# Developing Occupancy Grid with Automotive Simulation Environment
## 2 Materials and Methods
### Occupancy Grid Framework
#### 2.4.3 Fusing the Probability
The second crucial feature is the ability of the method to identify conflicting measurement information [11]. These advantages come with a cost that requires the definition of more than just the occupancy probability grid. Additional layers are needed to represent probability masses. Two mass matrices are needed--occupied \[m(O)\] and empty \[m(E).\] (15)The basis for computing of the cell probability is the Dempster-Shafer combination Formula (16). \[m_{12}(X)=(m_{1}\oplus m_{2})(X)= \begin{cases}\frac{(m_{1}\cap m_{2})(X)}{1-(m_{1 }\cap m_{2})(\emptyset)}&forX\
eq\emptyset\\ 0&forX=\emptyset\end{cases}  \tag{16}\]  
where  
\[(m_{1}\cap m_{2})(X)=\sum_{A,B\in\mathfrak{2}^{\Theta}|A\cap B=X}w_{1}m_{1}(A )\cdot w_{2}m_{2}(B) \tag{17}\]  
and the conversion of masses to probability is calculated by  
\[P(A)=\sum_{X\in\mathfrak{2}^{\Theta}}m(X)\frac{|A\cap X|}{|X|} \tag{18}\]  
The Dempster-Shafer combination formula applied to the occupancy grid gives the following formulas:  
\[m_{12}(O)=\frac{m_{1}(O)m_{2}(O)+m_{1}(O)m_{2}(\{O,E\})+m_{1}(\{O,E\})m_{2}\{O \}}{1-m_{1}(E)m_{2}(O)-m_{1}(O)m_{2}(E)} \tag{19}\]  
\[m_{12}(E)=\frac{m_{1}(E)m_{2}(E)+m_{1}(E)m_{2}(\{O,E\})+m_{1}(\{O,E\})m_{2}\{ E\}}{1-m_{1}(E)m_{2}(O)-m_{1}(O)m_{2}(E)} \tag{20}\]  
The equations provide a final probability value obtained by the following approach:  
\[P(O)=m(O)+\frac{m(\{O,E\})}{2} \tag{21}\]  
* Linear Opinion Pool (LOP) The method identified in the literature [8] allows for the resolution of conflicting sensor information [12] based on an opinion pooling methodology [13]. An example application of the family of opinion pooling methods is pooling opinions from individuals for drawing conclusions (such as expert opinions) [14]. \[p_{c}=\alpha\sum_{i}w_{i}p_{i}\] (22) This method assumes the existence of a weight \(w_{i}\) that can represent the contribution of certain sensors [12]. * Independent Opinion Pool (IOP) A method which is based on the assumption that the sources are considered to be independent is a modification of the IOP. \[p_{c}=a\prod_{i}p_{i}\] (23) \[a=(\prod_{i}p_{i}+\prod_{i}(1-p_{i}))^{-1}\] (24) The presented modification results in a geometric mean representation of the problem [8; 15]. \[\alpha=(\sum_{i}w_{i})^{-1}\] (25)
* Logarithmic Independent Opinion Pool (LIOP)Another generalization of prior methods is a weighting process on top of probability distributions, which comes as a result of splitting the process into two steps: first, a weighted average, and second, a geometric weighted average [8]. \[p_{c}=\alpha\prod_{i}p_{i}^{w_{i}} \tag{26}\]  
\[P(A)=\sum_{X\in\mathfrak{2}^{\Theta}}m(X)\frac{|A\cap X|}{|X|} \tag{18}\]0  
* De Morgan's Law (DeMorgan)  
Another approach proposed for fusing grids or measurement results inside grids is proposed by [PERSON] and has a tendency to favor higher probabilities [8]. This method has applications in text searching and it can be applied to simplify Boolean equations to the use of only AND and NOR gates. \[P(A)=\sum_{X\in\mathfrak{2}^{\Theta}}m(X)\frac{|A\cap X|}{|X|} \tag{18}\]1  
* Maximum Policy (Max)  
The simplest of all approaches proposed in the literature [16] assumes a conservative approach for taking the highest values of each cell [8]. \[P(A)=\sum_{X\in\mathfrak{2}^{\Theta}}m(X)\frac{|A\cap X|}{|X|} \tag{18}\]2  
An in-depth description of these methods, along with lookup table plots, can be found in the work of [PERSON] [8]. In the remaining sections, evaluations taking into consideration qualitative results are presented.