## 4 Methods
### Relevance of Different Size and Shape Occupancy Grids  
In addition to interpreting the basic context, it is also worth considering how to define the size of the elementary regions of space. Of course, increasing the resolution reduces the negative effects of discretisation, such as estimation error and degradation of control performance. On the other hand, the use of grids of larger elementary regions offers certain advantages, such as a reduction in computational complexity and the closely related reduction in computational resources and time. Considering all these, it seems reasonable to introduce a method that allows the interconnection and interoperability of different sized occupancy grids. This would make it possible to adapt the modelling framework for the different sizes of traffic actors, representing an important research advance in the modelling and management of complex traffic situations. In addition, providing interoperability between different occupancy grids also makes it possible to assign separate grids to different traffic directions, which may be beneficial in reducing the complexity of the traffic management. To examine the computational demands, it is useful to introduce the concepts used to describe the complexity of occupancy grids. When creating a grid, the plane is filled with polygons such that adjacent polygons cover the plane without gaps and there is no overlap between the adjacent polygons. In most cases, regular filling is used, so that the polygons concerned are congruent regular polygons. We will call these elementary polygons. In line with the above, we define the sequence of operations to create a grid as covering of the plane without gaps and overlaps. We will refer to each step of the covering process as modelling. We perform the covering process in steps such that each bounding edge of the constrained two-dimensional space domain modeled up to step \((r-1)\) is adjacent to an elementary polygon modeled at step \(r\). The constrained two-dimensional space modelled up to the \((r-1)\)-th step is collectively referred to as the kernel. We will refer to the set of elementary polygons modelled in the \(r\)-th step as the \(r\)-th layer. By extension of the kernel, we mean the addition of the layer modelled in the \(r\)-th step to the constrained two-dimensional space modelled up to the \((r-1)\)-th step of the covering process. In line with this, we form the kernel modelled in step \((r+1)\) as the last operation of the \(r\)-th step of the covering process by merging the kernel and the layer. The matching edges create a connection between the elementary polygons of the occupancy grid. The internal connection is the connection between the elementary polygons forming a layer. We describe the complexity of the occupancy grid \((UC)\) as the limit of the number of connections per unit area \((A)\) as a function of the number of steps of the covering process \((r)\)[48]. **Square grid**  
The most commonly used regular elementary polygon forming occupancy grids is the square. In this case, the elementary polygon can have a maximum of 4 neighbours, so we do not classify it as one of the more complex grids. It cannot handle diagonal vehicle movements because the polygons that do not touch at the sides are not adjacent. Similarly, there can be no vehicle movement between polygons that only touch at their vertices. For the occupancy grid under consideration, we express the total number of connections between adjacent squares \((N)\) as a function of the covering steps \((r)\) as described in Equation (10). \[N_{1}=0\ and\ N_{r}=N_{r-1}+2*(8*(r-1)-4)+2*(8*(r-1)),\ if\ r>1 \tag{10}\]  
where the second term of the summation is the number of connections between the kernel and the layer, and the third term is the number of connections between the elementary polygons within the layer. The number of elementary polygons \((M)\) forming the occupancy grid can be written as a function of \(r\) as follows.