6.2. Amplitude vs. phase relationship
Though the coherence parameter has no direct link to the phase, it is usually regarded as a quality descriptor of phase information. Though other parameters such as the phase derivative variance and the maximum phase gradient can also be used to measure phase quality [10
], the coherence is more widely accepted in SAR interferometry. In (3)
, the coherence seems to be independent of the amplitude. However, since lower amplitudes always correspond to lower SNR, uncorrelated noises will dominate the value of the coherence. Therefore, weak signals lead to low coherence.
Now we use the coherence-amplitude map to demonstrate the relationship between the amplitude and the phase of complex signals. 10,000 samples with same scattering characteristics are used to draw the 2-dimensional histogram between coherence and amplitude. Figure 1
shows the forest case as an example. Both the coherence and amplitude have 128 gray levels.
From the distribution we conclude that in most cases, the coherence of weak signals is low, and large amplitudes in general correspond to large coherence. Therefore, a larger amplitude implies a more reliable phase. Another experiment in [9
] also verifies this relationship.
6.3. Vegetation and bare ground
The scattering mechanism of the vegetation is very complicated due to its multiple components such as leaves, branches, trunks and the underlying ground. According to the vegetation scattering model based on physical properties, the total response of the vegetation is composed of the volume scattering (random or oriented) and the ground scattering (with or without the trunk) [11
]. Moreover, due to the repeat-pass interferometry mode, the temporal decorrelation can not be neglected, especially in the vegetation-covered area. Therefore, improving the phase quality is necessary for topography retrieval.
shows the sHH
of the test area (1,000×1,000 pixels), which includes several different kinds of targets, such as forest (F), road (R), bare ground (BG) and cropland (C). The corresponding optical image from Google Earth with the same resolution is given as Figure 2(b)
To demonstrate the effectiveness of the proposed method, two areas in white frame A and frame B containing typical targets are enlarged and processed.
The enlarged version of frame A is shown in Figure 3(a)
. The ground is mostly covered by forest with three roads through it. The black area is the bare ground. With noise and the effect of decorrelation, the phase noise in HH channel is so heavy that all details of the terrain are submerged (Figure 3(b)
). The averages of lower amplitudes of the selected area in HH, HV and VV channels are 0.3604, 0.1401, and 0.2578, respectively. Using the proposed method, the average of lower amplitude of the fused image pair is improved to 0.4768 and the amplitude of η1
is shown in Figure 3(c)
. It is obviously “whiter” than the amplitude of HH channel. The fused phase in (21)
, using 3 × 3 window, is shown in Figure 3(d)
. Noise is removed obviously and the phase fringes can be clearly observed. 99.76% residue points are removed. The improved phase between the forest area and the bare ground has no obviously boundary. It implies that the phases in the forest area can be regarded as those of the underlying topography, since the phase in the bare ground definitely corresponds to the topography.
To make a comparison, the phase result by the coherence optimization (CO2) method is also calculated and shown in Figure 3(f)
. The coherence is optimized close to 1 (shown in Figure 3(e)
, the white and the black colors mean 1 and 0, respectively) and many noises in the forested area are removed. However, the phase of the bare ground is still noisy, which is not in accordance with the “flat property” of the bare ground as shown in the optical image.
lists more comparisons between the amplitude optimization (AO) method and the coherence optimization (CO2) method. It includes the average of lower amplitude, the mean coherence, and the residue point number in both the single-look (SL) and multi-look (ML) cases.
shows the enlarged area of Frame B. Most parts of the ground are covered by lowvegetation like the crop and grass. The parallel straight lines are possible ridges of field. The dihedral angle between the ridge and the ground leads to strong responded signals. The phase in HH channel [Figure 4(b)
] is so noisy that it yields 6,550 residue points as shown in Figure 4(c)
(black points). The improved phase by the proposed AO method and the CO2 method are given in Figure 4(d) and (f)
, respectively. Obviously, the phase improved by the AO method has the better quality with less noise. 99.63% residue points are removed successfully as shown in Figure 4(e)
, demonstrating the effectiveness of the proposed approach.
