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In this paper, a method is proposed to improve the interferometric phase quality, based on fusing data from different polarimetric channels. Since lower amplitude implies less reliable phase in general, the phase quality of polarimetric interferometric data can be improved by seeking optimal fusion of data from different polarizations to maximize the resulting amplitude. In the proposed approach, for each pixel, two coherent polarimetric scattering vectors are synchronously projected onto a same optimum direction, maximizing the lower amplitude of the two projections. In the single-look case, the fused phase is equivalent to the weighted average of phases in all polarimetric channels. It provides a good physical explanation of the proposed approach. Without any filtering, the phase noise and the number of residue points are significantly reduced, and the interferometric phase quality is greatly improved. It is a useful tool to preprocess the phase ahead of phase unwrapping. The Cloude's coherence optimization method is used for a comparison. Using the data collected by SIR-C/X-SAR, the authors demonstrate the effectiveness and the robustness of the proposed approach.

Interferometric phase improvement is an important step for Interferometric Synthetic Aperture Radar (InSAR) applications. The original signals collected by a radar system are corrupted by heavy noise, which is caused by the system itself and the propagation. In the traditional SAR interferometry without polarimetric information, several phase filters have been proposed to reduce the noise and improve the phase quality [

In polarimetric SAR interferometry (PolInSAR), since the scattering element data of each pixel are composed of two scattering matrices or scattering vectors corresponding to two spatially separated antennae, it is possible to enhance the coherence and improve the phase between the signals received by both the antennae. In recent years, several algorithms have been proposed, such as the coherence optimization method with two vectors (CO2) [

In this paper, a novel method is proposed. First we provide a mathematical model to maximize the lower of the two amplitudes from the interferometric complex signal pair. Then the optimal solution is obtained in closed-form. Comparing with the CO2 method, we demonstrate that the proposed method has better performance.

This paper is organized as follows. In Section 2, the coherence optimization method proposed by Cloude

In SAR interferometry, for each scattering element, two complex scalar signals _{1} and _{2} are received from two spatially separated antennae. A 2×2 Hermitian semi-definite coherency matrix [

In polarimetric SAR interferometry, for each scattering element, there are two polarimetric scattering matrices, [_{1}] and [_{2}], or two scattering vectors
^{T} indicates the matrix transposition operation, and _{pq}_{HV} = _{VH}.

Similar to [^{H} denotes the complex conjugation and transpose.

To extend the scalar formulation into a vector expression, two normalized complex vectors _{1} and _{2} are introduced. Then two scattering coefficients _{1} and _{2} are defined as the projections of the scattering vectors _{1} and _{2} onto the vectors _{1} and _{2}, respectively,

Then the interferometric phase is derived as

The generalized vector expression for the coherence

To maximize the coherence

The maximum coherence value is then given by the square root of the maximum eigenvalue [_{1opt} and _{2opt}.

Finally, a sensible constraint is to require

In this method, the interferometric coherence _{1opt} and _{2opt}. The corresponding interferometric phase

Though the coherence might indicate the phase noise, however, it is usually estimated by using neighborhood information and not accurate. So for phase improvement, coherence optimization is not the best approach. Especially in weak signal area, the improved phase by coherence optimization is still noisy. Fortunately, the proposed method can be used to obtain a nearly noise-free phase result in the moderate noise case.

In SAR interferometry, only one polarimetric channel signal can be received, e.g., HH. For each scattering element, the amplitudes of the complex signals _{1} and _{2} vary with the terrain fluctuation and the scattering characteristic of the ground targets. In some areas, the amplitude of the received signals may be very low. When a complex noise is added to a weak signal, a considerable change in the signal phase may occur. In this case, the interferometric phase between two weak signals will be severely affected by noises and will be of low quality and unreliable. Therefore, a lot of residue points may exist to deteriorate the performance of phase unwrapping. In addition, weak signals usually imply a low signal-to-noise ratio (SNR). The noise components in _{1} and _{2} are totally irrelevant (in repeat-pass interferometry mode). According to _{1} and _{2} is corrupted by noise and the interferometric phase between two weak signals (or at least one weak signal) is not reliable, i.e., the quality is low.

The purpose of the proposed method is to fuse the interferometric signal pair in each polarimetric channel to augment the amplitude of the signals, especially in weak signal area. In general, except the effect of decorrelation, most residue points are caused by weak signals. Therefore, optimizing amplitude is necessary and effective to improve the phase quality and eliminate the residue points.

To improve the phase quality and remove the residue points, a feasible way is to augment the amplitudes of both coherent signals.

In polarimetric SAR interferometry, as mentioned above, each scattering element has two polarimetric scattering vectors _{1} and _{2}. To extend the scalar formulation into a vector expression, as a similar way to the CO2 method, a normalized complex vectors _{1} and _{2} are defined as the projections of the scattering vectors _{1}_{2}

The goal of the proposed method is to figure out an optimum vector _{1} and _{2} simultaneously. In other words, the lower amplitude between _{1} and _{2} is maximized. According to Section 3, if both the amplitudes are augmented, the interferometric phase quality can be improved.

Mathematically, the above optimization problem is described as follows:

To obtain the analytic solution of the above problem, it can be transformed into an equivalent problem as follows:

According to the quadratic programming theory, |^{H}_{1}|^{2}and |^{H}_{1}|^{2} are two quadrics in three dimensional complex space. Each of them has only one local maximum, which is also the global maximum. The solution has two following cases.

