Review of interferometric synthetic aperture sonar interferometric phase filtering methods

Interferometric synthetic aperture sonar (InSAS) is a novel three dimensional underwater mapping system, which is widely used in various fields. Interferometric phase filtering is a crucial process in InSAS signal processing. The study of interferogram filtering method has great significance to improve the signal-to-noise ratio (SNR) of interferogram, reduce the difficulty of phase unwrapping, and improve the accuracy of digital elevation maps. Firstly, this article introduces the sources of interferometric phase noise. Secondly, it illustrates the difference between real domain filter and complex domain filter by simulation experiment. Thirdly, the various methods of InSAS interferogram filtering are systematically studied, which can be classified into spatial domain filtering and transform domain filtering. The transform domain filtering can be further divided into frequency domain and wavelet transform filtering. Moreover, the advantages and disadvantages of various methods are analyzed theoretically. Finally, numerical simulation and real-data processing experiments are performed, and the advantages and shortcomings of several methods are compared using residue numbers, root mean square error (RMSE), and CPU time as evaluation criteria.


Introduction
The concept of interferometric synthetic aperture sonar (InSAS) was first introduced in 1983 by Spiess and Anderson 1 in their patent Wide Swath Precision Echo Sounder.InSAS became a very active research hotspot in the field of hydroacoustic signal processing in the late 1990s.Examples of its application include the synthetic aperture sonar (SAS) system by the National Defense Center of Excellence for Research in Ocean Sciences in the United States, 2 the synthetic aperture mapping and imaging SAS system in Europe, 3 kiwi-SAS system in New Zealand, 4 HISAS system in Norway, 5,6 InSAS system by the Naval University of Engineering in China, 7 and others. 8,91][12] Compared with SAS, [13][14][15][16][17][18][19] its advantage is that it has the ability of bathymetric.InSAS adds one or more receiver arrays based on SAS to obtain the height value of targets or the deformation of the seafloor by using the phase difference.InSAS combines the advantages of SAS resolution is independent of the distance and operating frequency, and the high accuracy of interferometric bathymetry; therefore, it is widely used in military and civilian fields. 20,21Its primary military applications include detection of mines and unexploded ordnance and reconnaissance of important sea areas, while its main civil applications include salvaging wrecks and sunken objects, 22 mapping underwater topographies, surveying submarine deposits, and finding submarine fiber optic cables. 23he InSAS workflow primarily consists of synthetic aperture imaging, [24][25][26][27][28][29] complex image registration, 30,31 interferometric map generation, unflattening, phase filtering, phase unwrapping, 32,33 and digital elevation model reconstruction.The interferometric phase map obtained after complex image registration and conjugate multiplication is generally not used directly for phase unwrapping because the interferogram SNR is low at this time, there is a lot of noise in the image, and the phase in the interferogram cannot reflect the real topography.If phase unwrapping is conducted at this time, the result will deviate substantially from the real phase value; especially, phase unwrapping cannot be developed at all under severe noise conditions.Therefore, interferogram filtering before phase unwrapping is particularly important.The purpose of interferometric phase filtering is to improve the SNR of interferogram, which is beneficial for phase unwrapping and the accuracy of digital elevation model.An ideal interferometric phase filtering algorithm should reduce the noise while maintaining details and edge information as much as possible, such as the interferometric fringe edge information in the image.
This paper provides an overview of interferometric phase filtering in InSAS.The main contribution of this paper is to illustrate the phase filtering should be carried out in complex domain, summarizing the existing methods of phase filtering and analyzing their advantages and disadvantages.
The remainder of this paper is organized as follows.The sources of phase noise is introduced in Section ''Sources of phase noise.''Section ''Real domain filter and complex domain filter'' discusses the difference of real domain filter and complex domain filter and emphasizes the phase filter should be proceeded in complex domain.Section ''Interferogram filtering methods'' reviews the interferogram filtering methods in spatial domain and transform domain.Evaluation criteria for interferometric phase filtering are given in Section ''Evaluation criteria for interferometric phase filtering.''In Section ''Experimental results and analysis,'' some interferogram filtering methods are used to illustrate their noise removing effects by simulation experiment and real InSAS data experiment.

