论文阅读笔记：Embedded Deformation for Shape Manipulation

论文阅读笔记：Embedded Deformation for Shape Manipulation

• 理解
• 变换公式
• 优化条件
• 旋转项
• 正则项
• 交互约束
• 目标函数

理解

变换公式

对边进行仿射变换，求p。反映了已变形点的仿射变换对待变形点的影响。
$\tilde{\mathbf{p}}=\mathbf{R}_j(\tilde{p}-\tilde{g}_j)+\tilde{g}_j+\mathbf{t}_j$p~​=Rj​(p~​−g~​j​)+g~​j​+tj​

周围多个已变形点对待变形点的影响
$\tilde{\mathbf{v}}_i=\sum_{j=1}^{m}w_{j}(\mathbf{v}_i)[\mathbf{R}_j(\mathbf{v}_i-\mathbf{g}_j)+\mathbf{g}_j+\mathbf{t}_j]$v~i​=j=1∑m​wj​(vi​)[Rj​(vi​−gj​)+gj​+tj​]
通过加权将影响待变形点的范围控制在knn范围内。
$w_j(\mathbf{v}_i)=\left\{\begin{matrix} (1-\frac{\left \| \mathbf{v}_i-\mathbf{g}_j \right \|}{d_{max}})^2 \\ 0 && 其余点 \end{matrix}\right.$wj​(vi​)={(1−dmax​∥vi​−gj​∥​)20​​其余点​
待变形点的法向量也发生了变换（如果为SO3表面刚性变换，法向量不变；如果为Sim3表面发生非刚性变换，法向量变化）
$\tilde{\mathbf{n}}_i=\sum_{j=1}^{m}w_j(\mathbf{v}_i)\mathbf{R}_{j}^{-T}\mathbf{n}_i$n~i​=j=1∑m​wj​(vi​)Rj−T​ni​

优化条件

旋转项

为了保留模型细节特征，变换应为SO3，即局部尽可能约束为刚性变换。

The objective function encodes detail preservation directly by specifying that the affine transformations should be rotations. Consequently, local features deform as rigidly as possible.

正则项

A second energy term serves as a regularizer for the deformation by indicating that the affine transformations of adjacent graph nodes should agree with one another.

交互约束

两个约束：handle constraints 和 fixed constraints。用于用户交互，用户调整一些点后，再对表面进行重新生成。

handle constraints, where a collection of model vertices are selected and become handles that are manipulated by the user, and fixed constraints, where a collection of model vertices are selected and guaranteed to be fixed in place.

目标函数

用扰动模型得到雅克比矩阵，用高斯牛顿优化非线性问题。

Cholesky factorization求增量状态