# SAR Interferogram Filtering of Shearlet Domain Based on Interferometric Phase Statistics
## 5 Discussion  
This paper presented a shearlet threshold filtering method for SAR interferograms based on the coherence and look number of the interference phase. The phase information and noise can be separated more easily in the shearlet domain. The standard deviation is a function of coherence and look number, so we modified the threshold filter to incorporate information contained in these two parameters.  
The proposed method was validated with both simulation data and real data. The simulated test shows that after applying the proposed model, RMSE values between true phases and the results of the Goldstein, the wavelet, the Lee and the proposed filters are 1.3715, 1.2196, 1.3439 and 1.0708 rad, respectively. The proposed filter got the closest results to the true phases in the simulation experiment. The real data results over the Etna volcano show that the residue reductions resulting from the Goldstein filter, wavelet filter and Lee filter are 45.73%, 71.87% and 76.61%, respectively, while the reduction resulting from the proposed filter is 92.09%. The real data results over Hong Kong show  
Figure 12: Deformation results of (**a1**) the Goldstein filter; (**b1**) the wavelet filter; (**c1**) the Lee filter; and (**d1**) the proposed filter; (**a2–d2**) Cross-section over the estimated deformation phase; the location of the cross-section is shown in (**a1–d1**). (Unit: rad).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Filter** & \begin{tabular}{c} **Positive** \\ **Residues** \\ \end{tabular} & \
\begin{tabular}{c} **Negative** \\ **Residues** \\ \end{tabular} &
\begin{tabular}{c} **Residue** \\ **Reduction (\%)** \\ \end{tabular} & **PSD (\(\times\)10\({}^{5}\))** & **SPD (\(\times\)10\({}^{5}\))** & **RMSE** \\ \hline Unfiltered & 9901 & 9905 & - & 6.1753 & 4.5503 & - \\ Goldstein & 3556 & 3559 & 64.08 & 3.4275 & 2.6937 & 0.1711 \\ Wavelet & 2248 & 2241 & 77.34 & 2.8216 & 2.3629 & 0.1499 \\ Lee & 2667 & 2671 & 73.04 & 3.3403 & 2.7315 & 0.1117 \\ Proposed & 1825 & 1826 & 81.56 & 2.5054 & 2.2158 & 0.0979 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Quantitative evaluation with real data over Hong Kong.  
that the residue reductions resulting from the Goldstein filter, wavelet filter, and Lee filter are 64.08%, 77.34%, and 73.04%, respectively, while that from the proposed filter is 90.45%. The deformation phase RMSE of the proposed method has the smallest value. Furthermore, we can see that interference fringes of the proposed filter is clearer and with lower noise level, especially in low coherence regions. This indicates that the proposed filter is significantly superior to the other two filters.  
The proposed filter is more consistent and robust because the shearlet filtering method has the time-frequency analysis capability and the filtering threshold is defined with the coherence and look number of the interference phase. Traditional wavelet filtering methods apply the wavelet coefficient, a rough estimate, to define the threshold in the interferogram, getting poor results in low coherence regions. Furthermore, shearlets' mathematical properties can form a tight frame of well-localized waveforms at various scales and directions, and are optimally sparse in representing images with edges. These properties can maintain the edge information. In addition, shearlets have the ability to deal with nonstationary signals. The premise of the Goldstein filter is the unchanged phase frequency or stable statistical properties. Actually, the observed interferometric phase usually contains both stationary and nonstationary components. So a more sophisticated algorithm, such as the shearlet transforms, should be applied.  
However, the proposed method also has drawbacks, such as the decomposition levels. The level of shearlet decomposition has great influence on denoising. We will therefore research the determination of decomposition levels in the future.