# Optical and SAR Image Registration Based on the Phase Congruency Framework
## 2 Proposed Method
### The Proposed Algorithm
#### 2.2.2 Feature Description  
It is well known that the PC-based descriptor is prevalent because of its robustness to nonlinear radiation. However, most of them are sensitive to scale and rotation differences. Thus, we propose a novel PC-based descriptor, which is compatible with large-scale and rotation differences between optical and SAR images. Here, we use the previously calculated response map \(A_{s,o}(x,y)\), which contains both orientation and amplitude information, to construct a novel descriptor. Above all, the amplitude and orientation are calculated under the condition that \(s\) and \(o\) are set to 4 and 6, respectively. The calculated 24 response maps as shown in Figure 3. We sum the response map value \(A_{s_{i},o_{j}}(x,y)\) in each scale \(s_{j}\), and obtain the response map \(A_{o_{j}}(x,y)\) in each orientation \(o_{j}\) as follows:  
\[A_{o_{j}}(x,y)=\sum_{i=1}^{4}A_{s_{i},o_{j}}(x,y) \tag{16}\]  
\[A_{o}(x,y)\!\!=\{A_{o_{1}}(x,y),\ldots,A_{o_{j}}(x,y),\ldots,A_{o_{6}}(x,y)\} \tag{17}\]  
As for each keypoint \((x,y)\), we obtain the maximum response map \(A_{o_{i}}^{\max}(x,y)\) as follows, and the index value \(o_{i}(x,y)\) represents the orientation of \(A_{o_{i}}^{\max}(x,y)\):  
\[A_{o_{i}}^{\max}(x,y)\!\!=\!\max\{A_{o_{1}}(x,y),\ldots,A_{o_{j}}(x,y),\ldots, A_{o_{6}}(x,y)\} \tag{18}\]  
Then, we construct the descriptor vectors based on the above method. On the one hand, we expect that the descriptor is unique. On the other hand, it is better to be insensitive to the geometric and radiometric differences between optical and SAR images. However, the two aspects are contradictory. To solve this problem, we use the novel descriptor construction framework to obtain the orientation histogram. As shown in Figure 4, the inner circle improves the descriptor uniqueness by the histogram statistic of the six orientations based on the four location bins, and the outer circle reduces the sensitivity to nonlinear differences by mapping the six orientations to three orientations. In detail, the steps of the proposed descriptor are as follows:  
Step 1: The descriptor framework is constructed in the polar coordinate system, and the main orientation is obtained in the region where one keypoint is located. The framework is rotated to the main orientation to ensure the rotation invariance of the framework. Among them, the main orientation is statistically obtained from the \(\theta(x,y)\) corresponding to Formula (13). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Multi-Harris** & **PC-Harris** \\ \hline \(\gamma\) (\%) & 13.62 & 15.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparisons on repeatability rates. Figure 2: Feature detection results based on the Multi-Harris and PC-Harris. (**a**) Multi-Harris. (**b**) PC-Harris. Step 2: As shown in Figure 4, two radial areas are composed of 4 and 8 orientation bins, respectively. \(o_{i}(x,y)\in[1\ 6]\) and the amplitude of each location bin is counted up in the six orientations. Each location bin corresponds to a 6-dimensional orientation histogram, and 12 groups of 6-dimensional vectors are obtained totally. Step 3: For the rotation invariance of the elements in one eigenvector, the orientation histogram in each location bin is rotated to the main orientation \(\Theta\) by the process in Figure 5 to ensure the rotation invariance of the elements. Figure 4: Descriptor construction based on the polar coordinate system. Figure 5: Rotation process of the descriptor elements. Figure 3: Constructions of the maximum response map and the orientation index map. Step 4: In the outer circle, the 6-dimensional orientation is mapped to the 3-dimensional orientation. In detail, the amplitudes in two orientations are added together to form the outer circle vector. It is a 24-dimensional vector which consists of 8 location bins and 3 orientations. In the inner circle, a 24-dimensional vector is counted up from the 4 location bins and 6 orientations. The two vectors are concatenated into a 48-dimensional descriptor.