# Infrared Small Moving Target Detection via Saliency Histogram and Geometrical Invariability
## 3 Inter-Frame Detection Based on Geometrical Invariability
### False Alarm Suppression
It is absolutely true that rotation cannot change the length of a vector, so Equation (17) can be further written as  
\[\left\|\mathrm{P}_{\mathrm{n}}^{\mathrm{s}}-\mathrm{P}_{\mathrm{m}}^{\mathrm{ s}}\right\|_{2}=\left\|\mathbb{N}(\Delta\Phi)\cdot\left(\mathrm{P}_{\mathrm{n} }^{\mathrm{f}}-\mathrm{P}_{\mathrm{m}}^{\mathrm{f}}\right)\right\|_{2}=\left\| \mathrm{P}_{\mathrm{n}}^{\mathrm{f}}-\mathrm{P}_{\mathrm{m}}^{\mathrm{f}} \right\|_{2} \tag{18}\]  
In contrast, if \(\mathrm{P}_{\mathrm{m}}^{\mathrm{s}}\) is a moving target point while \(\mathrm{P}_{\mathrm{n}}^{\mathrm{s}}\) is a background point, \(\mathrm{P}_{\mathrm{m}}^{\mathrm{s}}\) may have an extra displacement \(\Delta\rho_{\mathrm{m}}\) caused by self-motion, which means Equation (18) needs to be modified as  
\[\left\|\mathrm{P}_{\mathrm{n}}^{\mathrm{s}}-\mathrm{P}_{\mathrm{m}}^{\mathrm{ s}}\right\|_{2}=\left\|\mathbb{N}(\Delta\Phi)\cdot\left(\mathrm{P}_{\mathrm{n} }^{\mathrm{f}}-\mathrm{P}_{\mathrm{*}}+\Delta\rho\right)-\mathbb{N}(\Delta\Phi )\cdot\left(\mathrm{P}_{\mathrm{m}}^{\mathrm{f}}-\mathrm{P}_{\mathrm{*}}+ \Delta\rho+\Delta\rho_{\mathrm{m}}\right)\right\|_{2} \tag{19}\] \[=\left\|\mathbb{N}(\Delta\Phi)\cdot\left(\mathrm{P}_{\mathrm{n}}^{ \mathrm{f}}-\mathrm{P}_{\mathrm{m}}^{\mathrm{f}}-\Delta\rho_{\mathrm{m}}\right) \right\|_{2}\] \[=\left\|\mathbb{P}_{\mathrm{n}}^{\mathrm{f}}-\mathrm{P}_{\mathrm{ m}}^{\mathrm{f}}+\Delta\rho_{\mathrm{m}}\right\|_{2}\] \[>\left\|\mathrm{P}_{\mathrm{n}}^{\mathrm{f}}-\mathrm{P}_{\mathrm{ m}}^{\mathrm{f}}\right\|_{2}\]  
Therefore, a conclusion can be summarized that the distance between two background points is constant, i.e., there is a geometrical invariability between background points. However, if there is at least a moving target point among them, the distance is changed. Motivated by this regulation, we propose a geometrical invariability based false alarm suppression method below. For the n-th candidate point in the second map, an index \(\psi\) denoting the difference of relative position is developed to judge whether \(\mathrm{P}_{\mathrm{n}}^{\mathrm{s}}\) is a real moving target or a false alarm:  
\[\psi(\mathrm{P}_{\mathrm{n}}^{\mathrm{s}})=\sum_{\mathrm{m}=1}^{\mathrm{N}} \left\|\left|\mathrm{P}_{\mathrm{n}}^{\mathrm{s}}-\mathrm{P}_{\mathrm{m}}^{ \mathrm{s}}\right\|_{2}-\left\|\mathrm{P}_{\mathrm{n}}^{\mathrm{f}}-\mathrm{P }_{\mathrm{m}}^{\mathrm{f}}\right\|_{2}\right| \tag{20}\]  
where N is the number of trajectories in the second map. Lastly, the judge criterion is defined as follows:  
\[\mathrm{P}_{need to be detected are marked with red circles, and the fake targets, as well as the regions that may possibly generate false alarms, are marked with yellow circles. The sky background of Seq.1 is clear, but the cloud regions are of great inhomogeneity of intensity, which means interference of cloud edge is the main source of false alarms. Among these four groups, Seq.2 and Seq.3 contain dark IR small targets, improving the difficulty and complexity of detection to some extent. Those bright but still points existing around the hills and houses in Seq.2 are fake targets, and it is quite hard to distinguish these kinds of points in a single frame. The sea waves are also very challenging for small target detection because the edges of waves are difficult to be suppressed by filtering based algorithms, and the motion of waves also cause much difficulty for the inter-frame detection. In addition, a dead point caused by the detector itself always exists during the whole sequence, which is a tough task for us to address. Seq.4 has a moving background and the cloud layer is quite dense. In addition, there are a certain number of dead pixels due to the poor quality of imaging.