑡𝑝 𝑇 𝑛 , 𝑇 + 𝑥𝐹 𝑡ℎ𝑠 (𝑆) 1. Low isometric distortion 2. Area of 𝑆 approaches zero

Isometric energy • ARAP distortion metric [Liu et al. 2008] 𝑂 𝑔 2 𝐹 𝑗𝑡𝑝 𝑇 𝑛 , 𝑇 = 𝐵𝑠𝑓𝑏 𝑔 𝑗 ||𝐾 𝑗 − 𝑆 𝑗 || 𝐺 𝑗=1 𝑆 𝑗 is an orthogonal matrix

Shrink energy • Our novel rank-one energy 𝑂 𝑆𝑔 𝐵𝑠𝑓𝑏 𝑢 𝑗 ||𝐾 𝑗 − 𝐶 𝑗 || 𝐺 2 𝐹 𝑡ℎ𝑠 𝑆 = 𝑗=1 𝐶 𝑗 is a rank one matrix • Other choices 2 • Frobenius energy ||𝐾 𝑗 || 𝐺 • Determinant energy det 𝐾 𝑗 2 det 𝐾 𝑗 Input ||𝐾 𝑗 || 𝐺 rank-one

min 𝐹 𝑗𝑡𝑝 𝑇 𝑛 , 𝑇 + 𝑥𝐹 𝑡ℎ𝑠 (𝑆) Local-global solver 𝑡𝑢. 𝜖𝑆 = 𝜖𝑇 𝑛 and 𝑤 𝑛 , 𝑤 𝑆 ∈ 𝑁 Local step ： 𝑂 𝑔 𝐹 𝑗𝑡𝑝 𝑇 𝑛 , 𝑇 = 2 𝑈 𝐵𝑠𝑓𝑏 𝑔 𝑗 ||𝐾 𝑗 − 𝑆 𝑗 || 𝐺 𝑆 𝑗 = 𝑉 𝑗 𝑊 𝑗 𝑗=1 𝑂 𝑆𝑔 𝑈 2 𝐶 𝑗 = 𝑉 𝑗 𝑒𝑗𝑏𝑕 𝜏 𝑗 , 0 𝑊 𝐹 𝑡ℎ𝑠 𝑆 = 𝐵𝑠𝑓𝑏 𝑢 𝑗 ||𝐾 𝑗 − 𝐶 𝑗 || 𝐺 𝑗 𝑗=1 Global step ： 𝑜𝑓𝑥 = 𝑄(𝑤 𝑙 + 𝜀𝑤 𝑙 ) 𝑤𝑏𝑠𝑗𝑏𝑐𝑚𝑓𝑡 𝜀𝑤 𝑙 𝑗𝑜 tangent space 𝑤 𝑙 𝑘 𝑈 𝐺 𝑘 = 𝐺 𝑤 𝜀𝑤 𝑙 𝜀𝑤 𝑙 𝑗𝑜 𝐺 𝑘 𝑗𝑡 𝜀𝑤 𝑙 𝑙 𝑙

Some details Representations of M stalk locations

Suitable input

Unsuitable input

Iterative interaction Mapping Process Almost cover Final resulting cut Iterative design Cut Generation Interaction Process

Interaction place Prune and Unfold 𝑇 𝑛 and 𝑆 Decompose

Interaction 1: shape augmentation

Interaction 2: part deletion

Interaction 3: angle augmentation

Interaction 4: curvature reduction

Interaction 5: pre-processing Input Input Unprocessed Processed with specify area high distortion low distortion

Interaction 5: pre-processing Input with specify area Unprocessed: high distortion Align to initialize Processed: low distortion

Cut generation Mapped shape Resulting cut Simplify boundary

Real peeling

Real design

Real peeling

Experiments

Shapes designed by Yoshihiro Okada

Comparison to Yoshihiro Okada Okada’s Ours Dove Eagle Shrimp

Our results

More results

Conclusion • A computational tool for peeling art design and construction. • Unsuitable input 2D shapes are rectified by an iterative process.

Limitations: conservation principle … User input Interaction many times also cannot keep posture

Thank you