# SAR Interferogram Filtering of Shearlet Domain Based on Interferometric Phase Statistics
## 2 Phase Filtering of Shearlet Domain Based on Interferometric Phase Statistics
### The Proposed Filter  
The shearlet transform in the time domain and frequency domain has a good localization property, so it can remove the noise while keeping the edge information. The shearlet transform is a linear operator, and the additive noise model is retained within the shearlet domain. Applying the shearlet transform to the real and imaginary parts in (5), we obtain the following  
\[ \begin{split} Sh(\cos(\theta_{z}))&=N_{c}Sh(\cos( \theta_{x}))+Sh(v_{r})\\ Sh(\sin(\theta_{z}))&=N_{c}Sh(\sin(\theta_{x})) +Sh(v_{i})\end{split}  \tag{8}\]  
where \(Sh\) represents the shearlet transform operation. We apply a soft thresholding method to recover the InSAR phase, which is similar to that given in [31]. The shearlet threshold function \(T(j,l)\) is defined as  
\[T(j,l)=k\varepsilon_{j,l}\delta_{n} \tag{9}\]  
\(k\) is a constant and equals to 3, 3, 4 when \(j=1\), 2, 3, \(\varepsilon_{j,l}\) denotes the average energy distribution of white noise in shearlet coefficient on scale \(j\) and direction \(l\). \(\delta_{n}\) is the standard deviation of noise [32]. Obviously, the key to determine the threshold is the noise standard deviation \(\hat{\sigma}_{n}\)[33]. \[\hat{\sigma}_{n}=\frac{median(|y_{i}|)}{0.6745}\;y_{i}\in\text{subband }HH_{1}, \tag{10}\]  
where \(HH_{1}\) is the high frequency sub-band wavelet coefficient in the first level. However, the estimate of \(\hat{\sigma}_{n}\) derived from (10), is rough and changes with wavelet functions. More importantly, the threshold defined by (10) generates poor results in low coherence regions. As demonstrated in Section 2.1, the noise standard deviation is a function of coherence and look number. Therefore, we modify the threshold filter to incorporate information contained in these two parameters. We thus propose to replace \(\hat{\sigma}_{n}\) with the phase standard deviation, which is more reasonable. The standard deviation of a given coherence and multi-look number can be calculated by numerical integration [30]. Correspondingly, the phase standard deviation map can be calculated pixel by pixel with the coherence map and multi-look factor through numerical integration. However, this process is time-consuming, and will be intolerable if the interferogram is large. As the multi-look number is an integer and the coherence is between 0 and 1, we can create a look-up table for phase  
Figure 2: Frequency supports for different values of \(a\) and \(s\) (\(a\) is the scale parameter, \(s\) is the shear parameter). standard deviations versus coherences and multi-look numbers, shown in Table 1. The calculation process of the phase standard deviation map in a given interferogram is then simplified to match the coherence and multi-look number in the look-up table at the pixel basis, and finally the interferogram noise standard deviation matrix \(\sigma\) is generated. For a given interferogram, phase standard deviation \(\hat{\sigma}_{n}\) varies with coherence. To solve the second problem, we use a robust estimator, the median, in case the fine-scale wavelet coefficients contain a small proportion of strong \"signals\" mixed with \"noise\". The whole process of the proposed method can be listed as follows. The flow chart is shown in Figure 3.  
1. The real and imaginary parts of InSAR interferogram are processed by the shearlet transform, which are then decomposed into multiscale and multidirectional frequency bands \(c(j,l)\), where \(j\) is scale index, \(l\) is direction index. 2. The threshold is then determined by the following procedure. * Create a phase standard deviation look-up table between different coherence and multi-look number, as shown in Table 1. * For a given interferogram, the standard deviation map \(\sigma\) can be generated pixel by pixel using the look-up table of the coherence and multi-look. * Median robust estimator \(\hat{\sigma}_{n}=median(\sigma)\) * Calculate the threshold \(T(j,l)=k\varepsilon_{j,l}\hat{\sigma}_{n}\), \(k=3,3,4\) for \(j=1,2,3\).