【感知算法】Dempster-Shafer理论（下）

尝试DS理论应用到自动驾驶地图众包更新。

地图特征变化判断

a mass function is applied to quantify the evidence of the existence.
existence state: existenct、non-existent、tenative、conflict
$$\begin{gathered} \exists \\ \nexists \\ \Omega \\ \varphi \end{gathered}$$
mass function: quantify the evidence of the existence.

• mass functions of the measurement

$$\begin{gathered} mas{s}_{z_t}(\exists )=\lambda \\ mas{s}_{z_t}(\nexists )=0 \\ mas{s}_{z_t}(\varphi )=0 \\ mas{s}_{z_t}(\Omega )=1-\lambda \end{gathered}$$

• mass functions of the non-measurement

$$\begin{gathered} mas{s}_{z_t}(\exists )=0 \\ mas{s}_{z_t}(\nexists )=\lambda \\ mas{s}_{z_t}(\varphi )=0 \\ mas{s}_{z_t}(\Omega )=1-\lambda \end{gathered}$$

Inference of the map feature existence based Dempster Combination Rule

• mass functions of map features and new map features
  初始化使用第i个地图特征的先验置信度${\lambda }_{HD}$

m a s s H D 0 { i } ( ∃ ) = λ H D m a s s H D 0 { i } ( ∄ ) = 0 m a s s H D 0 { i } ( ϕ ) = 0 m a s s H D 0 { i } ( Ω ) = 1 − λ H D mass_{HD_{0\{i\}}}( \exist ) = \lambda_{HD} \\ mass_{HD_{0\{i\}}}( \not \exist ) = 0 \\ mass_{HD_{0\{i\}}}( \phi ) = 0 \\ mass_{HD_{0\{i\}}}( \Omega ) = 1 - \lambda_{HD} massHD0{i}​​(∃)=λHD​massHD0{i}​​(∃)=0massHD0{i}​​(ϕ)=0massHD0{i}​​(Ω)=1−λHD​

新增加的第j个地图特征，按下列式初始化
m a s s n e w 0 { j } ( ∃ ) = 0 m a s s n e w 0 { j } ( ∄ ) = 0 m a s s n e w 0 { j } ( ϕ ) = 0 m a s s n e w 0 { j } ( Ω ) = 1 mass_{new_{0\{j\}}}( \exist ) = 0 \\ mass_{new_{0\{j\}}}( \not \exist ) = 0 \\ mass_{new_{0\{j\}}}( \phi ) = 0 \\ mass_{new_{0\{j\}}}( \Omega ) = 1 \\ massnew0{j}​​(∃)=0massnew0{j}​​(∃)=0massnew0{j}​​(ϕ)=0massnew0{j}​​(Ω)=1

• Usd Dempster combination rule $\oplus$ to accumulate the measurement existence $mas{s}_{z_t}$to the each map feature existence at time $t-1$，

m a s s H D t { i } = m a s s H D t − 1 { i } ⊕ m a s s z t m a s s n e w t { j } = m a s s n e w t − 1 { j } ⊕ m a s s z t mass_{HD_{t\{i\}}} = mass_{HD_{t-1\{i\}}}\oplus mass_{z_t} \\ mass_{new_{t\{j\}}} = mass_{new_{t-1\{j\}}}\oplus mass_{z_t} massHDt{i}​​=massHDt−1{i}​​⊕masszt​​massnewt{j}​​=massnewt−1{j}​​⊕masszt​​

其中，
$$\begin{gathered} mas{s}_{1\oplus 2}(A)=\frac{mas{s}_{1\cap 2}(A)}{1-mas{s}_{1\cap 2}(\varphi )},\forall A\subseteq \Omega ,A\ne \varphi \\ mas{s}_{1\oplus 2}(\varphi )=0 \\ \forall A\subseteq \Omega ,mas{s}_{1\cap 2}(A)=\sum _{B\cap C=A|B,C\subseteq \Omega }mas{s}_1(B)mas{s}_2(C) \end{gathered}$$
注：公式$mas{s}_{1\oplus 2}(A)$即$mas{s}_1(A)\oplus mas{s}_2(A)$
求和条件中的$|$为并列含义，$\Omega$为超集$2^X$，
$$\begin{gathered} \exists \cap \exists =\exists \\ \exists \cap \Omega =\exists \\ \exists \cap \nexists =\varnothing \\ \varnothing \cap \exists =\varnothing \\ \varnothing \cap \nexists =\varnothing \\ \varnothing \cap \Omega =\varnothing \\ \varnothing \cap \varnothing =\varnothing \end{gathered}$$
集合运算满足交换律。