Phase unwrapping (PU) is the key step of digital elevation model (DEM) generation. The main difficulties of phase unwrapping are from noise and steep topography. Both the factors lead to the existence of huge amount of residue points. For path following based PU methods, branch cuts are used to balance the charge of the positive and negative residue points. In the case that a large numbers of dense residue points exist, several branch cuts based algorithms [13
] do not work.
Though Buckland [15
] proposed an algorithm based on the Hungarian algorithm from integer programming and declared the algorithm enables one to unwrap unfiltered speckle-interferometry phase maps at high point densities (0.1 points per pixel), the computation efficiency should be considered. It will take a long time to solve a large wrapped phase map with heavy noise. So it is significant to improve the phase quality before phase unwrapping.
shows a test area containing a mountainous area. The corresponding optical image is shown in Figure 5(b)
, which is also from Google Earth. The topography is not steep. The interferometric phase in HH channel is displayed in Figure 5(c)
. With 1,000×1,000 pixels, the phase map after flat-removal corresponds to 95,329 residue points, so the density of residue points is close to 0.1 sources per pixel. Though a reasonable result may be figured out by the algorithm in [15
], it is time-costly and the unwrapped phase is still noisy.
Fusing the information from each polarimetric channel, the average of lower amplitude of the image pair is enhanced from 0.3250 (in HH), 0.1310 (in HV) and 0.2831 (in VV) to 0.4574. The fused phase is shown in Figure 5(d)
. 99.96% residue points are removed. Using typical PU algorithms, the unwrapped phase can be obtained fast and accurately. The 3-D illustration is displayed in Figure 6
. From Figure 5(d)
, the topography becomes clearer and the detailed information is preserved well.
The CO2 method can be used to enhance the phase quality well in most mountainous areas with moderate and strong signals. But in flat ground area with weak signals, the fused phases still correspond to lots of residue points. Please pay attention to the area in the white frame in Figure 5(a)
that the amplitudes of the right half pixels are low. The improved phases obtained by the proposed AO method and the CO2 method are shown in Figure 7(a) and (b)
, respectively. Corresponding to the area with low amplitude, lots of residue points exist in the right half of Figure 7(d)
and the phase in Figure 7(b)
Based on the signal amplitude optimization, the phase result corresponds to very few residue points in both strong signal areas and weak signal areas shown in Figure 7(c)
. It demonstrates the robustness of the proposed method.
To demonstrate the denoising ability of the AO method further, we add some noise to the original PolInSAR data in the white frame in Figure 5(a)
for simulation. In each pixel, let
= 1, 2, then the covariance matrix by each antenna,
, is calculated. According to the noise statistics in [16
], the simulated complex noise vector ni
has complex Gaussian distribution N
]), where m
is a scalar between 0 and 1. The residue point number (before flat-removal) is used to measure the performance of both the methods. m
can be regarded as an indicator of the added noise intensity.
illustrates the denoising performance of the AO and CO2 methods when m
increases from 0 to 1. When the noise is comparatively weak, e.g., m
≤ 0.3, more than 99% residue points are removed by the AO method. Even in the strong noise case, e.g., m
≥0.8 (shown in Figure 9(a)
), the noise can be reduced effectively and the spatial distribution of the remained residue points is close to uniform (Figure 9(b)
), regardless of strong signal area (mountain) or weak area (bare ground). On the other hand, using the CO2 method, the remained residue points concentrate in bare ground area (Figure 9(c)
). It may be difficult to unwrap the phase with such dense residue points. For example, in the case of m
= 1, the average density of residue points is 0.135 points per pixel by the CO2 method (Figure 9(c)
). Most existing PU methods do not work in such an extreme situation. Using the AO method, the average density can be reduced to 0.049 points per pixel (Figure 9(b)
). Then the unwrapped phase can be obtained by several noise-immune methods.