Three-Component Decomposition Based on Stokes Vector for Compact Polarimetric SAR

In this paper, a three-component decomposition algorithm is proposed for processing compact polarimetric SAR images. By using the correspondence between the covariance matrix and the Stokes vector, three-component scattering models for CTLR and DCP modes are established. The explicit expression of decomposition results is then derived by setting the contribution of volume scattering as a free parameter. The degree of depolarization is taken as the upper bound of the free parameter, for the constraint that the weighting factor of each scattering component should be nonnegative. Several methods are investigated to estimate the free parameter suitable for decomposition. The feasibility of this algorithm is validated by AIRSAR data over San Francisco and RADARSAT-2 data over Flevoland.


Introduction
Compact polarimetric (CP) synthetic aperture radar (SAR) has been widely investigated over recent years. Compared with fully polarimetric (FP) SAR, CP SAR transmits only one polarization, thus avoiding the problems caused by high pulse repetition frequency (PRF), such as a low swath coverage, high data storage requirement and complicated system design. Several investigations demonstrate that CP SAR have the potential for a variety of remote sensing applications, such as soil moisture measurement [1], ship detection, oil spill identification [2,3], and vegetation height estimation [4].
According to the combination of polarization states, three typical CP SAR modes have been proposed, namely: π/4, circular transmission while linear reception (CTLR) and dual circular polarization (DCP). As the quantity of polarimetric information acquired by CP SAR is only half that of FP SAR, CP research has focused mainly on the extraction of scattering characterizations with similar finesse to that derived from FP systems [5].
Decomposition is an effective way to analyze the scattering data from a target. For CP SAR, several widely used decomposition methods have been proposed and improved, such as pseudo FP construction [6][7][8] and CP entropy/alpha decomposition [9][10][11]. In the paper [12] of Rui Guo et al., in 2014, a three-component decomposition for a CP configuration is derived from a series of algebraic calculations, without reconstructing the pseudo FP information. Another school of thought is based on Stokes vector (SV), which is completely constructed from CP data. By using the polarization degree and relative phase calculated from SV, Raney et al. proposed the m   decomposition [13,14]. Cloude extended this idea to a compact decomposition theory in SV form [15].
In this paper, we focus on a three-component decomposition based on SV under the CTLR and DCP modes. We first establish the three-component model from the relationship between the covariance matrix and SV. To solve the underdetermined equations for CP decomposition, the contribution of volume scattering is taken as a free parameter, thus giving a complete set of solutions in an explicit format. According to the constraint that all weighting factors should be nonnegative, the depolarization degree is taken as the upper bound of volume scattering contribution. To validate the effectiveness of this algorithm, San Francisco data from AIRSAR and Flevoland data from RADARSAT-2 are used for testing. Section 2 introduces the three-component model of SV. Section 3 and Section 4 present the deduction of decomposition for CTLR mode and DCP mode respectively. Section 5 compares this algorithm with Cloude CP and m   decompositions. Section 6 discusses methods to estimate the volume scattering contribution. Section 7 demonstrates the decomposition performance using real remote sensing data. Conclusions and future work are drawn in section 8.

Three-Component Model
Three stages are taken in turn to relate SV to decomposition theory: to begin with, we establish a three-component model of FP covariance matrix; this model is then transformed into the coherency matrix; finally the model is expressed by the SV under the CTLR and DCP modes respectively.

Three-Component Model of FP Covariance Matrix
The scattering characteristics of polarimetric SAR images can be evaluated by the second-order statistics of scattering matrix. Here we firstly focus on the covariance matrix where stands for the ensemble average in the data processing,   H means transposition and conjugation, and subscript L states for lexicographic scattering vector. For monostatic FP SAR, the scattering vector L k is defined as [16] 2 In three-component decomposition theory, the covariance matrix is modeled as the contribution of three scattering mechanisms: volume, double-bounce, and surface scatterings. According to [17], the three-component model of FP covariance matrix is given by where v f , d f and s f are weighting factors of each component.

