Table 1 Minimum K distance algorithm

Input: Number of observation stations NS; Number of targets NT; Bearing line \(L_{i j}(i=1,2, \ldots , N S ; j=1,2, \ldots , N T)\). Intersections set without preprocessing; Interested intersections pair K.

Output: Set of intersections after preprocessing P.

1 Mark all bearing lines as unprocessed

2 Randomly select an unprocessed bearing line \({L_{ij}}\) as the reference line, all intersections on \({L_{ij}}\)

form a set, which is record as \(\Lambda _{i j}\)

3 Mark \({L_{ij}}\) as processed

4 for all \(m, m=1,2, \ldots ,NS,m \ne i\) do

5 The set \(S_{m n, i j}=\left( X_{m 1, i j}, X_{m 2, i j} \cdots \right)\) of intersections formed by \({L_{ij}}\) and

\({L_{mn}}(n = 1,2, \ldots ,NT)\)

6 for all \({X_{mn,ij}} \in {S_{mn,ij}}\) do

7 Calculate the distance \(d_{m n, i j}\) between point \({X_{mn,ij}}\) and all other intersections X

(\(X \in \Lambda _{i j}, X \notin S_{m n, i j}\)), \(d_{m n, i j}=\min \left\{ \left\| X_{m n, i j}-X\right\| \right\}\)

8 end

9 Each loop in step 6 will generate a \(d_{m n, i j}\), sort all \(d_{m n, i j}\) generated in step 6, then select

the first K pairs of intersections with the smallest distance \(d_{m n, i j}\) to add to P

10 end

11 Repeat steps 2-10, until there are no unprocessed bearings

12 Remove the repeated intersections in P

13 Output P