This paper proposes a novel spherical parameterization to process the genus zero models. The method is based on the planar ARAP+ method and further research of the local/global parameterization. It consists of two main steps, local phase and global phase. In the local phase, we employ the planar ARAP+ method to optimize the Spring energy, and achieve the displacement of planar vertex around its 1-ring neighborhood, then map the new vertex to the 3D 1-ring neighborhood on the sphere. In the global phase, we add the spherical constrains to the planar ARAP+ method, and then obtain the final result iteratively according to the Newton method. Numerical results demonstrate that our method is efficient and convergent, and outperforms several popular methods in term of controlling the distortion measures (angle, area, rigidity). Furthermore, it achieves a better visualization performance in texture mapping.
国家自然科学基金(61432003,61572105,11171052,61328206)
我们特别感谢辽宁师范大学的亓万锋 博士和大连理工大学的孟兆良老师在本文研究过程 中提出的许多宝贵意见, 同时还要衷心感谢以色列理工学院的 Emil Saucan 教授和纽约州立大学的顾险峰教授与我们保持密切的科研合作.
Figure 1
(Color online) The five types of spherical parameterization. (a) Convex; (b) ARAP $^{[4]}$; (c) BLD
Figure 2
(Color online) Original mesh and initial parameterization
Figure 3
The processing of the local 1-ring neighborhood. (a) Local flattening of original mesh;(b) local optimization;(c) local flattening of initial mesh;(d) global solution;(e) spherical projection;(f) spherical parameterization
Figure 4
(Color online) The iterative results produced by our method. (a) Original mesh; (b) 1 iteration; (c) 2 iterations; (d) 3 iterations
Figure 5
(Color online) The post-processing result. (a) Original mesh; (b) before processing; (c) after processing
Figure 6
(Color online) The spherical parameterization and texture mapping. (a) Our conformal; (b) our isometric; mbox(c) our authalic
Figure 7
(Color online) The numerical convergence. (a) Our conformal; (b) our isometric; (c) our authalic
Figure 8
(Color online) Texture mapping of the multi-boundary model. (a) Original mesh; (b) planar parameterization; (c) planar texture mapping; (d) spherical parameterization; (e) cube; (f) spherical texture mapping
Figure 9
(Color online) Comparison of spherical parameterization (ARAP: the first and third rows of (a)–(c); ARAP+: the second and fourth rows of (a)–(c)) and texture mapping of ARAP and ARAP+. (a) 1 iteration; (b) 2 iteration; (c) 3 iteration; (d) texture mapping of ARAP; (e) texture mapping of ARAP+
Figure 10
(Color online) Comparison of spherical parameterization and texture mapping. (a) Convex; (b) ARAP; mbox(c) our conformal;d) our isometric; (e) our authalic
Figure 11
(Color online) The processing of high-curvature model. (a) Original mesh;(b) our conformal; (c) our isometric; (d) our authalic
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| $S^*=S_0^*$; // 初始化ŁOOP |
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// 局部优化 |
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$\omega_{i,j}$ $\Leftarrow$ ComputeWeights(RingNodes($p_i$)); |
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RingNodes($p^{\prime}_i$) $\Leftarrow$ ComputeWeights(RingNodes($p_i$)); |
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RingNodes($q^{\prime}_i$) $\Leftarrow$ ComputeWeights(RingNodes($q_i$)); |
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$L_{(i,j,j+1)}$ $\Leftarrow$ FittingMatrix($\triangle p^{\prime}_ip^{\prime}_jp^{\prime}_{j+1}$, $\triangle q^{\prime}_iq^{\prime}_jq^{\prime}_{j+1}$); |
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New($q^{\prime}_i$) $\Leftarrow$ Relocate2D($\triangle p^{\prime}_ip^{\prime}_jp^{\prime}_{j+1}$, $\triangle q^{\prime}_iq^{\prime}_jq^{\prime}_{j+1}$); |
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$\widetilde{\omega}_{i,j}$ $\Leftarrow$ MeanValue(RingNodes(New($q^{\prime}_i$))); |
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$q^{\prime\prime}_i$ $\Leftarrow$ Relocate3D($\widetilde{\omega}_{i,j}$,RingNodes($q_i$)); |
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$B_i$ $\Leftarrow$ MotionVector($q^{\prime\prime}_i$, $\widetilde{q}_i$); |
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$B_i=(0,0,0)$; // 后处理 |
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| $S^*_1$ $\Leftarrow$ ComputePara($\omega$, $B_i$); // 全局求解 |
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$S^*_0=S^*_1$; |
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$S^*=S^*_0$; |
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break; |
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| ENDLOOP |
| Method | $n$ | Child tiny (V:6368, F:12732) | Skull tiny (V:5007, F:10010) | ||||||
| Angle | Area | Rigidity | Time (s) | Angle | Area | Rigidity | Time (s) | ||
| 1 | 2.28 | 6.42 | 2.93 | 10.66 | 2.03 | 5.65 | 1.59 | | |
| ARAP | 2 | 2.52 | 6.23 | 2.85 | 21.69 | 2.11 | 4.89 | 1.58 | |
| 3 | 3.41 | 5.75 | 2.79 | 32.45 | 2.15 | 3.84 | 1.56 | | |
| 1 | | | | | | | | 6.91 | |
| ARAP+ | 2 | | | | | | | | 13.95 |
| 3 | | | | | | | | 21.43 | |
| Method | Man tiny (V:5785, F:11566) | Girl tiny (V:6610, F:13216) | ||||||
| Angle | Area | Rigidity | Time (s) | Angle | Area | Rigidity | Time (s) | |
| Convex | 2.13 | 9.45 | 2.35 | | 2.03 | 9.11 | 2.68 | |
| ARAP | 2.31 | 3.63 | 1.73 | 20.4 | 2.34 | 4.04 | 2.15 | 22.4 |
| Our conformal | | 9.61 | 2.16 | 17.4 | | 7.65 | 2.73 | 19.1 |
| Our isometric | 2.29 | 3.54 | | 17.1 | 2.15 | 3.71 | | 18.6 |
| Our authalic | 2.94 | | 4.04 | 19.6 | 2.81 | | 4.55 | 21.5 |
| Method | Kitten tiny (V:10200, F:20396) | Canonical tiny (V:2243, F:4482) | ||||||
| Angle | Area | Rigidity | Time (s) | Angle | Area | Rigidity | Time (s) | |
| Convex | 2.11 | 3.37 | 0.82 | | 2.01 | 4.65 | 2.14 | |
| ARAP | 2.15 | 2.62 | 0.58 | 72.1 | 2.11 | 3.35 | 1.65 | 4.8 |
| Our conformal | | 3.65 | 0.86 | 67.7 | | 4.59 | 2.12 | 3.4 |
| Our isometric | 2.12 | 2.56 | | 66.4 | 2.09 | 3.32 | | 3.2 |
| Our authalic | 2.54 | | 1.47 | 70.2 | 3.34 | | 2.51 | 4.3 |
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