# A Topology Based Automatic Registration Method for Infrared and Polarized Coupled Imaging
## 2 Methodology
### Region Skeletonization for Control Points  
The binary images of polarization parameter image and infrared image were obtained after the segmentation we introduced in Section 2.2.  
Considering that imaging is a mapping from a real target to a digital signal, the topological structure of the target does not change with the imaging modality. In our case, based on the characters mentioned above, we put forward that the endpoint of the morphology  
Figure 2: A graphical representation of the Gabor filter bank. Filter with scale in \(7\times 7\) and \(11\times 11\) from top to bottom, the angle from left to right are 0, 45, 90, and 135 degrees, gamma is set to 1 so the filter is round, psi is 0 so no phase shift. Warm colors indicate larger values and cool colors indicate smaller values.  
skeleton could be the registration basis in this study. Image skeletonization is the ultimate of image thinning, which is topology-preserving. It reduces the elongated parts to one-pixel-thick arcs or closed curves, without changing the connectivity properties of the object or its background [25]. It is widely used in gait recognition, motion tracking and fingerprint analysis [26; 27; 28; 29]. Taking the topological structure of the target as the registration basis can guarantee that the registration basis of the heterogeneous images remains invariable when the imaging sensor changes. We skeletonized the segmented binary images of infrared and polarization parameter simultaneously. Image skeletonization is represented by erosion and opening operation, and the formula [30] is as follows:  
\[\mathrm{S}(\mathrm{Q}_{\mathrm{in}})=\bigcup\limits_{\mathrm{k}=0}^{\mathrm{ K}}\mathrm{S}_{\mathrm{k}}(\mathrm{Q}_{\mathrm{in}}) \tag{2}\]  
where \(\mathrm{Q}_{\mathrm{in}}\) means the geometric region located inside the image after segmentation, and \(\bigcup\limits_{\mathrm{k}=0}^{\mathrm{K}}\) means a union set with elements numbered from 0 to K.  
\[\mathrm{S}_{\mathrm{k}}(\mathrm{Q}_{\mathrm{in}})=(\mathrm{Q}_{\mathrm{in}} \odot\mathrm{kB})-(\mathrm{Q}_{\mathrm{in}}\odot\mathrm{kB})\circ\mathrm{B} \tag{3}\]  
where B represents the structural element. Here, we used the built-in skeletonization lookup table from MATLAB, which contains a set of different structural elements, and which changes with the erosion condition. \(\circ\mathrm{B}\) means an opening operation by structural element B, and \(\odot\) B means an erosion operation by structural element B.  
Furthermore,  
\[(\mathrm{Q}_{\mathrm{in}}\odot\mathrm{kB})=\left(\left(\cdots(\mathrm{Q}_{ \mathrm{in}}\odot\mathrm{B})\odot\mathrm{B}\right)\
otin\cdots\right)\odot \mathrm{B}) \tag{4}\]  
represents multiple erosion, and k is the last iteration step before \(\mathrm{Q}_{\mathrm{in}}\) is eroded to an empty set, that is:  
\[\mathrm{K}=\max\{\,\mathrm{k}|\,(\mathrm{Q}_{\mathrm{in}}\odot\mathrm{kB}) \
eq\varnothing\}. \tag{5}\]  
The skeletonization of the image can not only maintain the topological information of the original structure but also has good resistance to slight deformation such as stretching and rotating, preserving consistency between skeletons from different imaging perspectives.