If the global maximum of |^{H}_{1}|^{2} is equal to ^{H}_{2}|^{2} is equal to _{1} or _{2}

If the global maximum of |^{H}_{1}|^{2} is greater than ^{H}_{2}|^{2} is greater than ^{H}_{1}|^{2}≤ |^{H}_{2}|^{2} and |^{H}_{1}|^{2}>|^{H}_{2}|^{2}. Therefore, when

In this situation, an eigendecomposition method can be used to obtain the analytic solution. The detailed process is described in _{1} > 0, _{2}_{1}_{1}^{H} − _{2}_{2}^{H}, with corresponding eigenvectors _{1} and _{2}, respectively.

In the single-look case, the fused phase is

The cross-correlation item
_{s} in

According to the definition of the scattering vector in _{pq}_{pq,1}| and |_{pq,2}| denote the amplitudes of the signals, the larger the product of them, the larger the weight. This is reasonable from the basic viewpoint in Section 3: the phase of strong signals is more reliable than that of weak signals in general. Since a larger weight is assigned to a more reliable phase of a given polarimetric channel, the noise of the improved phase is reduced effectively and the coherence is enhanced.

Here we used the single-look L-band experimental data consisting of fully polarimetric complex image pairs of the Tien Shan test site acquired by the SIR-C/X-SAR radar system on Oct. 8 and 9, 1994. The test area is close to the southern edge of Lake Baikal, Russia. It contains many different ground targets such as forest, cropland, bare ground and mountain. Without denoising, the interferometric phase is corrupted by heavy noise and lots of residue points exist.

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 [

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.

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 [

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) [

_{HH} 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

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 _{1} in

To make a comparison, the phase result by the coherence optimization (CO2) method is also calculated and shown in

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 [

Though Buckland [

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

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

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

To demonstrate the denoising ability of the AO method further, we add some noise to the original PolInSAR data in the white frame in _{i}_{i}

A novel interferometric phase improvement method has been proposed. The key point is to maximize the amplitude of the signals based on the relationship between the amplitude and the phase of a complex signal. In the single look case, the improved phase is equivalent to the average of information in each polarimetric channel with different weights which are proportional to the amplitude in each channel.

In the proposed method, we used one normalized complex vector instead of two, because the correlation information between both the interferometric channels is important, and the proposed method did not optimize the coherence directly. In one-vector case, the correlation information is used more sufficiently, contained in the eigenvectors of matrix _{1}_{1}^{H} − _{2}_{2}^{H}. Considering two-vector case, _{1} and _{2} can be figured out as the normalized version of _{1} and _{2}, respectively, with the only constraint

Using the PolInSAR data, the performance of phase improvement has been demonstrated. In the multi-look case, more than 99% residue points caused by moderate noise can be removed by the proposed method in both strong and weak signal areas. The detailed information of topography is observed more clearly, which makes phase unwrapping becomes easier and faster.

This work was supported by the National Natural Science Foundation of China (40571099). We acknowledge the manuscript revision of Yilun Alwyn Chen and Xinrong Yang.

In Case II, from

If _{1} = _{2}

If _{1} ≠ _{2}_{1} > 0, _{2} < 0 and _{3} = 0, and the corresponding eigenvectors are _{1}, _{2} and _{3}, respectively.

If _{3}, it completely meets _{3} = 0. But the space spanned by _{1} and _{2}, i.e., _{1},_{2}}, is equivalent to _{1}, _{2}}. Since _{3} is orthogonalized to both _{1} and _{2}, then |^{H}_{1}| = |^{H}_{2}| = 0. It does not satisfy the goal of

Thus, _{1} and _{2}

So

Substituting

According to the Cauchy inequality, the maximum of

After substituting

To determine how to choose ^{(1)} and ^{(2)}, we define two functions _{1}(|_{2}(|

Let

The corresponding solution of |

Because of the monotonicities of _{1} and _{2}, there are three sub-cases as follows (schematically shown in

If
^{(1)};

If
^{(1)};

Otherwise, ^{(1)},

In conclusion, the final solution of the model is

The sketch map of the three sub-cases of the model solution in the second case.

According to

From ^{(1)}, then
_{1}║^{2} or ║_{2}║^{2}. So _{s} and ^{(2)}, since _{1} is an eigenvector of the matrix

Similarly,

According to

Substituting _{s} and φ are equivalent.

The relationship between the coherence and amplitude. It is a 2-D histogram of the coherence and amplitude in a forest area (10,000 samples).

The unwrapped phase 3-D illustration corresponding to

The comparisons of average lower amplitude, average coherence and residue point number among the original HH/HV/VV channel data and the fused data processed by the AO and CO2 method in both the single-look and multi-look cases in

Lower Amplitude | Coherence | Residue point number | |
---|---|---|---|

| |||

0.3604 | 0.7843 | 8331 | |

0.1401 | 0.7077 | 12033 | |

0.2578 | 0.7450 | 10279 | |

0.4768 | 0.8523 | 1699 | |

0.2795 | 0.8858 | 1889 | |

--------- | 0.8228 | 20 | |

--------- | 0.9634 | 173 |