Polarimetric Model-Based Decomposition with Reﬁned Double-Bounce Orientation Angle and Scattering Model

: Oriented manmade targets can produce signiﬁcant cross-polarization power. The scattering mechanism interpretation of them is still challenging. Within the framework of traditional scattering models, the scattering mechanism of oriented manmade targets will be interpreted as volume scattering. Recently, many advanced approaches have been proposed to mitigate the cross-polarization terms of the coherency matrix or distribute the power of cross-polarization to new scattering models, such as orientation angle compensation and multiple scattering components decomposition. Among these methods, the general model-based decomposition with physically meaningful double-bounce and odd-bounce scattering models has been proposed by modeling their independent orientation angles and becomes a widely accepted method. However, the two vital parameters of generalized scattering models: double- and odd-bounce orientation angles are derived through nonlinear optimization procedure. These generalized models lead to a heavy computation burden for parameters inversion. In this paper, we disclose the latent relationship between the double-bounce orientation angle and polarization orientation angle by data ﬁtting experiments. With this simpliﬁed relationship, a reﬁned double-bounce scattering model is established. Then, the odd-bounce orientation angle can be derived through equations. In this way, the nonlinear optimization procedure can be converted to a linear solution. A fast generalized model-based decomposition is developed thereafter. The main contribution of this work is to inherit the generalized models while speeding up the parameter calculation procedure. The comparison studies are carried out with X-band airborne PiSAR, L-band spaceborne ALOS-2, and C-band spaceborne Radarsat-2 PolSAR datasets. Compared with the state-of-the-art approaches, the proposed decomposition achieves improved interpretation performance from both visual and quantitative investigations especially for oriented built-up areas.


Introduction
Polarimetric target decomposition is a powerful tool for polarimetric synthetic aperture radar (PolSAR) data interpretation and achieves plenty of successful applications [1,2]. Among them, model-based decompositions, which can provide physically meaningful solutions have been widely used. The Freeman-Durden decomposition is one of the pioneer model-based decompositions [3]. From then on, many advances have been reported to overcome the negative power issues, volume-scattering overestimation, and scattering mechanism ambiguity [4]. Non-negative eigenvalue constrain was proposed to avoid the occurrence of negative power [5,6]. Meanwhile, the power of the volume-scattering component is primarily determined by the cross-polarization term within the scope of conventional scattering models [3,7]. In this vein, the orientation angle compensation (OAC), which aims at cross-polarization element minimization has been incorporated in model-based decomposition to alleviate volume-scattering overestimation [8,9]. Moreover, double unitary transformations have been included to further enhance the performance of

