4.2.1. Case 1
Due to the large phase gradient, some deformation signals could not be derived correctly by phase unwrapping. However, different schemes or parameter settings for the same method often leads to different unwrapped results. Therefore, we require that an appropriate method is found that has better unwrapping performance on the CKF PU method.
Two tests were designed to evaluate the unwrapping effect including the number of multi-looks and quality index for path-guiding. While each index was tested, the other indexes remained constant. The detailed schemes for these tests can be listed as follows:
Test 1: Test on the effect of the number of multi-looks on the CKFPU method.
Scheme 1: FD was used as the path-guiding quality index, the single-looked differential interferogram was the wrapped objective.
Scheme 2: FD was used as the path-guiding quality index, the two-looked differential interferogram was the wrapped objective.
Scheme 3: FD was used as the path-guiding quality index, the four-looked differential interferogram was the wrapped objective.
The interferograms and coherence maps are shown in
Figure 2. It can be seen that as the number of multi-looks increases, there is a corresponding decrease in the speckle noise. However, some details were also smoothed out, which is unsatisfactory. From
Figure 2, we can observe that the single-looked interferogram shows more noise, fringes and detailed information. As the number of multi-looks increases, the spatial-resolution decreases, some detailed information is smoothed out and fringes become vague, which may introduce another problem—the number of fringe decreases and the maximum unwrapping ability will be weakened especially in areas with fast subsidence.
Figure 3 shows the unwrapped results obtained by the CKFPU algorithm.
Figure 3a–c are unwrapped maps and
Figure 3d–f are the corresponding rewrapped maps. Compared to the interferograms in
Figure 2, all of the rewrapped maps possess a distinct characteristic—noise was filtered out. However, the unwrapped results appear quite different from each other. For single-looking, there are several subsidence and upraised areas which are not continuous and do not reflect the actual situation of the study area. This is because the single-looked data includes high noise which plays a key role in these areas. For two-looking, the unwrapped map seems more continuous, since a higher Signal to Noise Ratio of the interferogram can be obtained by two-looking processing. Additionally, compared to single-looking and four-looking results, two-looking obtained the highest subsidence value and had the best unwrapping performance. Based on the collected coal mining logs, this working face was active during the time TerraSAR-X satellite visited the target area and the maximum subsidence of the active working face could reach up to 20–30 cm (80–100 radian in line of sight direction). However, the maximum subsidence achieved by some conventional unwrapping methods and software are all far less than the actual subsidence value. Here, we introduced a qualitative evaluation scheme to estimate the performance of these schemes, which was that the greater the maximum unwrapped phase, the better result, which is closer to the reality. For real data, the rewrapped map and original interferogram are generally compared to evaluate the performance of phase unwrapping methods. The rewrapped map with clearer fringes, less noise and more detailed information is considered to be the better one. From
Figure 3, it is clear that Scheme 2 is the best one. For Scheme 3, although the unwrapped map is relatively smoother than the other two, the maximum subsidence is less and the fringes are vaguer. This is because spatial resolution declines as the number of multi-looks increases. In summary, the two-looking method is the most appropriate selection for the given interferogram. It should be noted that how to select a proper number of multi-looks for specific data is not the emphasis in this paper. The main aim of this test is to prove that the number of multi-looks is important to the CKFPU method in study areas with high noise. Additionally, it is also verified that the new method can unwrap the interferograms while filtering out noise.
Figure 3.
Unwrapped maps (left) and rewrapped maps (right) under different numbers of multi-looks: (a) unwrapped map of 1 × 1 look; (b) unwrapped map of 2 × 2 looks; (c) unwrapped map of 4 × 4 looks; (d) rewrapped map of 1 × 1 look; (e) rewrapped map of 2 × 2 looks; (f) rewrapped map of 4 × 4 looks.
Figure 3.
Unwrapped maps (left) and rewrapped maps (right) under different numbers of multi-looks: (a) unwrapped map of 1 × 1 look; (b) unwrapped map of 2 × 2 looks; (c) unwrapped map of 4 × 4 looks; (d) rewrapped map of 1 × 1 look; (e) rewrapped map of 2 × 2 looks; (f) rewrapped map of 4 × 4 looks.
