Shape Morphing

要解决的问题

给定头尾模型，生成中间的模型，常用于做关键帧动画。

Input: two meshes source & target
• Frames at $t_0$ and $t_n$
Output: sequence of intermediate meshes
• Frames $t_1$ to $t_{n‐1}$
• For each point on source/target model specify location at time $t_i$ consistent with source & target

Wraping VS Morphing

Warping: Unary Op
• Given Object A and F(t), find Object B

Morphing: Binary Op
• Given Object A and Object B, find F(t)

Rules for Good Morphing

Natural
- Keep as much as possible of the two shapes during the transformation
  - Volume, curvature, area, etc...
- Subjective aesthetic criteria
User control
- intuitive
- not too heavy
- can be adapted to user's knowledge

Two Sub‐Problems

Correspondence problem
- Compatible meshes
- For each point on source/target meshes find corresponding point on second mesh = Parameterization
Path problem
- Inbetween shapes
- Specify trajectory in time for each point
  - For mesh – specify vertex trajectory

1. Vertex Correspondence

目标

Each vertex on source mesh mapped to vertex on target (and vice versa)

1.4 Decomposition Based

[Shlafman et al. 2002]

分解成部分，每个部分分别对应。
大问题分解成小问题

1.7 Error‐Bounded Compatible Remeshing

Yang et al. Error‐Bounded Compatible Remeshing. Siggraph 2020.

• Optimization based method

More: Correspondences between planar shapes – Matching

• Physically Based Method [Sederberg et al. 1992]

• Curve Aligning [Sebastian et al. 2003]

$$ \mu[g]=\int_{C}\left|\frac{\partial}{\partial s}(\bar{C}(\bar{s})-C(s))\right|^{2} d s+R \int_{C}(\kappa(s)-\bar{\kappa}(\bar{s}))^{2} d s $$

• Perceptually Based Method [Liu et al. 2004]

2. Vertex Path (Trajectory)

Input:
• All vertices on source & target have one‐to‐one correspondence with each other
• Each vertex has two 3D coords vFc1 (source) and vFc2 (target)
Output: generate the intermediate shapes from two shapes

Simplest Method: Linear Interpolation

Linear interpolation between corresponding points

优点：Work well for many cases，Simple and easy
缺点：Shrinkage

Intrinsic Approach

Sederberg et al. 1993

不插值顶点，而是插值多边形的边长和夹角

$$ \begin{array}{l} \begin{bmatrix}x(t) \\y(t) \end{bmatrix} \\ =\begin{bmatrix}a_0 \\c_0 \end{bmatrix}+ {\textstyle \sum_{k=1}^{\infty }} \begin{bmatrix} a_k &b_k \\ c_k &d_k \end{bmatrix}\begin{bmatrix}\cos(2\pi kt) \\ \sin(2\pi kt) \end{bmatrix} \ \end{array} $$