# Real-Time Geo-Localization for Land Vehicles Using LIV-SLAM and Referenced Satellite Imagery
## 3 Materials and Methods
### Fast Cross-Modal Matching  
To address the distortions illustrated in Figure 1, we proposed a fast cross-modal matching algorithm based on DSPD [11]. DSPD is a feature descriptor based on structural orientation of images, which can effectively adapt to nonlinear intensity variations across different imaging modalities. Although the DSPD feature is robust to cross-modal distortions, it does not consider the invalid regions caused by the projection as shown in Figure 1. Besides, the matching algorithm proposed in [11] suffers from high computational cost, as its similarity metric requires full searching in spatial domain and the affine searching, which in this algorithm leads to excessive search redundancy for the matching between local projection image and the referenced image, which only has translational uncertainty.  
To enhance the computational efficiency, we optimize the similarity metric and employ translation full searching, which can be realized with fast correlations in the frequency domain. Specifically, the dissimilarity metric proposed in [11] can be written to a similarity metric defined as follows:  
\[S=\sum_{P}\Big{|}\frac{\pi}{2}~{}-~{}|\alpha_{P}~{}-~{}\beta_{P}|\Big{|},~{}~{} ~{}\alpha_{P},\beta_{P}~{}\epsilon\Big{[}-\frac{\pi}{2},\frac{\pi}{2}\Big{]} \tag{4}\]  
where \(\alpha_{P}\) denotes the DSPD at a pixel of the projection image, and \(\beta_{P}\) denotes the DSPD at the corresponding pixel of the referenced image.  
To incorporate fast correlation in the frequency domain, we modify the similarity metric in (4) by introducing a double-angle cosine similarity as below:  
\[S=\sum_{P}cos(2\alpha_{P}~{}-~{}2\beta_{P}) \tag{5}\]  
Both metrics give minimal similarity when the two DSPDs are mutually orthogonal and maximal similarity when they are parallel. Actually, Equation (5) can be derived from Equation (4) according to Fourier expansion. The detailed derivation of the equations is provided in the Supplementary Materials.  
Furthermore, the translation full searching-based matching with the double-angle cosine similarity metric can be expressed in the form of two correlations, as shown in (6):  
\[S=\sum_{P}m_{P}cos2\alpha_{P}cos2\beta_{P}+\sum_{P}m_{P}sin2\alpha_{P}sin2 \beta_{P} \tag{6}\]  
where \(m_{P}\) is the mask of the projection image, which can be obtained during the projection process by binarizing the weight map. It is used to eliminate the influence of invalid regions in the projection image on the matching performance, as illustrated in Figure 1.  
According to the convolution theorem, the full searching in the correlation of spatial domains can be efficiently implemented in frequency domain with Fast Fourier Transform (FFT) [35].  
The translation full-searching with the similarity metric presented in Equation (4) has to be realized in the spatial domain. Given a projection image of size N \(\times\) N and a search area of size M \(\times\) M, the computational complexity of the full searching in the spatial domain is \(O\left(N^{2}(M~{}-~{}N+1)^{2}\right)\). In contrast, the translation full-searching with the similarity metric presented in (6) can be realized in the frequency domain, where the FFT dominates the computational cost, resulting in an overall complexity of \(O\left(2M^{2}logM\right)\)[35]. The comparison of matching algorithms in ablation experiments (Section 4.3) demonstrates that the optimization significantly enhances the computational efficiency while maintaining the matching accuracy.