# An InSAR Interference Fringe-Matching Algorithm Based on Mountain Branch Points
## 2 Methods
### Extraction of Mountain Branch Points  
To accomplish the image matching task that follows, it is necessary to convert the extracted mountain range line features into point features. As shown in Figure 6b, the mountain range line features have multiple branches within each line. These branch points correspond to the bifurcation points of the real mountain range and indicate that the topography of the real mountain range changes at that point. Branch points with different distributions represent different topographic features, while interference fringes for the same mountain region always extract the same mountain branch points. Therefore, the interference fringes obtained in real time correspond to the branch points of the mountain range extracted from the interference fringes in the inversion of the base terrain library. Compared with the keypoints extracted using traditional matching algorithms, these branch points are highly specific and are used as feature points for subsequent image matching. For a binary image with a line feature width of 1 pixel within the image, the branch point is defined as shown in Figure 7, using a 3*3 window to find the branch points that match the conditions. A total of four possible structures are shown in the diagram, and it is worth noting that the structures shown in the diagram need to be rotated by \(90^{\circ},180^{\circ}\), and \(270^{\circ}\) during implementation of the algorithm; because the mountains in the interference fringe image have many orientations, rotations are required for each structure, i.e., there is a total of 16 possible scenarios. Figure 6: Mountain line extraction results. (**a**) Interference fringe demonstration area. (**b**) Results of interference fringe mountain line extraction. Figure 7: Branch point detection template. In the demonstration area of interference fringes with phase mutation shown in Figure 9a, mountain branch points were detected separately. The red circles represent false detection points that are about to be eliminated, while the blue circles represent correct mountain branch points. Figure 10a shows the detection result of the valley line branch points, and Figure 10b shows the detection result of the ridge line branch points. It can be observed from the red circles in the figure that they are all located near the phase mutation region. It can be proven that by setting a threshold, false detection points caused by phase mutation can be eliminated completely. In addition, the correctly detected mountain branch points are distributed relatively evenly, which satisfies the requirements of subsequent feature matching. Figure 10: Verification of branch point detection algorithm. (**a**) Detection results of valley branch points of interference fringe with phase mutation. (**b**) Detection results of ridge branch points of interference fringe with phase mutation. Figure 9: Analysis of the influence of phase winding. (**a**) The demonstration area of interference fringes with phase winding. (**b**) Interference fringe eigenvalue map. (**c**) Slice 1, with calculation of the absolute value. (**d**) Slice 2.  
branch point descriptors have been extracted, a normalization operation is performed to increase the efficiency of matching. \[radius=\frac{3\sqrt{2}\delta(d+1)+1}{2} \tag{9}\]  
This improved descriptor has two advantages. Based on the analysis above, SIFT extracts descriptors on a gradient basis, which is applicable to corner points but not to points on ridge lines and valley lines and should be replaced with the eigenvalue as well as the orientation of the Hessian matrix when describing the surrounding pixels. The eigenvalue and eigenvector maps of the interference fringes have already been calculated in Section 2.3 and do not need to be calculated again in this section, thus saving computing time. To quantitatively compare the advantages of the improved descriptors, the following experiments were carried out. Figure 13 shows the branch points detected in the real interference fringes as well as in the reference interference fringes.