Transforming Covariance Matrix to Coherency Matrix
The coherency matrix of monostatic FP SAR is based on the Pauli vector where the Pauli vector is defined as The relation between the covariance matrix and the coherency matrix is then derived from Equations (2) and (5) According to Equations (3) and (6), we obtain the three-component model of the coherency matrix 2 2

Mapping Coherency Matrix to Output SV under CTLR Mode
For CTLR mode, assuming the transmitted polarization is right hand circular, the normalized polarization of the transmitted wave is [16] Correspondingly, the SV of the transmitted wave is given by The SV of the scattered wave is related to that of the incident wave by Mueller matrix The Mueller matrix can be expressed by Huynen parameters as [18,19] 0 0 Therefore, the SV of the scattered wave is written as Notice that the coherency matrix can also be expressed by Huynen parameters as [18,19] Using Equations (13) and (14), the SV of the scattered wave is related to the coherency matrix Re( ) Im( ) The three-component model based on SV under the CTLR mode is derived from Equations (8) and (15) as

Mapping Coherency Matrix to Output SV under DCP Mode
For the DCP mode, we also assume the transmitted polarization as the right hand circular. The scattering vector in this case is For simplicity of expression, the SV elements under DCP mode are also rewritten as 0 g , 1 g , 2 g and 3 g . From Equation (18), the following relationships are obtained as Therefore, by exchanging the first and third elements in the SV under the CTLR mode, we derive the SV under the DCP mode Re( ) Im( ) The three-component model based on SV under the DCP mode is derived as

Explicit Expressions of Three-component Decomposition for CTLR Mode
Essentially, the three-component decomposition of SV for CP SAR implies solving underdetermined equations. From Equation (16), there are only four constraint equations, while the number of unknowns is seven. According to Freeman and Durden's algorithm [17], the number of unknowns can be reduced to five by setting 1   or 1   , however, one free parameter still remains. In this paper, we take the volume scattering component as the free parameter. Setting 2 v xf  and substituting it into Equation (16), we obtain the Rest Scattering Model (RSM) with double-bounce and surface scattering components as According to Freeman and Durden's algorithm, we fix Because both 2 g and 3 g are real, we have

Calculation of Unknowns When 3 0
g  According to the discussion above,  is fixed as −1 when 3 0 g  , and Equation (29) becomes 2 0 To eliminate d f and s f , take the ratio to give Thus  is given by Now we concern the value range of x. With the constraint that all weighting factors are nonnegative, x must satisfy the following inequalities The second inequality in Equation (37) is quadratic and the corresponding value range for x is Because x cannot be larger than 03 gg  according to the first inequality in Equation (37), the value in 2 ( , ) x  is not acceptable. Based on the analysis above, the value range of x is given by It is easy to show that 1 0 3 x g g  , and thus Equation (42) is rewritten as The contribution of each scattering mechanism is estimated with elements of the SV v Px  It is easy to verify

Calculation of Unknowns When
With a similar method, contributions of each scattering mechanism are obtained as v Px  (50) It is interesting to notice that  

Explicit Expressions of Three-component Decomposition for DCP Mode
Setting 2 v xf  and substituting into Equation (28), we obtain the RSM with double-bounce and surface scattering components as

Comparison with Cloude CP and  m  Decompositions
This section compares the proposed algorithm with other two SV based decompositions. For simplicity, only the CTLR mode is taken for analysis.

Cloude CP Decomposition
According to [15], for a general rank-1 symmetric scattering mechanism, the SV of the scattering wave under the CTLR mode can be written as   Here the depolarized component is regarded as the contribution of volume scattering. d P and s P are estimated from the polarized component and 3 g .

m   Decomposition
In the m   decomposition [13,14], the contribution of volume scattering is also estimated as the depolarized component, while the relative phase  is taken as the factor to split the polarized component into d P and s P

Difference Analysis
Different from the above two methods, the decomposition proposed in this paper takes the depolarized component as the upper bound of volume scattering contribution. Three conclusions are obtained from Equations (44)-(47) and (50)-(54): (a) In our case, we consider volume scattering can be less than the depolarized component that [13][14][15] used (b) Besides the volume scattering, the combined effect of double-bounce and surface scatterings also contributes to depolarization (c) When the depolarized component is only caused by volume scattering i.e., x = x 1 this algorithm degrades to a two-component decomposition.

