## 4 Multiple Hypothesis Testing Technique
### Multiple Hypothesis Testing Technique  
For most motorways and less dense road networks, the moving targets' interferometric phases can easily be determined using their azimuth displacement. For dense road networks, the roads are so close that the azimuth displacement intervals overlap. A moving target in such an area may have several possible azimuth displacement and interferometric phase estimates because the \(\Delta a\) and \(\overset{\rightharpoonup}{a}(\psi)\) estimations are linked by the position and orientation of the roads. The wrong road assignments and steering vector estimates may lead to false detections, velocity estimation ambiguities, and false relocations. Since the moving target steering vector estimation is linked to the position and orientation of the roads, they can be exploited as a likelihood ratios test (LRT) function indicating the probability of moving target detection and the true corresponding road that the detected targets belong to. The multiple hypothesis testing technique [23; 33; 34] is proposed in this paper to robustly resolve potential ambiguities in the moving target phase estimation. Define the \(M+1\) hypotheses as H\({}_{0}\) and H\({}_{k,k=1.....M}\). H\({}_{0}\): clutter plus noise. H\({}_{k}\): the \(k\)th signal plus clutter and noise, and the target belongs to the \(k\)th road. Therefore, the signal model in Equation (1) can be modified as,  
\[ \begin{array}{l}\text{Hypothesis H}_{0}:\overset{\rightharpoonup}{Z}(m,n )=\overset{\rightharpoonup}{C}(m,n)+\overset{\rightharpoonup}{N}(m,n)\\ \text{Hypothesis H}_{k}:\overset{\rightharpoonup}{Z}(m,n)=\overset{ \rightharpoonup}{S}_{k}(m,n)+\overset{\rightharpoonup}{C}(m,n)+\overset{ \rightharpoonup}{N}(m,n),\ k=1\ldots M\end{array}  \tag{27}\]  
In Equation (27), \(\overset{\rightharpoonup}{S}_{k}(m,n)=\beta\overset{\rightharpoonup}{a}_{k}( \psi)\odot\overset{\rightharpoonup}{S}_{Nor}(m,n)\). \(\overset{\rightharpoonup}{a}_{k}(\psi)\) is the moving target's spatial steering vector and the target belongs to the \(k\)th road. \(\overset{\rightharpoonup}{a}_{k}(\psi)\) can be estimated from road network database and radar parameters using Equations (12)-(14). A priori probability of hypothesis \(H_{k}\) is denoted by \(P(H_{k})=\text{Probability}\{H_{k}\}=1\ldots M\). We assume that \(P(H_{k})\), \(k=1\ldots M\) are unknown, but those pertaining to the target classes are all the same. As such, \(P_{k}=(1-P_{0})/M\), \(k=1\ldots M\), where \(P_{0}\) is the probability of the hypothesis \(H_{0}\). Figure 2: ROC(Pd versus Pfa in log scale) for AMF and ABORT. In this paper, we propose a detection, classification, and relocation scheme, which can detect the presence of a target and correctly classify it to corresponding roads. Based on road network database and SAR image, the detected and classified targets can be relocated to corresponding locations on the road. We propose to exploit the multiple hypothesis testing strategy, which needs to calculate all the likelihood ratios between each pair of hypotheses and compare each likelihood ratio with a proper threshold. For the MHT strategy, to compute the \(H_{0}\) versus \(H_{j}\) and \(H_{j}\) versus \(H_{j^{\prime}}\), we may exploit the \(AMF\) technique as Equation (19) and the ABORT technique as Equation (20).