Sources of phase noise
The sources of phase noise in InSAS are manifold, and it can be assumed that the interferogram contains mainly the following types of noise: (1) Ambient noise.Underwater acoustic wave propagation is affected by the water background noise, and the SNR of the signal received by the receiver is reduced owing to the multipath effect.(2) System thermal noise.This corresponds mainly to the noise generated by the sonar system during the processes of transmitting and receiving signals as well as signal transmission.This noise reduces the SNR to a certain extent and decreases the correlation between two complex images.(3) Speckle noise.Such noise is inherent to interferometric imaging systems because there are many scattering points in a resolution unit of InSAS system.The echoes from these scattering points overlap with each other, thereby leading to the generation of speckle noise.(4) Complex image registration noise.In the complex image registration process, due to the poor selection of control points or the registration area locates in the low quality region caused by shadow or because geometric transformation and resampling methods are inappropriate, the registration results cannot achieve high accuracy and in some places, the registration error will be large, leading to the generation of low quality areas in an interferogram, which manifest as noise in an interferometric fringe image.(5) Decorrelation noise.Unlike the interferometric synthetic aperture radar, as InSAS adopts a dual-antenna operation mode, both master and slave images can be obtained in one measurement.Therefore, in InSAS, the temporal decorrelation noise can be ignored; it mainly contains the spatial decorrelation noise.
For the abovementioned types of noise, the generation of first two types cannot be controlled artificially.The speckle noise can be reduced by some speckle suppression algorithms, but it cannot be eliminated completely.The registration noise can be reduced by improving the accuracy of registration algorithm, and the spatial decorrelation noise can be reduced by improving the platform stability and the motion compensation.

Real domain filter and complex domain filter
Phase itself is not a signal, it is rather a property of a signal.The interferometric complex image obtained after the conjugate multiplication of master and slave image includes both amplitude and phase information.Therefore it is the complex signal itself that should be filtered and not the phase.In other words, interferometric phase filtering should be carried out in the complex domain before the phase is extracted.That is, instead of directly filtering the phase value, the real and imaginary parts should be filtered separately before extract the phase value.In this paper, we take the simplest mean filtering as an example to illustrate the difference between real domain filter and complex domain filter.
The real domain filter is performed directly using the interferometric phase, which can be defined by the conventional formula where k corresponds to the filtering window size, u(i, j) corresponds to the phase value of the pixel (i, j) inside the window, and u(m, n) is the filtered phase at the pixel (m, n), which is the center of the window.Smoothing the phase values directly will cause the interferogram to fail in maintaining a steep shape at the stripe jumps.This is because in the region of interferogram fringe jumps, the phase changes from p to Àp.If the filter window is located in those region, it will contain both types of pixels, ones that have positive phase values close to p and others that have negative phase values close to Àp.After summing and then averaging the phase values directly, the positive and negative phase values will cancel with each other.Therefore, the phase value of the pixel after filter may be close to 0, which deviates from its true value.
The complex domain filter is performed as the following equation where (i, j) is the pixels lies in window size and (m, n) is the center pixel of k 3 k windows.In equation ( 2), the phase data first must be mapped to vectors exp (ju i, j ) in the complex domain.Then, these vectors can be averaged and the resulting phase extracted by the arctangent operator.In fact, the right hand side of equation ( 2) is first filtered separately for the numerator and denominator, and the averaging operation in k 3 k windows is eliminated.
A simple simulation is performed to illustrate the problem.Figure 1(a) shows a simple example consisting of the wrapped phase of a tilted plane without noise, and Figure 1(b) shows the phase corrupted by Gaussian noise.Figure 1(c) presents the result of direct filtering of the interferometric phase according to equation (1).The fringes, or boundaries between the Àp and p have been blurred.This is guaranteed to produce a disastrously incorrect result when the phase is unwrapped.The correct averaging operation defined by equation ( 2) produces the result of Figure 1(d), which smoothes the real and imaginary parts separately, and then obtaining the phase by the arctangent operator.Comparing Figure 1(c) with 1(d), it can be seen that the interferometric fringes in Figure 1(c) gradually varied from p to Àp, but not changes sharply.On the contrary, the fringes in Figure 1(d) remain abrupt and the edges are more smother.Figure 2(a) to (d) show plots of a cross section through the four phase images of Figure 1(a) to (d). Figure 2(a) shows the sawtooth plot of a slice through the wrapped phase of Figure 1(a), and Figure 2(b) shows the Gaussian noise corrupted result.When the real domain filter is applied, the result is Figure 2(c).This plot is no longer the wrapped phase of a line but rather the wrapped phase of a skewed sawtooth function.It cannot be unwrapped because it already is.Figure 2(d) gives the result of the application of the correct filter defined by equation ( 2).As in Figure 2(a) it is the wrapped phase of a line.It can be clearly seen from the profile of Figure 2(c) and (d) that the results of the two filtering methods are obviously different.Thus, the filtering should be performed in the complex domain before extracting the phase.