Test 2 was designed to evaluate the effect of different path-guiding indexes on the CKFPU method, using the same data as used as Test1. Scheme 2 of this test is the same scheme used in Test 1.
Scheme 2: FD was used as the path-guiding quality index; the wrapped objective was the two-looked differential interferogram.
Scheme 4: MC was used as the path-guiding quality index; the wrapped objective was the two-looked differential interferogram.
Scheme 5: PDV was used as the path-guiding quality index; the wrapped objective was the two-looked differential interferogram.
Figure 4 shows the three phase quality maps with each of the different quality indexes.
Figure 4a–c are the quality maps calculated by FD, MC and PDV, respectively. A darker color in these graphs represents worse quality, and
vice versa. At first glance, there are more similarities overall between
Figure 4a,b, but there are also obvious differences. On the whole,
Figure 4a is brighter than
Figure 4b. This is because the values calculated by FD are mainly distributed in the lower range while the values of MC have a better dispersion. In addition, some pixels show low quality in
Figure 4a but high quality in
Figure 4b. Taking a typical pixel as an example, pixel A is an artificial Corner Reflector and is homologous in the subgraphs. However, pixel A exhibits quite different qualities in each of the three graphs. In
Figure 4a, pixel A is very dark which means that the pixel is low quality. This is because FD reflects similarities between each pixel and its surroundings. In this example, the surrounding area of the Corner Reflector is covered by ground objects which are seriously uncorrelated, so the similarity is very low and the FD value is much higher. However, in
Figure 4b, the opposite effect is shown. Since MC is used to indicate the correlation between each pixel in the two SLC images, more stable ground objects always lead to a greater coherence. Additionally, there is a slower change from a poor quality area to high quality area in
Figure 4c. In other words, it is insensitive to different objects. Equation (29) can be interpreted as meaning that PDV only reflects the statistical characteristics of pixels in a given window without emphasizing the information of the current pixel, so some details are smoothed out. In summary, FD and MC are more sensitive than PDV. Compared with MC, the FD method considers both spatial diversity and coherence randomness, and is more robust and comprehensive for guiding the path of phase unwrapping.
Figure 4.
Three phase quality maps with FD, MC and PDV: (a) FD quality map; (b) MC quality map; (c) PDV quality map.
Figure 4.
Three phase quality maps with FD, MC and PDV: (a) FD quality map; (b) MC quality map; (c) PDV quality map.
Figure 5 shows the corresponding unwrapped and rewrapped results guided by MC and PDV. This can be compared with
Figure 3b,e, which show the unwrapped and rewrapped results where path tracking is guided by FD. If we focus on the unwrapped maps, it can be seen that the FD method (shown in
Figure 3) obtains a more continuous unwrapped phase and a greater maximum subsidence. This fits the subsidence map and interferogram better—the area with dense fringes has large subsidence. This is because the FD index combines the phase derivative with the expected phase variance (based on coherence), which is more robust and suitable for mountainous areas with less persistent scatters. From the rewrapped maps, the MC method retains most of the details, due to the fact that MC only accounts for the correlation without considering the spatial similarity of the differential interferogram. The highly correlated pixels are usually permanent scatters with little noise, so working along this index may lead to a clearer rewrapped map. However, it is not reliable in areas with fast subsidence that are covered in vegetation. For example, the Corner Reflector pixel located in a fast subsidence area is usually treated as high quality pixel by the MC method, due to the high coherence value. This is clearly not reasonable, since fast subsidence can lead to signal aliasing, preventing the real situation of the subsidence from being reflected. If it is used as a reference in earlier steps, there will be more error propagation. The PDV method takes the spatial difference into account without considering the coherence, which may weaken the overall reliability. Therefore, a continuous but low maximum unwrapped phase result is obtained by PDV. In summary, the FD method is more suitable for guiding the path of CKFPU algorithm where there is high noise and large phase gradients.
Figure 5.
Unwrapped maps (left) and rewrapped maps (right) based on MC and PDV: (a) unwrapped map of MC; (b) unwrapped map of PDV; (c) rewrapped map of MC; (d) rewrapped map of PDV.
Figure 5.
Unwrapped maps (left) and rewrapped maps (right) based on MC and PDV: (a) unwrapped map of MC; (b) unwrapped map of PDV; (c) rewrapped map of MC; (d) rewrapped map of PDV.