Refined Double-Bounce Orientation Angle and Scattering Model
Incorporated with double-bounce orientation angle, the traditional double-bounce scattering model can be transformed into a generalized double-bounce scattering model [18], which can be represented as: |α| 2 α cos 2θ dbl −α sin 2θ dbl α * cos 2θ dbl cos 2 2θ dbl − 1 2 sin 4θ dbl −α * sin 2θ dbl − 1 2 sin 4θ dbl sin 2 2θ dbl   (1) where f d is the double-bounce scattering power coefficient, and θ dbl is the DBOA of scattering model. α is the model parameter.
In this way, the cross-polarization and off-diagonal term induced by oriented doublebounce structures will be reallocated to double-bounce scattering instead of volume scattering. In the former research of [18], double-bounce orientation angle is derived through the nonlinear least square optimization procedure as an unknown parameter. Experimental studies show that the DBOA is highly correlated with POA. A number of researchers have Remote Sens. 2021, 13, 3070 3 of 18 been devoted to exploring the relationship between polarization orientation angle and orientation angles of different scatterings [20][21][22]. However, POA describes the rotation of a coherency matrix and is a mixture orientation effect of separate scattering mechanisms. The oriented buildings will induce orientation of single-bounce and triple-bounce scattering components more than double-bounce scattering. In other words, the theoretical relationship between POA and DBOA is difficult to describe, even without existence.
Based on aforementioned facts, this work aims at disclosing their underlying statistics relationships and establishing a robust analytical expression through data fitting experiments on different PolSAR datasets such as X-band PiSAR, C-band Radarsat-2, and L-band ALOS-2 over oriented manmade areas. The experiment results of X-band PiSAR are shown below as an example.
The X-band Pi-SAR 8708 data were acquired on 12 February 2005, over Sendai, Japan, containing buildings with different orientation angles. The resolution is 2 m in the ground range direction at the scene center and 1.5 m in the azimuth direction. The full scene size is 5 km × 5 km. Three representative regions with variously oriented buildings of X-band PiSAR data are shown in Figure 1. The error bar plots are shown in Figure 2 with a POA interval of 5 degrees. The red dots indicate the mean values while the error bars represent the standard deviation. From Figure 2, it is observed that the error bar plots are rotational symmetry. Additionally, the relationship between DBOA and POA can be divided into 5 zones, as follows: Remote Sens. 2021, 13, x FOR PEER REVIEW 4 of 19 In this vein, a piecewise linear function is proposed to fit the relationship between POA and derived dbl θ , as follows: The optical images, Pauli images, polarization orientation angles, and double-bounce orientation angles of three patches selected from airborne X-band PiSAR data. Three patches of buildings are oriented in different directions. The orientation angle of (a1) is around 5 degree, (a2) is 35 degree, and (a3) is 45 degree.

Model-Based Decomposition Framework
This section proposes a model-based decomposition framework. The main scheme is to convert the nonlinear model inversion as a linear solution with the proposed model and nonnegative eigenvalue constrain. Combined with the refined double-bounce scattering model, the framework of the fast model-based decomposition is  [18]. The optimal volume-scattering model parameters are determined by the residual matrix minimization.
The proposed model-based decomposition is shown in Figure 3. • Zone A: Negative linear zone (−15 • ≤ POA ≤ 15 • ). According to the statistical distribution of POA and DBOA, the major part of the pixels are included in this interval. As shown in Figure 1(c1,d1), the relationship between POA and DBOA is approximate negative linear. As illustrated in [9], when the dominant polarization angle of a built-up area |D POA | ≤ 22 In this vein, a piecewise linear function is proposed to fit the relationship between POA and derivedθ dbl , as follows:

Model-Based Decomposition Framework
This section proposes a model-based decomposition framework. The main scheme is to convert the nonlinear model inversion as a linear solution with the proposed model and nonnegative eigenvalue constrain. Combined with the refined double-bounce scattering model, the framework of the fast model-based decomposition is where T vol ,T dbl θ dbl ,T odd (θ odd ),T hel represent the coherency matrixes of volume, refined double-bounce, generalized odd-bounce, and helix scattering models, and f v , f d , f s , f c are their scattering power coefficients separately.θ odd is the odd-bounce orientation angle. T res is the residual matrix. To maintain the generality of the volume scattering, the proposed method adopts 4 different volume-scattering models as the volume-scattering lookup table, which is similar with [18]. The optimal volume-scattering model parameters are determined by the residual matrix minimization.
min T res 2 (4) The proposed model-based decomposition is shown in Figure 3.

Volume-Scattering Model
Volume scattering from vegetation is often described as randomly oriented dipoles with different probability density functions (pdf) p(θ) with respect to orientation angle and particle anisotropy [18]. The coherency matrix of the general volume-scattering model can be expressed as where the elements a, b, c are real-valued, while d, e, f are complex-valued. The volumescattering model lookup table is shown as follows.
In the classic Freeman-Durden three-component decomposition, the pdf is considered as a uniform distribution (known as random dipoles model). The corresponding coherency matrix is Another two usual pdfs are sine and cosine distribution, proposed in the Yamaguchi four-component decomposition method, and are usually named as horizontal and vertical dipoles models.
There are also another representative models such as the highest entropy model.

