回顾

$R^3$中的参数曲面：link
参数化(Parameterization)：link
参数化的重要性：link

参数化期望保持的几何性质

• 保角映射(angle‐preserving)：conformal（共形）
• 保面积映射(area‐preserving)：authalic
• 等距映射(isometric)：conformal + authalic

👆 $f$ is approximated by piecewise linear maps between pairs of triangles

参数化引起的扭曲

扭曲来源

R 把3D三角形旋转为平面三角形。
$\phi $是两个平面三角形之间的变形。扭曲都来自$\phi $

$\phi 是奇异矩阵，因此2D是3×3.扭曲都来自L(2×2)$
线性变换都可以用transformation matrix表示，都可以通过矩阵的性质来分析变换特点

扭曲度量

$$ L=U\begin{pmatrix}\sigma_1 & 0 \\0 &\sigma _2 \end{pmatrix}V^* $$

$$ \sigma _2\ge \sigma _1 $$

$L 是 2×2，做SVD, \sigma_{1}和\sigma_{2}$是奇异值。　

• angle‐preserving (conformal) $\sigma _1=\sigma _2$

• area‐preserving (authalic) $\sigma _1\sigma _2$

• length‐preserving (isometric) $\sigma _1=\sigma _2=1$

Local Injectivity

Distortion (Flip/Foldover)

Methods of Mesh Parameterization

Tutte’s method and its variants

Tutte’s embedding method

Tutte’s method [Tutte 1963; Floater 1997, 2003]

link

外部点：映射到 convex boundary 上。
内部点：1 邻域点的线性组合，权自己定义。通过求解稀疏方程组确定点的位置。
优点：简单、不翻转。
缺点：扭曲大。

Variants of Tutte’s embedding method

Geometry‐based optimization methods

• Representation based methods [Sheffer and Sturler 2001; Sheffer et al. 2005; Chien et al. 2016b; Fu and Liu 2016]
• ARAP [Sorkine and Alex 2007; Liu et al. 2008]
• Bounded distortion methods [Lipman 2012; Aigerman et al. 2014; Kovalsky et al. 015]

Angle Based Flattening (ABF) & ABF++

[Sheffer and Sturler 2001; Sheffer et al. 2005]

基于角度的展开，把角度当作变量，求解参数化的网格

目标:minimize (relative) deviation of angles

$$ F(\alpha )=\sum_{i,j}w_i^j(\alpha _i^j-\beta _i^j)^2 $$

Initial choice for weights:

$$ w_i^j=\beta _i^{j^-2} $$

Constraints：To avoid flipped triangles

$$ g^1(\alpha )\equiv \alpha _i^j\ge \varepsilon $$

$$ g^2(\alpha )\equiv \alpha _i^1+\alpha _i^2+\alpha _i^3=\pi $$

$$ g^3(\alpha )\equiv \sum _k\alpha _i^j(k)=2\pi $$

$$ g^4(\alpha )\equiv \prod _k\sin(\alpha _i^{j(k)-1})-\prod _k\sin(\alpha _i^{j(k)+1})=0 $$

$$ \frac{l_1}{l_2} =\frac{\sin(\alpha _1) }{\sin(\alpha _2) } $$

$$ \frac{l_1}{l_2} \cdots \frac{l_2}{l_1}=\frac{\sin(\alpha _1) }{\sin(\alpha _2) }\cdots \frac{\sin(\alpha _i) }{\sin(\alpha _j) }
$$

求解：用 Lagrange 算法解带约束的优化问题

Simplex Assembly [Fu and Liu 2016]

优化三角形变换的系数

✅Instead of vertex positions, treat the affine transformation as variables
✅Use a barrier function to prevent the inversion
✅No theoretically guaranteed to avoid foldovers

[Fu and Liu. Computing Inversion‐Free Mappings by Simplex Assembly. Siggraph Asia 2016]

Foldover free guaranteed optimization methods

[Smith and Schaefer 2015; Kovalsky et al. 2016; Jiang et al. 2017; Claici et al. 2017; Rabinovich et al. 2017; Shtengel et al. 2017; Zhu et al. 2018]

以 Tuttle 算法为基础，调整边界点位置，减少扭曲

Flip‐free parameterization methods

Start with a flip‐free (valid) initialization
Reducing the distortion while guaranteeing the validity
• Generally non‐convex nonlinear optimization

红色表示扭曲大，白色表示扭曲少

调整的同时移动顶点，减少扭曲，关键是如何度量扭曲

度量扭曲的方法[22:43]（最后一个最常用）

目标：

$$ \min_{V} E(V)=\sum _{t\in T}(\sigma _1^2+\frac{1}{\sigma _1^2} +\sigma _2^2+\frac{1}{\sigma _2^2}) $$

s.t $\sigma _1\sigma _2>0,$ $\forall t$

• The cost function is highly nonlinear and nonconvex
• The constraints are nonlinear
• The Heissian matrix is highly non‐definite

Computationally expensive for large scale meshes!

求解：

Input: a valid parameterization initialization $𝑥_0$
Repeat
$p=-H^{-1}\nabla E(x)$
How to find a good decent direction?

$𝛼_{max}$← injective maximal search step
$𝛼$ ←line search by backtracking from $𝛼_{max}$
$x ← 𝒙 + 𝛼p$

Until converged
Output: a locally injective parameterization

局限性：非线性、非凸问题、且参数多
因此优化效率低

❗ 后面没展开讲的方法，就不记了，没讲清楚，记了也没用

AQP：利用离散 Laplacian 作为 Hessian 的近似
SLIM: $\dots ，加权 \dots $
AKVF:$\dots 向量场算子\dots$
CM：使用隐式 Laplacian 矩阵，二阶方法、速度更快
各种方法都是找更好的H近似，使得优化过程更快更稳定。

Accelerated Quadratic Proxy (AQP)

Kovalsky et al. Accelerated Quadratic Proxy for eometric Optimization. Siggraph 2016.

Scalable Locally Injective Mappings (SLIM)

Rabinovich et al. Scalable Locally Injective Mappings. Siggraph 2017.

Isometry‐Aware Preconditioning (AKVF)

Claici et al. Isometry‐Aware Preconditioning for Mesh Parameterization. SGP 2017.

Composite Majorization (CM)

Shtengel et al. Geometric Optimization via Composite Majorization. Siggraph 2017.

Blended Cured Quasi‐Newton (BCQN)

Zhu et al. Blended Cured quasi‐Newton for Distortion Optimization. Siggraph 2018.

Progressive Paramerization

• Near‐degenerate triangles (i.e., large distortion) in the initializations by Tutte’s method
• Small iterative step and slow convergence in existing methods
• Key: even one extremely large distortion term can restrict the line search step size!

[Ligang Liu et al. Progressive Parameterizations. Siggraph 2018]

问题描述：
[图28:52]以脖子为边界。头挤在中间，中间扭曲很大⇒能量大⇒下降慢
解决方法：
[图29:29] 不优化整体，或寻找更好的H逼近，而是中间那区域
*个人感觉不 make sense *
[图30:35]