# Three-Component Power Decomposition for Polarimetric SAR Data Based on Adaptive Volume Scatter Modeling

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Double Transformation of Coherency Matrix

#### 2.1. Orthogonal Transformation (Rotation) of Coherency Matrix

^{H}denote the complex conjugate and the conjugate transpose, respectively; ${\mathit{k}}_{p}=\frac{1}{\sqrt{2}}{[{S}_{HH}+{S}_{VV},{S}_{HH}-{S}_{VV},2{S}_{HV}]}^{T}$ is the Pauli vector. Rotation of

**T**by the orthogonal transformation is defined as [7]:

**R**

_{1}(θ) is an orthogonal matrix given by:

_{33}(θ) term. Specifically, according to Equations (2) and (3), T

_{33}(θ) becomes:

_{33}(θ) to zero yields:

_{23}) = (T

_{22}– T

_{33})tan4θ/2. By substituting Re(T

_{23}) into Equation (4), and after some necessary simplification, T

_{33}(θ) can be rewritten as:

_{33}(θ), it follows from Equation (6) that T

_{22}– T

_{33}and cos4θ should be of the same sign. Then according to Equation (5), we have:

_{22}– T

_{33}> 0; and k = 1 if T

_{22}– T

_{33}< 0.

**T**(θ). Firstly, T

_{22}(θ) is always larger than T

_{33}(θ), i.e., T

_{22}(θ) > T

_{33}(θ); second, T

_{23}(θ) becomes purely imaginary and T

_{23}(θ) = jIm(T

_{23}). As will be shown in Section 3, the former property is a necessary condition that guarantees non-negative powers in the decomposition results.

#### 2.2. Unitary Transformation of Coherency Matrix

**T**(θ) is further processed by a unitary transformation, that is:

**R**

_{2}(φ) is a unitary matrix given by

_{33}(φ) and by a similar reasoning. It is:

_{22}(θ) > T

_{33}(θ), as the result of the orthogonal transformation by

**R**

_{1}(θ), so that no ambiguity correction is needed as in Equation (7).

**T**(φ), the inequality T

_{22}(φ) > T

_{33}(φ) still holds because of the minimization of T

_{33}(φ); in addition, it can be easily proved that the element T

_{23}(φ) is forced to be completely zero, i.e., T

_{23}(φ) = 0.

**R**

_{1}(θ) and the unitary transformation

**R**

_{2}(φ) are related to the normalized circular-pol correlation coefficients proposed by Ainsworth et al. [8] which are used to characterize man-made structures in urban areas. Here the transformation angle θ and φ respectively account for the orientation angle and helicity in their paper. This again explains that distinction of man-made structures can be improved by such a double transformation.

**T**′ =

**T**(φ) which will be hereafter used for power decomposition.

## 3. Adaptive Volume Scattering Model and Power Decomposition

_{11}< 2T′

_{33}in the transformed coherency matrix. In order to prevent such a problem, we relax the volume scattering model as follows:

**A**and

**B**as Tr(

**A**

^{H}

**B**) where Tr(.) denotes the trace, then it is easy to derive the solution of Equation (13) as follows:

^{*}has been determined, the remaining procedure is the same to that of the Freeman-Durden decomposition [1]. However, it can be shown that as a consequence of the adaptive volume model Equation (14), the negative power problem will be completely avoided. After subtracting the volume scattering from

**T**′, the remaining coherency matrix becomes:

_{22}− T′

_{33}always holds as a result of the double transformation introduced in Section 2. It is also easy to verify that T′

_{11}− γ

^{*}T′

_{33}> 0 as a result of Equation (14). In other words, both diagonal elements remain positive, which prevents any possible negative solutions.

_{S}, f

_{D}, α, and β are unknowns to be determined. According to Equations (15) and (16), one needs to solve the following equations:

_{11}− γ

^{*}T′

_{33}) (T′

_{11}− T′

_{33}) and |T′

_{12}|

^{2}. If (T′

_{11}− γ

^{*}T′

_{33})(T′

_{11}− T′

_{33}) ≥ |T′

_{12}|

^{2}, then at least one physically meaningful (non-negative power) solution exists to Equation (17). In fact, the common treatment by assuming there is a dominant scattering mechanism [1] gives one a reasonable result. Another solution can be also derived by the eigen-decomposition approach [4].

_{11}− γ

^{*}T′

_{33})(T′

_{11}− T′

_{33}) < |T′

_{12}|

^{2}, it can be proved that no solution to Equation (17) exists (Appendix 1). In this case f

_{S}, f

_{D}, α, and β and are found by:

_{D}= 0) and vice versa.

## 4. Experimental Results

## 5. Discussion

_{HH}${S}_{\text{VV}}^{*}$) < 0. An explanation of this phenomenon is that the volume scattering within urban areas can be affected by randomly orientated double-bounce scatterers. Our experiments showed that discrimination of urban areas is improved by such an assumption.