Value Estimation of x
The difficult part of our algorithm is to estimate the value of the unknown parameter x. Based on the analysis above, three preliminary methods for value estimation are proposed: (a) Assuming the depolarization is only caused by the volume scattering Correspondingly, the estimation of 2 PCI indicates the quality of CP decomposition from another aspect, as shown in Table 2. In comparison, the PCI when p = 0.65 is more similar to that under FP mode, which indicates a better decomposition performance than the case when p = 1.
Since ADI is a single value which is convenient to quantify the performance of CP decomposition, we take it as the major criterion for measurement. Besides that, CDC and PCI are also regarded as supplementary criteria.

AIRSAR Data over San Francisco
The image over San Francisco was acquired by AIRSAR at L-band, with the image size of 900 × 1024. This region contains three typical terrain types: vegetation areas, man-made structures and the sea. Figure 1a shows the pseudo color image of the decomposition result under the FP mode.
In Figure 1a, the three colors red, green and blue correspond to d P , v P and s P respectively.
Classifying each pixel with the largest component, the results are shown in Figure 1b. The SVs for each pixel under the three CP modes are built from FP data, and then the algorithm described in Section 3 and 4 is adopted for decomposition. As mentioned in Section 6, the value estimation of volume scattering component is critical for our decomposition algorithm. To get an initial impression, we first assume that the depolarization is only caused by the volume scattering i.e., p = 1 in Equation (76), the corresponding decomposition results are shown in Figure 2.
It is obvious to observe that a certain number of double-bounce scatters are misclassified as volume scatters when p = 1. In order to select a value of x suitable for decomposition, we calculate ADI under different values of p, as shown in Figure 3.   We also process the data with Cloude CP and m   decompositions, as shown in Figures 6 and 7. The corresponding numerical comparisons are given in Tables 3 and 4.

RADARSAT-2 Data over Flevoland
The image over Flevoland was acquired by RADARSAT-2 at C-band, with the image size of 1513 × 1009. This region contains four major terrain types: forests, man-made structures, the lake and farms. Figure 8 shows the decomposition result under the FP mode by using the improved Freeman-Durden decomposition algorithm.
Covariance matrixes under CP modes are calculated from the FP data, and then decomposed with the proposed algorithm. We first test the decomposition performance when the value of p is taken as 1, as presented in Figure 9. Similar to Figure 2, the volume scattering component is overestimated in this case. Calculating ADI under different values of p, results are depicted in Figure 10.
Similar to the last demonstration, a satisfactory ADI can be obtained when 0.5

p 
, and a maximum is reached when p = 0.65, as shown in Figure 11. Decomposition results after reconstruction are given in Figure 12. We also process the data with Cloude CP and m   decompositions, as shown in Figures 13 and 14. The corresponding numerical comparisons are given in Tables 5 and 6.

Conclusions
This paper formulates a three-component decomposition algorithm for CTLR and DCP modes based on SV, the explicit expressions of decomposition results are derived based on setting the volume scattering component as a free parameter within a series of algebraic calculations. Different from Cloude CP and m   decompositions, this algorithm considers that the combined effect of double-bounce and surface and scatterings may also contribute to depolarization, thus taking the depolarized component as the upper bound of volume scattering, rather than the volume scattering component itself.
Two typical polarimetric SAR data sets are used to demonstrate the feasibility of the proposed decomposition algorithm. If the whole depolarization is taken as volume scattering component, the performance of the proposed algorithm is similar to that of Cloude  Two studies are suggested for future work: one is to extend this algorithm to a four-component decomposition for CP SAR, and the other is to improve the accuracy of reconstruction by using different empirical formulae in different areas.