Model Parameters Inversion
Traditionally, the helix scattering power is calculated from the imaginary part of 23 T as ( ) However, the matrix c hel P ′ = − T T T after helix scattering components derived cannot always guarantee positive semidefinite, and negative scattering power will be introduced. According to the nonnegative eigenvalue constrain [5,6,23], the maximum possible power of helix scattering can be derived through where ( ) det denotes the matrix determinant. Note that there is only one solution as (13) is a one-order equation. Thus, the helix scattering power c P can be obtained as Similarly, volume-scattering power v P is determined with the eigen equation The volume-scattering power v P can be derived as  11 12 13  *  12  22  23  *  *  13 23 33

Odd-Bounce Scattering Model
The odd-bounce scattering model adopted by the vast majority of decomposition methods is the Bragg surface model, which is generated from Freeman-Durden decomposition [3,7]. The coherency matrix of odd-bounce scattering is where β is the parameter of odd-bounce scattering model. Modeled with the odd-bounce orientation angle θ odd , the generalized odd-bounce scattering model can describe polarization basis rotation and the cross-polarization induced by the terrain slopes.

Double-Bounce Scattering Model
The double-bounce scattering is modeled by scattering from a dihedral corner reflector, including ground-wall backscatters, where the reflector surfaces are made of different dielectric materials. The corresponding coherency matrix is where α is double-bounce scattering model coefficiet. α is complex valued, and |α| < 1.
As illustrated in Section 2, the coherency matrix of refined double-bounce scattering model T dbl θ dbl is shown as (1).

Helix Scattering Model
The helix scattering model is proposed by Yamaguchi et al. to fit the reflection asymmetry condition [7]. The coherency matrix is

Model Parameters Inversion
Traditionally, the helix scattering power is calculated from the imaginary part of T 23 as P c = 2|Im(T 23 )|. However, the matrix T = T − P c T hel after helix scattering components derived cannot always guarantee positive semidefinite, and negative scattering power will be introduced. According to the nonnegative eigenvalue constrain [5,6,23], the maximum possible power of helix scattering can be derived through where det() denotes the matrix determinant. Note that there is only one solution as (13) is a one-order equation. Thus, the helix scattering power P c can be obtained as Similarly, volume-scattering powerP v is determined with the eigen equation The volume-scattering power P v can be derived as where λ 1 ,λ 2 and λ 3 are the eigenvalues of (15). After subtracting the helix and volume-scattering components, the remainder matrix T " is the summation of double-bounce and odd-bounce components Then, we can get the following equations: where T mn is the m row, and n column element of the matrix T " . According to (18), we can derive following equations.
Note that there are four solutions for θ odd . The appropriate solution is available with the following constrains.
The branch condition used in [8] is adopted to determine the dominance of doublebounce and odd-bounce scattering mechanisms, such as If C 0 > 0, the odd-bounce scattering mechanism is dominant, and parameter α is set to zero. Otherwise, the double-bounce scattering mechanism is dominant, and parameter β is set to zero. Once α or β is decided, their scattering power coefficients f d and f s can be derived and their scattering power are obtained as follows.