Interferogram filtering methods
The idea of interferometric phase filtering was first proposed by Goldstein et al. 34 They used a phase estimation process to reduce the number of residuals in the phase unwrapping, thereby improving the reliability of phase unwrapping.This was the original form of interferometric phase reduction, which further evolved into phase filtering.Li and Goldstein 35 developed a mathematical model of phase noise and triggered a hot research of phase noise models.Since then, the mathematical model of phase noise has become a separate research topic.According to the difference of signal processing domains, interferometric phase filtering methods can be divided into two main categories: spatial domain filtering and transform domain filtering.
Among the spatial domain methods, the simplest are circular period mean filtering 36 and circular median filtering. 37These methods employ vectors for processing, which avoid the 2p period ambiguity problem.It works well in case of Gaussian additive noise, however, their edge holding ability is inadequate when the terrain is undulating.Lee et al. 38 proposed an adaptive interferometric phase filtering method based on the local slope.This method involves first calculating the phase variance of 16 directional windows and then finding the directional window with the largest variance, followed by filtering along that directional window.The method has the advantage of directional adaptive selection, but it is computationally intensive owing to the need of pre-estimated slope values and local phase unwrapping for each sliding filter window.Xu et al. 39 proposed an interferometric phase diagram K-means filtering method with the advantage that the window size can be adjusted by the variance of the current region.However, the local variance becomes large due to steep terrain and therefore, it will have less noise suppression.Moreover, regions with large noise variances are not filtered well.Wu et al. 40 improved the method by Lee and proposed a filtering algorithm to find the normal direction of the stripe through local frequency estimation and then construct a directional filter window accordingly with the window direction being perpendicular to the stripe normal direction.In contrast to Lee's method, this algorithm does not require local phase unwrapping and the window direction is no longer limited to 16 directions, making it much faster than Lee filtering.Yu et al. 41 introduced an adaptive contoured window filter, which is determined by tracing along the local fringe orientation.And window sizes are determined adaptively by the fringe density.Yin et al. 42 present a new InSAR phase filtering method based on optimal integration, which uses statistical method to determine the number of windows used for the filtering.This method determined directional windows by the principle of minimum variance of local area, which increases the robustness of Lee filter, but its computations are relatively complex than Lee filter.Guo et al. 43 proposes a new adaptive noise suppressing method for interferometric phase images.Due to the minimum phase difference, the proposed method selects the filtering samples that obey the independent and identically distributed assumption more accurately, thus improving the filtering performance.But it does not consider the fringe frequency.Fu et al. 44 proposed a directionally adaptive filter, which has a filter window whose direction is able to vary continuously with the fringe direction.After determine the window size and direction, median or mean filtering is carried out, thus it does not realize adaptive intensity filtering.Chao et al. 45 presents a refined filter based on the Lee adaptive complex filter and the improved sigma filter, adaptively filter the interferometric phase according to the local noise level to minimize the loss of signal or a particular pattern of fringes.Li et al. 46 presented a coherence-weighted optimum interferometric phase filtering method, which estimated the coherence by reference DEM and a group of directional windows.Yi et al. 47 proposed an adaptive interferogram filtering method, in this method, the kernel pixel's iteration formula was weighted by coherence and anisotropic gradients between surrounding pixels of the window and the kernel pixel.Xue and Feng 48 proposed an adaptive complex phase filtering method based on local topographic phase compensation and anisotropic Gaussian filtering (AGF).The topography induced phase is approximately measured by local frequency estimation and removed from the original phase to eliminate the effect of the terrain topography.Herein, the time complexity is high because of the need to estimate the local terrain phase and parameters of the adaptive AGF.Li et al. 49 proposed an adaptive slope compensated interferogram filtering method, the window size is selected adaptively according to the correlation coefficient and variance, and good results are achieved.Yang et al. 50roposed an interferogram filtering method with multilayer feature fusion neural networks.Huang et al. 51 proposed an adaptive interferometric phase filtering method combined with quality maps, which adaptively determines the filter window size based on the correlation coefficient map, achieving good results and fast processing speeds.Among transform domain filter methods, Goldstein and Werner 52 proposed a method to filter interferometric phase noise in the frequency domain.The method involves Fourier transform on complex values to obtain a power spectrum, which is smoothed via an exponential operation, and a