# A Novel Mosaic Method for Spaceborne ScanSAR Images Based on Homography Matrix Compensation
## 3 Traditional Feature-Based Mosaic Process for SAR Images
### Homography Matrix Compensation  
To improve the efficiency of the mosaic, we initially tried to downsample the images for stitching and upsample the stitched image to the original size. Although multilooking or image downsampling can greatly reduce the amount of data and effectively speed up image mosaicking [17], the problem of ScanSAR with a lower resolution than stripmap SAR worsens. Neither of these methods were helpful in maintaining the original resolution or as many image details as possible. One aspect considered significant was that the image projection accuracy was directly determined by the homography matrix. In that case, we switched our attention from image scaling to homography matrix scaling and proposed a novel method. The extracted overlapping areas were downsampled for registration, so keypoints would be detected and matched quickly. After being calculated based on matched keypoints, the homography matrix should be compensated, which depends on the multiple of downsampling. Then, the compensated matrix could be applied to the projection of the original-size images. In that case, the resolution could be preserved successfully in the final mosaic result. To ensure the accuracy of projection, the scaling compensation for the homography matrix should be precisely deduced. As shown in Figure 5, the rows and columns of the reference image \(I_{\alpha}\) and the projected image \(I_{\beta}\) were scaled to \(1/n\). The homography matrix used to complete this step is as follows:  
\[\mathbf{H}_{1}=\text{diag}(1/n,1/n,1) \tag{8}\]  
the homography matrix \(\mathbf{H}\) used for the projection of downsampled image \(I_{\beta}\) to downsampled image \(I_{\alpha}\) was calculated as shown in Equation (1). After being projected based on \(\mathbf{H}\), the rows and columns should be scaled by \(n\) times to approximate the projection of the original-size images. The homography matrix used in amplification is:  
\[\mathbf{H}_{2}=\text{diag}(n,n,1) \tag{9}\]  
\(\left(x_{\beta},y_{\beta}\right)\) are the coordinates of any pixel in image \(I_{\beta}\), \(\left(x^{\prime}{}_{\beta},y^{\prime}{}_{\beta}\right)\) are the coordinates of any pixel in downsampled image \(I_{\beta}\), \(\left(x^{\prime\prime}{}_{\beta},y^{\prime\prime}{}_{\beta}\right)\) are the coordinates of any pixel in projected downsampled image \(I_{\beta}\), and \(\left(x^{\prime\prime}{}_{\beta},y^{\prime\prime}{}_{\beta}\right)\) are the coordinates of any pixel in projected original-size image \(I_{\beta}\). Thus, the relationship among these coordinates can be expressed as:  
\[\left[ \begin{array}{c}x^{\prime}{}_{\beta}\\ y^{\prime\prime}{}_{\beta}\\ 1\end{array} \right]=\mathbf{H}_{1}\left[ \begin{array}{c}x_{\beta}\\ y_{\beta}\\ 1\end{array} \right] \tag{10}\]  
\[\left[ \begin{array}{c}x^{\prime\prime}{}_{\beta}\\ y^{\prime\prime}{}_{\beta}\\ 1\end{array} \right]=\mathbf{H}\left[ \begin{array}{c}x^{\prime}{}_{\beta}\\ y^{\prime}{}_{\beta}\\ 1\end{array} \right] \tag{11}\]  
\[\left[ \begin{array}{c}x^{\prime\prime}{}_{\beta}\\ y^{\prime\prime}{}_{\beta}\\ 1\end{array} \right]=\mathbf{H}_{2}\left[ \begin{array}{c}x^{\prime\prime}{}_{ \beta}\\ y^{\prime\prime}{}_{\beta}\\ 1\end{array} \right]=\mathbf{H}_{2}\mathbf{H}\left[ \begin{array}{c}x^{\prime}{} _{\beta}\\ y^{\prime}{}_{\beta}\\ 1\end{array} \right]=\mathbf{H}_{2}\mathbf{H}\mathbf{H}_{1}\left[ \begin{array}[] {c}x_{\beta}\\ y_{\beta}\\ 1\end{array} \right] \tag{12}\]  
Figure 4: Overlapping area coarse extraction. Therefore, the homography matrix used to project the original image \(I_{\beta}\) to the original image \(I_{a}\) can be shown as:  
\[ \begin{split}&\mathbf{H}_{0}=\mathbf{H}_{2}\mathbf{H}\mathbf{H}_{1} \\ &=\left[\begin{array}{ccc}n&0&0\\ 0&n&0\\ 0&0&1\end{array}\right]\left[\begin{array}{ccc}a_{x}&b_{x}&c_{x}\\ b_{y}&a_{y}&c_{y}\\ 0&0&1\end{array}\right]\left[\begin{array}{ccc}1/n&0&0\\ 0&1/n&0\\ 0&0&1\end{array}\right]\\ &=\left[\begin{array}{ccc}a_{x}&b_{x}&nc_{x}\\ b_{y}&a_{y}&nc_{y}\\ 0&0&1\end{array}\right]\end{split}