Comparison with X-Band Pi-SAR Data
The decomposition results from X-band Pi-SAR data are shown in Figure 4. The major portion of the scene are buildings and city blocks. Four patches labeled A~D are selected for further comparison, and their enlarged decomposition results are shown in Figure 5. Patch A is an urban area with buildings parallel to the flight direction. Patch B is an oriented urban area, and the major orientation angle is around 45 degrees. Patch C is a forest area, and Patch D is a ground area. Quantitative analysis of dominant scattering components is demonstrated in Table 1. For urban area, the dominant scattering mechanism is double bounce. Nevertheless, for oriented urban area, double-bounce scattering is interpreted as volume scattering in the results of Y4D and G4U. The volume-scattering overestimation is obvious. The GMD has an improved result, while the percentage of dominant volume scattering (36.26%) is still larger than that of dominant double-bounce scattering (23.37%). The PSD and proposed method reached the maximum percentage of dominant double-bounce scattering (40.26%, 42.88%). What is more, the proposed method has a minimum percentage of dominant volume scattering (11.11%) among five methods over both the urban area and oriented urban area. For the forests area, the percentage of dominant volume scattering of the proposed method is lower than that of dominant double-bounce. However, compared with the PSD method, its volume-scattering component is improved. As for patch D, for the major part of the ground area, the percentage of dominant odd-bounce scattering (92.52%) in the proposed method is the highest. From Figure 5(a4,d4), we can find that there are oriented buildings around the ground area. In the decomposition results of (a4) and (b4), the dominant scattering of the boundaries is volume scattering, while in the result of the proposed method, the boundaries are interpreted as double-bounce scattering.
with the PSD method, its volume-scattering component is improved. As for patch D, for the major part of the ground area, the percentage of dominant odd-bounce scattering (92.52%) in the proposed method is the highest. From Figure 5(a4,d4), we can find that there are oriented buildings around the ground area. In the decomposition results of (a4) and (b4), the dominant scattering of the boundaries is volume scattering, while in the result of the proposed method, the boundaries are interpreted as double-bounce scattering.
In summary, the proposed decomposition method exhibits equivalent performance with the state-of-the-art methods over ground area and urban area with small orientation angles. Furthermore, the proposed method achieves improved performance over urban areas with large orientation angles, since the percentage of dominant double-bounce scattering increases, and the dominant volume scattering is the lowest.       In summary, the proposed decomposition method exhibits equivalent performance with the state-of-the-art methods over ground area and urban area with small orientation angles. Furthermore, the proposed method achieves improved performance over urban areas with large orientation angles, since the percentage of dominant double-bounce scattering increases, and the dominant volume scattering is the lowest.   From visual inspection, the percentage of dominant double-bounce scattering over the urban area in Figures 6f and 8f is increased compared with other decompositions. For manmade areas such as urban areas, the proposed method shows further improved performance. According to the quantitative analysis, the decomposition results of oriented urban areas have the largest percentage of dominant double-bounce scattering and the smallest percentage of dominant volume scattering. Note that all these model-based decompositions exhibit scattering mechanism ambiguity over Patch B for both RadarSat-2 and ALOS-2 data, which require further efforts to mitigate. Over natural areas, the five methods have similar decomposition performance. The forest areas are entirely volumescattering dominant, and ocean areas have 100% dominant surface scattering.

Comparison with C-Band Radarsat-2 Data and L-Band ALOS-2 Data
As a result, the decomposition performance of the proposed method is verified with different PolSAR datasets. In natural areas and urban areas of small orientation angles, the five methods have comparative decomposition performance. For oriented urban area, the proposed method achieves further improved performance.

Residual Examination
The proposed decomposition uses residual minimization as a measure to determine the optimal volume-scattering models. The minimized residuals are normalized by the corresponding measured coherency matrices for further examination The histograms of the normalized minimum residuals Rmin for the GMD and proposed method used Pi-SAR, RadarSat-2, and ALOS-2 PolSAR datasets are shown in Figure 10. It is obvious that the residuals are relatively small. There is a long tail in the small value region. For the proposed method, the normalized residual corresponding to peak values of pixels numbers is 10 −2 , 10 −4 , and 10 −4 , respectively. However, for the GMD decomposition, they are 5 × 10 −3 , 10 −3 , and 3 × 10 −3 . The normalized residuals of the proposed method demonstrate a relatively low level compared with GMD. Theoretically, the major difference between the proposed method and GMD lies in the orientation angle derivation procedure. With refined double-bounce orientation angles and scattering models, the proposed method appears to perform well and can fit mixed scenes well. However, there are still high residuals for some pixels. The four volume-scattering models used here may not be sufficient for these pixels, producing high residuals. Therefore, further efforts are needed to include more representative volume-scattering models to the volume-scattering lookup table.

