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 |
$\omega_{i,j}$ $\Leftarrow$ ComputeWeights(RingNodes($p_i$)); |
RingNodes($p^{\prime}_i$) $\Leftarrow$ ComputeWeights(RingNodes($p_i$)); |
RingNodes($q^{\prime}_i$) $\Leftarrow$ ComputeWeights(RingNodes($q_i$)); |
$L_{(i,j,j+1)}$ $\Leftarrow$ FittingMatrix($\triangle p^{\prime}_ip^{\prime}_jp^{\prime}_{j+1}$, $\triangle q^{\prime}_iq^{\prime}_jq^{\prime}_{j+1}$); |
New($q^{\prime}_i$) $\Leftarrow$ Relocate2D($\triangle p^{\prime}_ip^{\prime}_jp^{\prime}_{j+1}$, $\triangle q^{\prime}_iq^{\prime}_jq^{\prime}_{j+1}$); |
$\widetilde{\omega}_{i,j}$ $\Leftarrow$ MeanValue(RingNodes(New($q^{\prime}_i$))); |
$q^{\prime\prime}_i$ $\Leftarrow$ Relocate3D($\widetilde{\omega}_{i,j}$,RingNodes($q_i$)); |
$B_i$ $\Leftarrow$ 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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$S^*=S^*_0$; |
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 |