一、声明

• 本帖持续更新中
• 有纰漏还望指正！

二、无符号距离函数 (Unsigned Distance Function, UDF)

  无符号距离函数（Unsigned Distance Function）是计算物体表面距离的一种数学函数。它通常用于计算机图形学中的体素渲染、几何形状重建和碰撞检测等应用。图1是一些常见无符号距离场的原理示意图。

图1：不同距离原理可视化对比 ^[1]

Wasserstein Distance / Earth Mover’s Distance ☆

$$D_{was}(x)=\sqrt{\frac{1}{K}\sum _{p\in N_k(x)}\Vert x-p{\Vert }^2}$$
$N_k(x):$ the set of $K$ nearest neighbours to $x$
$K$: a tradeoff between robustness and accuracy
不够精确，只能用来确定大致的区域。

欧氏距离 (Euclidean Distance)

$$D_E(p,q)=\sqrt{(p-q{)}^2}$$
$p,q$： they are two points from ${\mathbb{R}}^N$

马氏距离 (Mahalanobis Distance)

• Distance between two probability measures
  $$W_p(\mu ,v)={\left(\underset{\gamma \in \Gamma (\mu ,v)}{\inf }{E}_{(x,y)\sim \gamma }d(x,y{)}^p\right)}^{1/p}$$
  $\mu ,v$: two probability measures on $M$
  $\Gamma (\mu ,v)$: the set of all couplings of $\mu$ and $v$

• Distance between a point and one probability measure
  计算一个点 $\mathbf{x}$ 到一个点集或分布 $D$ 的马氏距离 $D_M(\mathbf{x})$ 的公式通常表达为：
  $$D_M(\mathbf{x})=\sqrt{(\mathbf{x}-\mu {)}^T{\Sigma }^{-1}(\mathbf{x}-\mu )}$$
  其中:
  $\mathbf{x}$： 是要计算的点。
  $\mu$： 是点集 $D$ 的均值向量。
  ${\Sigma }^{-1}$： 是点集 $D$ 的协方差矩阵的逆，$\Sigma =\frac{1}{n-1}(\mathbf{x}-\mu {)}^T(\mathbf{x}-\mu )$。
  ${}^T$： 表示向量的转置。

带权重的最近邻距离（Weighted Nearest Neighbor Distance）

$$d(p)=\sum _{j=1}^k{\omega }_{i}d(p,p_j)$$
$p$: the query point
$p_i$: the $j$-th nearest neighbour of point $p$
$d(p,p_j)$: the distance of point $p$ and $p_j$
${\omega }_i$: the weight of point $p_i$, it can be some common distance functions, e.g. ${\omega }_i=-e^{\frac{d(p,p_i{)}^2}{2{\sigma }^2}}$, $\sigma (p_i)=s{\sum }_{j=1}^k\Vert p_i-p_{i,j}\Vert$
$k$: the number of neighbours for calculating distance

Chamfer Distance

Hausdorff Distance

Chebyshev Distance

City Block (Manhattan) Distance

Correlation Distance

Cosine Distance

Jensen-Shannon Distance

Minkowski Distance

Standardized Euclidean Distance

Squared Euclidean Distance

无名1

源自文章
[1] 2011, Chazal et al., Geometric Inference for Probability Measures. Foundations of Computational Mathematics
[2] 2013, Noise-adaptive shape reconstruction from raw point sets. CGF

$$d_{\mu ,m}^2:{\mathbb{R}}^n\to \mathbb{R},x\mapsto \frac{1}{m}{\int }_{B(x,r_{\mu ,m}(x))}\Vert x-y{\Vert }^{2}d\mu (y)$$
$x:$ a query point
$\mu :$ a probability distribution in ${\mathbb{R}}^n$
$r_{\mu ,m}(x):$ the minimum radius such that the ball centered at $x$ with radius $r$ encloses a mass of at least $m$.
$m:$ $m\in (0,1]$, a user-defined parameter.

三、有符号距离函数 (Signed Distance Function, SDF)

最常见距离场

$$\Phi (\mathbf{x})=(\mathbf{x}-{\mathbf{p}}_i)\cdot {\mathbf{n}}_i$$
$\mathbf{x}:$属于${\mathbb{R}}^3$空间
${\mathbf{p}}_i:$是点$\mathbf{x}$的最近邻点
则表面为距离场$\mathbf{\Phi }$的零等值面，该距离场对法向很敏感。

截断有符号距离函数（Truncated Signed Distance Function, TSDF）

四、参考文献

[1] [读论文]点云表面重建： SDF, TSDF, MLS, RBF