## 4 Multiple Hypothesis Testing Technique
### Multiple Hypothesis Testing Technique
In Figure 3, for the first detection step, the steering vector estimation results are directly integrated into the MHT technique with Equation (28) or (29), which are derived from the interferometric phase map. After the first detection step, when \(\max\left(DT_{0,1(\bullet)},DT_{0,2(\bullet)},\ldots,DT_{0,M(\bullet)}\right)<0\) for no target is declared, the detection strategy goes to the next image pixel. When \(\max\left(DT_{0,1(\bullet)},DT_{0,2(\bullet)},\ldots,DT_{0,M(\bullet)}\right)>0\) for one or more targets is declared, the second step is classification to decide the true corresponding road that the detected targets belong to with Equation (30) or (31). For the third step, the detected targets are relocated to their real positions, based on the detection and classification results. To evaluate the performance of the proposed scheme, we calculate the \(P_{fn}\), the \(P_{d}\), and the probability of correct classification (\(P_{c}\)). The \(P_{fn}\) is defined as the probability to decide one of the \(M\) hypotheses when \(H_{0}\) is true. The \"total\" \(P_{fn}\) can be expressed using the inclusion-exclusion principle [35; 36],  
\[\begin{array}{l}P_{fa}=P\Big{\{}\Big{(}\Lambda_{0,1(\bullet)}>\eta_{0,1( \bullet)}|H_{0}\Big{)}\cup\Big{(}\Lambda_{0,2(\bullet)}>\eta_{0,2(\bullet)}|H_{ 0}\Big{)  
\[ \begin{split} P_{d}&=\frac{p\big{\{}\big{(}\Lambda_{0, 1(\bullet)}>\eta_{0,1(\bullet)}|H_{1}\big{)}\cup\ldots\cup\big{(}\Lambda_{0,M( \bullet)}>\eta_{0,M(\bullet)}|H_{1}\big{)}\big{\}}P(H_{1})+\ldots+P\big{\{} \big{(}\Lambda_{0,1(\bullet)}>\eta_{0,1(\bullet)}|H_{M}\big{)}\cup\ldots\cup \big{(}\Lambda_{0,M(\bullet)}>\eta_{0,M(\bullet)}|H_{M}\big{)}\big{\}}P(H_{M}) \\ &=\frac{p\big{\{}\big{(}\Lambda_{0,1(\bullet)}>\eta_{0,1(\bullet)} |H_{1}\big{)}\cup\ldots\cup\big{(}\Lambda_{0,M(\bullet)}>\eta_{0,M(\bullet)}|H_{ 1}\big{)}\big{\}}P(H_{1})+\ldots+P\big{\{}\big{(}\Lambda_{0,1(\bullet)}>\eta_ {0,1(\bullet)}|H_{M}\big{)}\cup\ldots\cup\big{(}\Lambda_{0,M(\bullet)}>\eta_{0,M(\bullet)}|H_{M}\big{)}\big{\}}P(H_{M})}{P(H_{1})+P(H_{2})+\ldots+P(H_{M})} \end{split}  \tag{34}\]  
Under the assumption \(P(H_{1})=P(H_{2})=\ldots\ldots=P(H_{M})\), Equation (34) can be expressed as,  
\[P_{d}=\frac{1}{M}\sum\limits_{l=1}^{M}P_{d|H_{l}} \tag{35}\]  
\[ \begin{split} P_{d|H_{l}}&=P\Big{\{}\Big{(}\Lambda_ {0,1(\bullet)}>\eta_{0,1(\bullet)}|H_{l}\Big{)}\cup\Big{(}\Lambda_{0,2(\bullet )}>\eta_{0,2(\bullet)}|H_{l}\Big{)}\cdot\ldots\cdot\cup\Big{(}\Lambda_{0,M( \bullet)}>\eta_{0,M(\bullet)}|H_{l}\Big{)}\Big{\}}\\ &=\sum\limits_{i=1}^{M}P\Big{(}\Lambda_{0,i(\bullet)}>\eta_{0,i( \bullet)}|H_{l}\Big{)}-\sum\limits_{1\leq i<j\leq M}P\Big{(}\Lambda_{0,i( \bullet)}>\eta_{0,i(\bullet)},\Lambda_{0,j(\bullet)}>\eta_{0,j(\bullet)}|H_{l }\Big{)}\\ &+\sum\limits_{1\leq i<j<k\leq M}P\Big{(}\Lambda_{0,i(\bullet)}> \eta_{0,i(\bullet)},\Lambda_{0,j(\bullet)}>\eta_{0,j(\bullet)},\Lambda_{0,k( \bullet)}>\eta_{0,k(\bullet)}|H_{l}\Big{)}-\ldots\ldots\cdot\\ &+(-1)^{M-1}P\Big{(}\Lambda_{0,1(\bullet)}>\eta_{0,1(\bullet)}, \ldots,\Lambda_{0,M(\bullet)}>\eta_{0,M(\bullet)}|H_{l}\Big{)}\end{split} , \,l=1,\ldots,M\\\end{split} \tag{36}\]  
We can use the notation as follows [35; 37],  
\[PL_{k}=\sum\limits_{1\leq i_{1}<\ldots<i_{k}<\ldots\leq M}P\Big{(}\Lambda_{0, i_{1}(\bullet)}>\eta_{0,i_{1}(\bullet)},\ldots,\Lambda_{0,i_{k}(\bullet)}>\eta_{0,i _{k}(\bullet)}|H_{l}\Big{)},\,k=2,\ldots,M \tag{38}\]  
Then the \"local\" \(P_{d}\) can be expressed as,  
\[P_{d|H_{l}}=\sum\limits_{i=1}^{M}(-1)^{i-1}PL_{k} \tag{39}\]  
To simplify the computation of Equations (36) and (39), we can use the expression as follows [38; 39],  
\[ \begin{split} P_{d|H_{l}}&=P\Big{\{}\big{(}\Lambda_ {0,1(\bullet)}>\eta_{0,1(\bullet)}|H_{l}\big{)}\cup\Big{(}\Lambda_{0,2(\bullet )}>\eta_{0,2(\bullet)}|H_{l}\Big{)}\cdot\ldots\cdot\cup\big{(}\Lambda_{0,M( \bullet)}>\eta_{0,M(\bullet)}|H_{l}\big{)}\Big{\}}\\ &=1-P\big{(}\Lambda_{0,1(\bullet)}<\eta_{0,1(\bullet)},\ldots, \Lambda_{0,M(\bullet)}<\eta_{0,M(\bullet)}|H_{l}\big{)}\end{split}