# SAR Interferogram Filtering of Shearlet Domain Based on Interferometric Phase Statistics
## 3 Validation with Simulation Data  
To examine the filtering performance of the proposed method, the simulation data (592 \(\times\) 592 pixels) are used in this section. The DEM is first simulated by the multifractal technology. Then based on the European Remote Sensing (ERS) 1/2 imaging geometric parameters and a given vertical baseline length, a noise-free phase is simulated by the DEM [13], see Figure 4a. The coherence map shown in Figure 4b is simulated based on three principle errors: the thermal noise, geometric decorrelation and temporal decorrelation. On the basis of the simulated coherence map and a multi-look number, the phase noise can be simulated. Finally, the noisy phase in Figure 4c is generated by adding the phase noise to the noise-free phase.  
Figure 3: Flow chart of the proposed method.  
The results obtained by the Goldstein filter [11], wavelet filter and Lee filter [6] methods are also compared (Figure 5). The main parameters of the Goldstein filter are as follows: the filter parameters \(\alpha\) is 0.5, the patch size is 32 \(\times\) 32 pixels, the overlaps between patches is 15 pixels to avoid discontinuities. The scale parameter of the wavelet filter is 3. The main parameters of the proposed method are as follows: scale parameter of shearlet is 3, and the shear parameter is (1, 1, 2) for \(j=1,2,3\) scale.  
Figure 4: (**a**) Simulated noise-free phase; (**b**) corresponding coherence map; (**c**) simulated noisy phase. A and B, labeled by white rectangles, indicates the enlarged areas.  
Figure 5: Filtered results of simulated data with (**a1**) Goldstein; (**b1**) wavelet; (**c1**) Lee; (**d1**) the proposed method; (**a2–d2**) Close-up of results of area A in Figure 4; (**a3–d3**) Close-up of results of area B in Figure 4; the black line in (**a1–d1**) indicates the extracted section. (Unit: rad).  
From the coarse view of Figure 5(a1-d1), we can observe that noise is suppressed in all filtered phases. For clarity and detailed comparison, two areas, A and B, labeled by white rectangles in Figure 4, are enlarged in Figure 5(a2-d2) and Figure 5(a3-d3), respectively.  
The results of the [PERSON] filter (Figure 5(a1-a3)) and [PERSON] filter (Figure 5(c1-c3)) show large numbers of phase noise in almost all of the whole phase image, especially in areas of low coherence and high fringe density. The wavelet filter performs better than the [PERSON] and Lee methods. As Figure 5(b1-b3) shows, in high coherence areas, most noises have been filtered out. However, in some low coherence areas, there is still some phase noise. The result of the proposed filter is smooth and with all details, as most noise is filtered out in regions of low coherence, improving the spatial resolution, see Figure 5d1-d3. It is clear that the filtered phases of the proposed filter in Figure 5(d1) are the closest to the true phases in Figure 4a.  
For further analysis on filtering effect, quantitative assessments are calculated in terms of the root mean squared error (RMSE). RMSE values between true phases and filtered phases of the Goldstein, wavelet, Lee filter and the proposed filters are 1.3715, 1.2196, 1.3439 and 1.0708 rad respectively. Obviously, the proposed filter is more consistent and robust.  
A small cross-section marked by the black line in Figure 5(a1-d1) is extracted to compare the filtering effect and the results are shown in Figure 6. Clearly, the filtered phase values of the proposed filter appear less noisy than those of the other three filters. Furthermore, the proposed filter gets smoother curves while effectively maintaining edge details. RMSE values between true phases and filtered phases of the Goldstein, wavelet, Lee and the proposed filters over the cross-section are 0.6457, 0.3607, 0.4528 and 0.2859 rad respectively.