denoised interferogram is subsequently obtained via an inverse Fourier transform.The Goldstein filtering algorithm is simple to implement, fast, and has a good filtering effect in regions of high SNR.However, as the frequency domain weighting function in the algorithm takes the form of a fixed power exponent, the way of selecting the power exponent value has a great impact on the filtering effect.In 2003, based on the original Goldstein filtering algorithm, Baran et al. 53 introduced a correlation coefficient control power exponent to construct a frequency domain weighting function using the correlation coefficients between image pairs.Zhu et al. 54 proposed a filter based on local spatial frequency estimation, which can adjust the parameters according to terrain changes and can effectively overcome the shortcomings of traditional multi-visual filters that destroy dense interferometric fringes and can therefore achieve better experimental results.Sun et al. 55 proposed an improved Goldstein adaptive interferogram filtering method based on SNR, which uses a local SNR to determine the filtering parameters of Goldstein, with strong filtering for low SNR regions and weak filtering for high SNR regions.Yan et al. 56 proposed an improved Goldstein filtering method by constructing adaptive frequency domain weighting coefficients to replace the original artificially-set empirical values and by calculating their power exponents using local correlation coefficients, which solved the over-filtering or under-filtering problem caused by consistently processing the original algorithm on the face of complex interferometric fringe maps.Suo et al. 57 proposed an improved InSAR phase filter in frequency domain.It uses different window sizes to suppress the phase noise, and the window size is determined by the coherence of the central processing pixel.Meanwhile it uses fringe frequency compensation to eliminate the effect of the terrain topography.Feng et al. 58 improved the Goldstein method by reducing the effect of noise on streak frequency estimation using variable window pre-filtering before local frequency estimation.Thereafter, local stripe frequency estimation was performed, which was followed by Goldstein filtering.Notably, some methods combine frequency domain filtering with other methods, such as empirical modal decomposition 59 and the regional growth method. 60eanwhile, the wavelet analysis is also a transform domain method.Lopez-Martinez and Fabregas 61 proposed a wavelet domain noise filtering method; it takes the advantages of the coefficient characteristics of the complex interferogram in the wavelet domain, which can better maintain the edge details while filtering.He and Wang 62 combined the wavelet transform method and median filtering method to consider the directionality of high frequency coefficients and adopted different filtering processes for the edges and nonedges of the high frequency detail part but their filter window size was fixed.Jin et al. 63 proposed vector-separated wavelet soft-threshold filtering.Li and Zhu 64 proposed two stages interferogram filtering based on a SNR threshold judgment, in which the ratios larger than the threshold are processed in two stages using two discrete wavelet transforms or static wavelet transforms; however, no specific threshold selection method is given.Fan et al. 65 used a dual-tree complex wavelet for interferometric filtering, which has a strong ability to suppress interferogram noise and retain fringes and detail information of the interferogram to a large extent.Abdallah and Abdelfattah 66 proposed wavelet interferometric phase filtering based on coherence maps.Li and Ren 67 proposed an adaptive weighted median filtering method for noise coefficients after the wavelet transform, which can remove noise adaptively.He et al. 68 proposed a phase filtering algorithm by combining a shear-wave transform and the interferometric phase standard deviation.The algorithm combines the interferometric phase statistics with shear-wave threshold filtering and corrects the filter threshold using the phase standard deviation to improve the filtering effect.The disadvantage of wavelet transform filtering is that interferometric phase statistics are not taken into account, resulting in unsatisfactory filtering effects in low coherence regions, and that the wavelet transform does not optimally represent the image.
0][71] However, its biggest drawback is that the process is complex and it requires circular iterations, which takes a long time, and it is difficult to use for real-time processing, although it can be used for post-processing.
Notably, interferometric phase filtering has also been performed using morphological methods. 72Zhang and Li 73 proposed an improved morphological interferogram filtering method to optimize the structural element selection for preserving detailed information of the phase by improving an adaptive gradient estimation method.However, the edge direction from local gradient estimation may be affected by noise and pseudo edges may appear, and the time consumption is also relatively large.
Some scholars also combined two or more filter methods together by using their advantages, such as wavelet-Wiener filter 74 and contoured median and Goldstein two-step filter method. 75For the previous method, the selection of the wavelet threshold needs intelligent decision.For the latter method, it introduces the pseudo-coherence of interferogram to improve the adaptiveness of the Goldstein filter, but the contour lines extraction in the strong noise area needs further study.

