This document discusses a method for estimating the polarization orientation angle (OA) from polarimetric synthetic aperture radar (PolSAR) data by maximizing the Hellinger distance between the coherency matrix elements. The proposed estimation technique was tested on an airSAR image and demonstrated similar results to previous methods, with average OA values of -4° and 6° in different regions. The study emphasizes the importance of compensating for OA in geophysical parameter estimation and classification.

1. Orientation Angle Estimation from PolSAR Data using a
Stochastic Distance
Avik Bhattacharya1, Arnab Muhuri1, Shaunak De1, Alejandro C. Frery2
1Centre of Studies in Resources Engineering, IIT-Bombay
2LaCCAN
International Geoscience and Remote Sensing Symposium (IGARSS 2014)
Qu´ebec, Canada

2. Introduction
The angle of rotation of any object about the line of sight (LOS) is known as the
polarization orientation angle (OA) θ0.
OA is found to be non zero for undulating terrains and man-made targets oriented
away from the radar LOS. This effect is more pronounced at lower frequencies (eg.
L- and P- bands).
OA shift is not only induced by azimuthal slope but also by range slope.

3. Introduction
The OA shift increases the cross-polarization (HV) intensity
and subsequently the covariance or the coherency matrix
becomes reflection asymmetric.
Compensating this OA prior to any model-based
decomposition technique for geophysical parameter
estimation or classification is crucial.
In this paper we propose to estimate the OA (θ0) by
maximizing the Hellinger distance between the un-rotated
and rotated diagonal elements of the coherency matrix.

4. Stochastic Distance
The stochastic distance (divergence) is any symmetric, non-negative function of
two probability measures.
For two random matrices X and Y having densities fX (Z , θ1) and fY (Z , θ2)
respectively, the (h, φ) divergence between fX and fY is defined by
Where A is the support matrix, φ : (0, ∞) → [0, ∞) is a convex function and
h : (0, ∞) → [0, ∞) is a strictly increasing function with h(0) = 0

5. Stochastic Distance
dh
φ(X, Y) =
Dh
φ(X,Y)+Dh
φ(Y,X)
2
dh
φ : A × A → R are distances over A
The Hellinger distance is defined by:
dH(X, Y) = 1 −
√
fXfY

6. Methodology
The effect of the OA (θ) obtained from Lee and Ainsworth (2011) as on the three
diagonal elements of the coherency matrix states that:
1. T11 = |HH + VV |2
/2 is roll invariant for any θ
2. T22 = |HH − VV |2
/2 always increases or remains the same after the OA
compensation
3. T33 = 2 |HV |2
always decreases or remains the same after OA compensation.
OA (θ) obtained from Lee and Ainsworth (2011)
θ =
1
4
tan−1 −2Re(T23)
T33 − T22
(1)
The orientation angle θ is in the range [−π
8 , π
8 ]
J.-S. Lee and T. Ainsworth. The effect of orientation angle compensation on coherency matrix and polarimetric target decompositions. Geoscience
and Remote Sensing, IEEE Transactions on, 49(1):53–64, Jan 2011. ISSN 0196-2892. doi: 10.1109/TGRS.2010.2048333.
J.-S. Lee, D. Schuler, and T. Ainsworth. Polarimetric sar data compensation for terrain azimuth slope variation. Geoscience and Remote Sensing,
IEEE Transactions on, 38(5):2153–2163, Sep 2000. ISSN 0196-2892. doi: 10.1109/36.868874.

7. Methodology
In PolSAR data analysis, z is often assumed to obey multivariate complex
Gaussian distribution f (z; Σ) with zero mean (2),
f (z; Σ) =
1
πp |Σ|
exp(−z∗T
Σ−1
z) (2)
The p × p Hermitian positive definite matrix A = nZ follows a complex Wishart
distribution given by (3)
f (A) =
|A|L−p
exp[−Tr(Σ−1A)]
Γp(L) |Σ|L
(3)
Γp(L) = π
p(p−1)
2
p
i=1
Γ(L − i + 1) =
A
|A|(L−p)
exp[Tr(A)] dA (4)

8. Methodology
The Hellinger distance between two central complex Wishart distribution for
Σ1 = Σ2 (different covariance matrices) and L1 = L2 = L (same number of looks)
is given by (5),
DH = 1 − f1(A)f2(A) = 1 −
|Σ1|L/2
|Σ2|L/2
Σ1
2 + Σ2
2
L (5)

9. Methodology
The method proposed in this paper estimates the OA by first maximizing the
Hellinger distance over the range [−π
4 , π
4 ] between un-rotated T22, T33 and
rotated T22(θ), T33(θ) elements respectively leading to two candidate angles,
φ3 = argmax
−π/4<θ≤π/4



1 −


2 σ2
3σ2
3(θ)
σ2
3 + σ2
3(θ)


L



φ2 = argmax
−π/4<θ≤π/4



1 −


2 σ2
2σ2
2(θ)
σ2
2 + σ2
2(θ)


L



(6)

10. Methodology
Two maxima are found at φ = φ{3,2} and φ = φ{3,2} ± π/4, and the OA is chosen such
that the Hellinger distance (DHφ3 ) corresponding to σ2
3 is greater than the Hellinger
distance (DHφ2 ) corresponding to σ2
2 either at φ = φ{3,2} or at φ = φ{3,2} ± π/4.
Figure: Estimated angle from Hellinger distance (Red: T33, Blue: T22)

11. Methodology
This condition exactly corresponds to the case where the cross-polarized (HV )
component is minimized
The other peak corresponds to the situation where the HV component is
incorrectly maximized.
The OA (θ0) is obtained in the range [−π
8 , π
8 ] by
θ0 =



φ + π/4, if φ < −π/8
φ − π/4, if φ > π/8
φ, otherwise
(7)

12. Results
Figure: (a) Pauli RGB (R=< |SHH − SVV | > , G=< 2|SHV | >, B=< |SHH + SVV | >), (b)
Estimated OA angle from Lee et al. (2000), (c) Estimated OA angle from the proposed method

13. Match with Lee et al. (2000)’s Method
The results from the proposed method and the one described in Lee et al. (2000)
show a high degree of similarity
Difference between the two results is almost zero, as shown:
Theta
Figure: Histogram of difference in θ0 computed from the proposed and that described
in Lee et al. (2000)

14. Results
Figure: Comparison of OA estimated from Lee et al. (2000) and the proposed method (Blue:
Lee et. al, Red: Proposed method)

15. Results
Figure: Real orientation angle (a) real orientation angle obtained by the proposed method, (b)
real orientation angle obtained by the method of (Lee and Ainsworth, 2011), (c) comparison of
the estimated orientation angle for the given transect, (d) histogram of the difference in
orientation angle computed for the entire scene by the two methods.

16. Summary
The proposed method is tested on an AIRSAR San-Francisco L-band image with
some built-up areas oriented from the LOS (A and B)
The comparison of OA estimated from Lee et al. (2000) and the proposed method
along the horizontal profile (Red line) in is shown.
As it can be seen the proposed OA estimation exactly matches the OA estimated
from Lee et al. (2000).
The average OA estimated over region A is −4◦ and the average OA over region
B is 6◦.
The region C is perpendicular to the LOS and hence the OA estimated is ∼ 0.
The proposed method gives an insight into the behavior of the T22 and T33
elements with the change in the orientation angle.
The usefulness of the Hellinger distance as a polarimetric descriptor will be further
investigated.