## 6. Conclusion

## Acknowledgments

## References

- Freeman, A.; Durden, S.L. A three-component scattering model for polarimetric SAR data. IEEE Trans. Geosci. Remote Sens
**1998**, 36, 963–973. [Google Scholar] - Yamaguchi, Y.; Moriyama, T.; Ishido, M.; Yamada, H. Four-component scattering model for polarimetric SAR image decomposition. IEEE Trans. Geosci. Remote Sens
**2005**, 43, 1699–1706. [Google Scholar] - An, W.; Cui, Y.; Yang, J. Three-component model-based decomposition for polarimetric SAR data. IEEE Trans. Geosci. Remote Sens
**2010**, 48, 2732–2739. [Google Scholar] - Van Zyl, J.J.; Arii, M.; Kim, Y. Model-based decomposition of polarimetric SAR covariance matrices constrained for nonnegative eigenvalues. IEEE Trans. Geosci. Remote Sens
**2011**, 49, 1104–1113. [Google Scholar] - Arii, M.; Van Zyl, J.J.; Kim, Y. Adaptive model-based decomposition of polarimetric SAR covariance matrices. IEEE Trans. Geosci. Remote Sens
**2011**, 49, 1104–1113. [Google Scholar] - Singh, G.; Yamaguchi, Y.; Park, S.-E. 4-Component Scattering Power Decomposition with Phase Rotation of Coherency Matrix. Proceedings of 2011 IEEE International Geoscience and Remote Sensing Symposium, Vancouver, BC, Canada, 25–29 July 2011.
- Yamaguchi, Y.; Sato, A.; Boerner, W.M.; Sato, R.; Yamada, H. Four-component scattering power decomposition with rotation of coherency matrix. IEEE Trans. Geosci. Remote Sens
**2011**, 49, 2251–2258. [Google Scholar] - Ainsworth, T.L.; Schuler, D.L.; Lee, J.-S. Polarimetric SAR characterization of man-made structures in urban areas using normalized circular-pol correlation coefficients. Remote. Sens. Environ
**2008**, 112, 2876–2885. [Google Scholar] - Sato, A.; Yamaguchi, Y.; Singh, G.; Park, S.-E. Four-component scattering power decomposition with extended volume scattering model. IEEE Geosci. Remote Sens. Lett
**2012**, 9, 166–170. [Google Scholar] - An, W.; Zhang, W.-J.; Yang, J.; Hong, W.; Cao, F. On the similarity parameter between two targets for the case of multi-look polarimetric SAR. Chin. J. Electron
**2009**, 18, 545–550. [Google Scholar] - Freeman, A. Fitting a two-component scattering model to polarimetric SAR data from forests. IEEE Trans. Geosci. Remote Sens
**2007**, 45, 2583–2592. [Google Scholar]

## Appendix

_{11}− γ

^{*}T′

_{33}) (T′

_{11}− T′

_{33}) < |T′

_{12}|

^{2}, then there exists no solution to Equation (17). Specifically, we prove that if Equations (17a) and (17b) hold, Equation (17c) will be not satisfied. According to Cauchy-Schwarz inequality, we have:

_{S}, f

_{D}, α, and β, |f

_{S}β + f

_{D}α| < |T′

_{12}|

^{2}, which contradicts Equation (17c). Thus no solution to Equation (17) can be found.

_{11}− γ

^{*}T′

_{33}) (T′

_{11}− T′

_{33}) < |T′

_{12}|

^{2}. According to the triangular inequality and Equation (A1), we have:

_{D}= 0; on the other hand, if δ > 1, note that |β| < 1, according to Equation (A3b) f

_{S}= 0. The rest of the procedure is straightforward and the final result is given in Equation (19).

**Figure 2.**(

**a**) Original Freeman-Durden decomposition result for AIRSAR data; (

**b**) Proposed adaptive decomposition result for AIRSAR data; (

**c**) Map of γ.

**Figure 3.**(

**a**) Power distribution of original Freeman-Durden decomposition in vegetated area; (

**b**) Power distribution of proposed adaptive decomposition in vegetated area.

**Figure 4.**(

**a**) Original Freeman-Durden decomposition result for ALOS-PALSAR data; (

**b**) Proposed adaptive decomposition result for ALOS-PALSAR data; (

**c**) Map of γ.

**Figure 5.**(

**a**) Zoomed result by the original Freeman-Durden decomposition; (

**b**) Zoomed result by the proposed adaptive decomposition

**Figure 6.**(

**a**) Power distribution of original Freeman-Durden decomposition in urban area; (

**b**) Power distribution of proposed adaptive decomposition in urban area

## Share and Cite

**MDPI and ACS Style**

Cui, Y.; Yamaguchi, Y.; Yang, J.; Park, S.-E.; Kobayashi, H.; Singh, G.
Three-Component Power Decomposition for Polarimetric SAR Data Based on Adaptive Volume Scatter Modeling. *Remote Sens.* **2012**, *4*, 1559-1572.
https://doi.org/10.3390/rs4061559

**AMA Style**

Cui Y, Yamaguchi Y, Yang J, Park S-E, Kobayashi H, Singh G.
Three-Component Power Decomposition for Polarimetric SAR Data Based on Adaptive Volume Scatter Modeling. *Remote Sensing*. 2012; 4(6):1559-1572.
https://doi.org/10.3390/rs4061559

**Chicago/Turabian Style**

Cui, Yi, Yoshio Yamaguchi, Jian Yang, Sang-Eun Park, Hirokazu Kobayashi, and Gulab Singh.
2012. "Three-Component Power Decomposition for Polarimetric SAR Data Based on Adaptive Volume Scatter Modeling" *Remote Sensing* 4, no. 6: 1559-1572.
https://doi.org/10.3390/rs4061559