Evaluation criteria for interferometric phase filtering
The purpose of InSAS interferometric phase filtering is two-fold: first, to remove noise from the interferometric phase map effectively and second, to try maintaining the detail information (i.e. to enhance or at least maintain the sharpness of interferometric fringes).In this paper, the advantages and disadvantages of the denoising algorithm are mainly evaluated based on two aspects: denoising ability and edge preserving capacity.The following evaluation criteria are commonly used for interferometric phase filtering.

Number of residual points
In the InSAS interferogram, the residue point is defined as follows where W is mod 2p operation, u is the wrapped phase, if the value of q is not 0, the pixel (i, j) is called residue point, which is shown in Figure 3.
A high number of residual points indicates that the interferometric phase diagram is seriously contaminated by noise; therefore, the purpose of phase filtering is to reduce the number of residual points as much as possible.However, this number should not be too small because if the filtering is extensive, then the number of residual points is indeed less but the detail information (such as information on small target points) may be lost and it may be difficult to distinguish again in case of another excessive filtering.

Root mean square error
Root mean square error (RMSE) can be used to evaluate the deviation of the interferogram before and after filtering and is defined as follows: where u 0 is the ideal interferometric phase value without noise and u is the interferometric phase value after filtering.The smaller the RMSE after filtering, the better the denoising ability of the filtering method.

Experimental results and analysis
In this section, some interferogram denoising methods were used to illustrate their noise removing effects by simulation experiment and real InSAS data experiment.All the experiments were carried out in software MATLAB 2021a and the CPU is Intel Xeon Gold 128R with @2.1 GHz and @2.1 GHz.

Simulation experiment
The ideal simulation interferogram without noise is given in Figure 4(a), we add two different levels of noise to Figure 4(a), which is shown in Figure 4(b).In Figure 4(b), it can be seen that the noise in the bottom half is stronger than the noise in the top half.The denoising results using pivoted mean filter, Lee filter, Goldstein filter, PDE filter, and quality map combined filter were given from Figure 4(c) to (g), respectively.From Figure 4(c), it can be seen clearly that the noise removing ability of pivoted mean filter is high in fringe sparse domain, the fringe clarity after filter is increased dramatically, however, the fringes are overlapped in the dense area.This is because the noise suppression of pivoted mean filter is with the cost of spatial resolution loss.When the spatial resolution is reduced to a certain extent, the phase difference between adjacent pixels in the fringe dense region is greater than p, which resulting in overlap.The result of Lee filter (see Figure 4(d)) is better than the pivoted mean filter, especially in the fringe dense regions, which is because the windows of Lee filter have 16 directions.Due to the Goldstein filter (see Figure 4(e)) used the same coefficient in the whole area, the filter effect is more obvious in the area of lower phase noise, while it is not obvious in the strong phase noise region.The result of PDE filter (see Figure 4(f)) has clearer fringe edge due to it is based on anisotropic nonlinear diffusion which can change the filtering strength self-adaptively and it has the advantage of preserving details.In the result of quality map combined method (see Figure 4   In order to intuitively see the effect of different filtering methods, the residue maps of those filtering methods are given in Figure 5(a) to (f), which correspond to Figure 4(b) to (g), respectively.From Figure 5, we can see the distribution of residue numbers about different methods, which can reflect the noise removing capability.
Aiming to compare the results of those methods quantitatively, we select residue numbers, RMSE, and computing time as the criteria, which is given in Table 1.From Table 1, we can see that the residue numbers and RMSE of those method, which is consistent with the results in Figures 4 and 5.

