# An InSAR Interference Fringe-Matching Algorithm Based on Mountain Branch Points
## 2 Methods
### Extraction of Mountain Line  
A monotonic relationship exists between the DEM and the interference fringes, allowing for the design of an interference fringe mountain range line extraction algorithm based on DEM mountain range line extraction [31; 32]. Edge detection is a common means for detecting line features in images. The main principle is to find the first-order derivative or gradient [33]. However, the gradient method allows only detecting boundaries with high and low phase values in interference fringe images and not those positioned directly in phase extremity regions. Figure 4a gives the function curve for the first-order derivative of a continuous function with a high value. The first-order derivative is not located directly at the high value of the function but rather in the region on either side of the high value of the function. In the interference fringe image, the lines of the mountain ranges we are interested in are represented as a collection of points with very large and very small phase values, the physical significance of which represents ridges and valleys, respectively. In the field of digital image processing, the detection of such thin lines that are darker or brighter  
\begin{table}
\begin{tabular}{c c c c} \hline \(\lambda\)**/(cm)** & **\(H\)/(m)** & **\(B\)/(m)** & **\(\alpha\)/(\({}^{\circ}\))** \\ \hline
2.075 & 1500 & 0.5 & 0 \\ \hline \end{tabular}
\end{table}
Table 1: Simulation parameters.  
Figure 3: Interference phase versus elevation values. **(a)** InSAR system geometric model. **(b)** Simulation of the relationship between the interference phase and elevation value.  
than the surrounding area is defined as ridge detection [34]. Ridge detection usually involves finding the first-order derivative over the zero point, or second-order derivative, for the image. Figure 4a similarly shows the function curve for the second-order derivative of a continuous function, and it can be seen that the extreme values of the second-order derivative of the function correspond to where the extreme values of the original function occur. Figure 4b simulates a ridge with a bright line, and Figure 4c,d show the effect after taking the first- and second-order derivatives of the image, respectively. The first-order derivatives are presented as two bright columns distributed on either side of the true ridge position. The bright line of the second-order derivative coincides with the original ridge at its location. The second-order derivative is the tool that provides the basis for the extraction of mountain range line features for interference fringes.  
In an interference fringe image, the ridge and valley lines correspond to the set of very large and very small phase value points, respectively, which also represent the curvature extrema of the surface on which the point is located. For a discrete two-dimensional digital image \(I\), the curvature of a point \(p(x,y)\) on the image can be represented by a Hessian matrix [35]:  
\[H=\begin{bmatrix}L_{xx}&L_{xy}\\ L_{yx}&L_{yy}\end{bmatrix} \tag{7}\]  
\(L_{xx}\), \(L_{xy}\), \(L_{yx}\) represent the second-order derivatives of the point. Considering that digital images inevitably have noise points, the second-order derivative of the image can be obtained by convolving the original image with a Gaussian function \(G(\alpha,\beta,\delta)\) having scale \(\delta\), as in Equation (8), where \(\alpha,\beta\) denote the combination of the derivatives of \(x,y\). Thus, a certain amount of noise can be filtered while acquiring the derivative of the image.  
\[\begin{cases}L_{\alpha\beta}=p(x,y)\otimes G(\alpha,\beta,\delta)&L_{\alpha \beta}\in\{L_{xx},L_{xy},L_{yx},L_{yy}\}\\ G(\alpha,\beta,\delta)=\frac{1}{2\pi\beta^{2}}e^{\frac{-(\alpha^{2}+\beta^{2}) }{2\beta^{2}}}\end{cases} \tag{8}\]  
Figure 4: Ridge detection in relation to edge detection. (**a**) Difference between the first derivative and second derivative. (**b**) Simulated ridge. (**c**) Edge detection. (**d**) Ridge detection.