Residual Examination
The proposed decomposition uses residual minimization as a measure to determine the optimal volume-scattering models. The minimized residuals are normalized by the corresponding measured coherency matrices for further examination The histograms of the normalized minimum residuals Rmin for the GMD and proposed method used Pi-SAR, RadarSat-2, and ALOS-2 PolSAR datasets are shown in Figure 10. It is obvious that the residuals are relatively small. There is a long tail in the small value region. For the proposed method, the normalized residual corresponding to peak values of pixels numbers is 10 −2 , 10 −4 , and 10 −4 , respectively. However, for the GMD decomposition, they are 5 × 10 −3 , 10 −3 , and 3 × 10 −3 . The normalized residuals of the proposed method demonstrate a relatively low level compared with GMD. Theoretically, the major difference between the proposed method and GMD lies in the orientation angle derivation

Conclusions
The scattering mechanism interpretation of oriented manmade targets, especially those with large POAs, is still a challenging task. The cross-polarization and off-diagonal terms induced by oriented double-and odd-bounce scattering are totally allocated to volume scattering in previous model-based decomposition. By modeling the double-bounce and odd-bounce structures with their independent orientation angles, general modelbased decomposition mitigates this issue and achieves improved performance. However, the model inversion is a nonlinear optimization and is time consuming. To inherit the advantage of these generalized models while speeding up the decomposition parameters inversion, this paper proposed a refined double-bounce scattering model with simplified double-bounce orientation angle. The main breakthrough point lies in disclosing the latent relationship between the derived double-bounce orientation angle and the POA. However, there are difficulties in directly deriving the double-bounce scattering orientation angles from actual oriented buildings, and the theoretical relationship between DBOA and POA is not clear. Therefore, this paper proposed a piecewise linear function to approximately fit the relationship between DBOA and POA. Then, a refined double-bounce scattering model was developed. With the refined double-bounce scattering models, the proposed model-based decomposition derives the scattering power of four components with linear solutions. Comparison investigations have been carried out with three kinds of PolSAR datasets, including PiSAR, RadarSat-2, and ALOS-2. Compared with the stateof-the-art decomposition methods including the Y4D, G4U, GMD, and PSD, the proposed method achieves better performance especially for oriented urban areas. Future work will be focused on the volume-scattering components derivation procedure.
Author Contributions: S.C. and Y.L. conceived the study. G.W. designed the experiment and analyzed PolSAR datasets in consultation with S.C. and Y.L. S.C. and G.W. wrote the paper. All authors

Conclusions
The scattering mechanism interpretation of oriented manmade targets, especially those with large POAs, is still a challenging task. The cross-polarization and off-diagonal terms induced by oriented double-and odd-bounce scattering are totally allocated to volume scattering in previous model-based decomposition. By modeling the double-bounce and odd-bounce structures with their independent orientation angles, general model-based decomposition mitigates this issue and achieves improved performance. However, the model inversion is a nonlinear optimization and is time consuming. To inherit the advantage of these generalized models while speeding up the decomposition parameters inversion, this paper proposed a refined double-bounce scattering model with simplified double-bounce orientation angle. The main breakthrough point lies in disclosing the latent relationship between the derived double-bounce orientation angle and the POA. However, there are difficulties in directly deriving the double-bounce scattering orientation angles from actual oriented buildings, and the theoretical relationship between DBOA and POA is not clear. Therefore, this paper proposed a piecewise linear function to approximately fit the relationship between DBOA and POA. Then, a refined double-bounce scattering model was developed. With the refined double-bounce scattering models, the proposed model-based decomposition derives the scattering power of four components with linear solutions. Comparison investigations have been carried out with three kinds of PolSAR datasets, including PiSAR, RadarSat-2, and ALOS-2. Compared with the state-of-the-art decomposition methods including the Y4D, G4U, GMD, and PSD, the proposed method achieves better performance especially for oriented urban areas. Future work will be focused on the volume-scattering components derivation procedure.
Author Contributions: S.C. and Y.L. conceived the study. G.W. designed the experiment and analyzed PolSAR datasets in consultation with S.C. and Y.L., S.C. and G.W. wrote the paper. All authors have read and agreed to the published version of the manuscript.