Real data experiment
The real InSAS data was obtained in Qiandao Lake located in Zhejiang province in 2010.The raw interferogram with noise of real InSAS data is shown in Figure 6(a), Figure 6(b) to (f) are the noise removing results of pivoted mean filter, Lee filter, Goldstein filter, PDE filter, and quality map combined filter, respectively.
From Figure 6, we can see that all these methods have achieved good noise reduction effect, but it is very difficult to find the difference intuitively just from Figure 6.So we select the cross section of the Figure 6, which are given in Figure 7.The smoother the lines in Figure 7, the better the filtering effect.From Figure 7, it can be seen that the result of quality map combined filter has the more smoother line, which indicated that it has the better denoise capability.The residue numbers of those methods are given in Table 2.Because of there is no raw interferogram without noise, so the Table 2 has not shown the RMSE result, which is different with simulation experiment in Table 1.Compare the evaluation criteria data in Table 2 and Table 1, it can be seen that all the results of those methods are consistent.

Conclusions
Interferometric phase filtering is a crucial step for InSAS.It is of great significance for phase unwrapping and digital elevation maps.Other than increasing the SNR of interferogram, phase filtering is useful for reducing the number of residues, which can make the phase unwrapping process considerably easier and improve the accuracy of digital elevation maps.Interferometric phase filtering method has always been a research hotspot for scholars at home and abroad.The purposes of interferometric phase filtering are remove the phase noise as soon as possible and maintain details information.In this article, firstly, the development history of InSAS and its work flow are briefly introduced, and the importance of phase filtering is emphasized.Secondly, the familiar sources of phase noise is analyzed, there are mainly five types of phase noise, such as ambient noise, system thermal noise, speckle noise, complex image registration noise, and decorrelation noise.Thirdly, we described the difference between real domain filter and complex domain filter by using simulation experiment.It is declared that the filter should be carried out in complex domain rather than in real domain, then extract the phase information.Then, we detailedly dicussed the problem of interferometric phase filtering in InSAS.We survey and summarize the most commonly of the phase noise removing methods of domestic and foreign, and classified them into two main categories that is spatial domain and transform domain filtering.Meanwhile, the advantages and disadvantages of these methods are analyzed from usage scenario and time efficiency.Four commonly used evaluation criteria of filtering effect are given.Lastly, we choose pivoted mean filter, Lee filter, Goldstein filter, PDE filter, and quality map combined filter as the representative for simulation experiment and real data experiment.The experiments results and quantitative evaluation indicators are given to illustrate their filtering effect.
Noise removing ability and time cost is a pair of contradictions, and it is very difficult to determine the filter strength accurately.In all filtering methods, it is difficult to find one method which has the best filtering result and lest computing time at the same time.In the future, we will devote for the better filtering algorithm.The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the Natural Science

Figure 1 .
Figure 1.Comparison of real domain filter and complex domain filter: (a) simulated interferogram without noise, (b) phase with Gaussian noise, (c) result of real domain filter using equation (1), and (d) result of complex domain filter using equation (2).

Figure 2 .
Figure 2. Cross section of Figure 1: (a) the phase of a line with positive slope, (b) phase with Gaussian noise, (c) result of real domain filter using equation (1), and (d) result of complex domain filter using equation (2).
(g)), it is more clearer in the strong noise area in the bottom of this figure.

Figure 4 .
Figure 4. Results of different filtering methods using simulation data: (a) simulated interferogram without noise, (b) noised interferogram, (c) result of pivoted mean filter, (d) result of Lee filter, (e) result of Goldstein filter, (f) result of PDE filter, and (g) result of quality map combined filter.

Figure 5 .
Figure 5. Residue maps of different filtering methods: (a) residue map of noised interferogram, (b) residue map of pivoted mean filter, (c) residue map of Lee filter, (d) residue map of Goldstein filter, (e) residue map of PDE filter, and (f) residue map of quality map combined filter.

Figure 6 .
Figure 6.Results of different filtering methods using real InSAS data: (a) original interferogram, (b) result of pivoted mean filter, (c) result of Lee filter, (d) result of Goldstein filter, (e) result of PDE filter, and (f) result of quality map combined filter.

Figure 7 .
Figure 7. Cross section of the results in Figure 6: (a) original interferogram, (b) result of pivoted mean filter, (c) result of Lee filter, (d) result of Goldstein filter, (e) result of PDE filter, and (f) result of quality map combined filter.

Table 1 .
The comparison of different filtering methods with simulation experiment.

Table 2 .
The comparison of different filtering methods with real data experiment.