4.2.2. Case 2
Two datasets (Dataset A and Dataset B) were used to measure the performance of CKFPU in high noise areas. Additionally, the test was also designed to compare the proposed CKFPU algorithm with the MCF method. It should be noted that FD was also used in the MCF method. The pre-filtering steps for the MCF method was conducted using GAMMA software with adaptive filtering and the thresholds were set at 0.1 and a default value of 0.25, respectively. The CKFPU method employed the same settings as Scheme 2 in Case 1.
For Dataset A, the same 2 × 2 multi-look interferograms and coherence maps were used for both CKFPU and MCF. By way of explanation, the interferograms and coherence maps are shown in
Figure 2b,e. The corresponding results of CKFPU are shown in
Figure 3b,e.
Figure 6 shows the results of the MCF method only.
Figure 6a,b are the unwrapped and rewrapped maps when the pre-filtering threshold is set at 0.1.
Figure 6c,d are the unwrapped and rewrapped maps when the pre-filtering threshold is set at the default value 0.25.
Figure 6.
Results based on MCF method: (a) unwrapped map with pre-filtering filter threshold set at 0.1; (b) rewrapped map of (a); (c) unwrapped map with adaptive filter threshold set at 0.25; (d) rewrapped map of (c).
Figure 6.
Results based on MCF method: (a) unwrapped map with pre-filtering filter threshold set at 0.1; (b) rewrapped map of (a); (c) unwrapped map with adaptive filter threshold set at 0.25; (d) rewrapped map of (c).
Figure 7 shows the differential interferogram, coherence map and the experimental results for Dataset 2. The interferogram and coherence map are shown in
Figure 7a,b. The corresponding results for CKFPU are shown in
Figure 7c,d.
Figure 7e,f are the unwrapped and rewrapped maps based on MCF when the pre-filtering threshold is set at 0.1.
Figure 7g,h are the unwrapped and rewrapped maps based on MCF when the pre-filtering threshold is set at the default value 0.25.
From the initial interferograms of Dataset A and Dataset B, it can be seen that the unfiltered interferograms contain high noise content. Pre-filtering must be implemented before the MCF phase unwrapping. However, the unwrapped results based on MCF vary under different filtering thresholds, so it is difficult to select a proper threshold for different interferograms. On the one hand, the exact phase information cannot be unwrapped if there is insufficient pre-filtering. On the other hand, detailed phase information may be lost if there is too much pre-filtering. In contrast, the CKFPU method can get a similar or superior phase unwrapping result without pre-filtering, as shown in
Figure 3b,e and c,d. This method not only removes a lot of noise but also achieves a satisfactory unwrapped map. In other words, the CKFPU method can avoid the pre-filtering dilemma to some extent. This is a typical advantage of the CKFPU method. In addition, there is some difference between the results of Datasets 1 and 2. For Dataset 1, the maximum subsidence calculated by CKFPU is much larger than the MCF method. According to the real measurements (introduced in
Section 4.1.1 Dataset A), the former is closer to reality. For Dataset 2, the unwrapped results are similar for both CKFPU and MCF. This is most likely due to the fact that the interferogram for Dataset A has dense fringes, which are sensitive to pre-filtering and cause loss of detailed information.
Figure 7.
Experimental results for Dataset 2: (a) original differential interferogram; (b) corresponding coherence map; (c) unwrapped map of CKFPU; (d) rewrapped map of CKFPU; (e) unwrapped map of MCF when the pre-filtering threshold is set at 0.1; (f) rewrapped map of (e); (g) unwrapped map of MCF based on MCF when the pre-filtering threshold is set at 0.25 and (h) its rewrapped map.
Figure 7.
Experimental results for Dataset 2: (a) original differential interferogram; (b) corresponding coherence map; (c) unwrapped map of CKFPU; (d) rewrapped map of CKFPU; (e) unwrapped map of MCF when the pre-filtering threshold is set at 0.1; (f) rewrapped map of (e); (g) unwrapped map of MCF based on MCF when the pre-filtering threshold is set at 0.25 and (h) its rewrapped map.
In summary, from the results of these two datasets, the CKFPU method shows similar or even better phase unwrapping performance compared with MCF and it can avoid or relieve the dilemma of parameter setting for pre-filtering to some extent.