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Ship Detection Using Polarization Cross-Entropy Jiong Chen, Student Member, IEEE, Yilun Chen, Student Member, IEEE, and Jian Yang, Member, IEEE

Abstract—In this paper, the polarization cross-entropy is introduced based on the eigen-decomposition of the polarimetric coherency matrix. Then the new parameter is employed for ship detection. From experimental results, it is derived that the distribution of the polarization cross-entropy in ocean region can be well approximated by a generalized exponential distribution. Then a CFAR ship detection method is proposed based on the distribution of the polarization cross-entropy. Using experimental results, the authors demonstrate the effectiveness of the new method in ship detection.

exponential distribution. Then we employ a constant falsealarm rate (CFAR) method to detect ships in an ocean region. The usefulness and effectiveness of the proposed method is validated through detection results using full polarimetric data from NASA/JPL AirSAR. II. S HIP

DETECTION USING POLARIZATION CROSS - ENTROPY

A. Eigen-decomposition of Polarimetric Coherence matrix I. I NTRODUCTION A polarimetric aperture synthesis radar (SAR) has been widely adopted in the earth surveillance due to its all-day and all-weather capability. Ship detection, an important application in earth surveillance, has been deeply studied for years. Standard polarimetric detectors, such as the Polarimetric Whitening Filter (PWF) [1] and the Intensity Likelihood Ratio Test (ILRT) [2] have been successfully applied in ship detection. The basic idea under these standard detectors is to reduce multi-channels of polarimetric data to single decision criteria, in order to perform a detection process. In 2003, a more general approach to detection using the full coherency matrix has been developed [3][4] by Schou et al, which uses the Wishart distribution and CFAR detector for edge detection. Extracting appropriate parameters from full polarimetric data is also very essential for target detection in polarimetric SAR images. Cameron proposed the coherent target decomposition method [5] in 1990, which was then applied to ship detection by Robert Ringrose [6] in 1999, using SIR-C singlelook complex image. In 1996, Cloude and Pottier took a review of target decomposition theory and proposed the concept of polarization entropy [7]. Although there have been many available polarimetric parameters proposed, most of them are introduced for measuring the scattering characteristics of a target. For example, polarization entropy describes the polarimetric scattering randomness within a neighborhood of a given target and the angle describes its possible scattering mechanism; the Cameron decomposition results describe how the given target behaves like a specified known target. For target detection, it is necessary to introduce a discriminative parameter, which can be used to enhance the difference between an interested target and its local clutter. In Schou’s method, the full coherence matrix with Wishart distribution is used to discriminate the difference between two targets. Eigenvalues and entropy are important parameters extracted from the full coherency matrix. In this paper, the Polarimetric Cross-Entropy (PCE) is introduced for ship detection. The new parameter is capable of measuring the polarimetric scattering difference between targets and local clutter. Furthermore, we conclude that the distribution of PCE over an ocean region can be well described by a generalized

For a reciprocal case, the scattering vector (Pauli basis) of a target is ~k = √1 [shh + svv , shh − svv , 2shv ]T 2

(1)

where h and v denotes horizontal polarization basis and vertical polarization basis, respectively. The coherence matrix is then defined as D E T = ~k · ~k H (2) which can be decomposed as

T = U · Λ · UH

(3)

The diagonal matrix Λ contains three eigenvalues of the coherence matrix T , expressed by   λ1   λ2 Λ= (4)  λ3

where λ1 ≥ λ2 ≥ λ3 ≥ 0. The unitary matrix U contains eigenvectors of T . h i U = ~u1 ~u2 ~u3 (5) where the vectors ~ui , i = 1..3 can be formulated as iT h jγ jδ ~ui = cos αi sin αi cos βi e i sin αi sin βi e i

(6)

The definition of αi , βi , δi and γi can be found in [7]. Based on the eigen-decomposition results shown above, Cloude and Pottier [7] defined three important parameters for the representation of target information, which are the polarization entropy H, alpha angle α, and anisotropy A. These parameters are proved to be effective in terrain classification [8]. H is defined as follows: H=−

3 X i=1

pi log3 pi , pi =

λi 3 P λi

(7)

i=1

For the ocean and ships have different scattering characteristics, it is reported that polarization entropy H can be useful to detect ships in oceans [9], [10], even ship wakes [11].

2

B. Polarization cross-entropy Most current available polarimetric parameters are descriptive. For example, the polarization entropy describes the randomness of the scattering mechanism of targets. However, for the purpose oriented to target detection, a parameter is desired to reflect the scattering mechanism difference between a target and its local clutter. In this section, we propose the polarization cross-entropy based on the eigen-decomposition of the polarimetric coherence matrices. The definition of the polarization cross-entropy is also based on the eigen-decomposition of coherence matrices. Suppose that the polarimetric coherency matrices of target and clutter are Tt and Tc , respectively. Decompose Tt and Tc as follows,   λ1   H λ2 T t = Ut  (8)  Ut λ3 µ1



µ2

 T c = Uc 

µ3



 H  Uc

(9)

Fig. 1. Structured cell window for calculating polarization cross-entropy. The guard area ensures no target cells are included in the clutter statistics estimation.

(a) Span image 1

(b) Span image 2

(c) Span image 3

(d) HV image 1

(e) HV image 2

(f) HV image 3

(g) PCE image 1

(h) PCE image 2

(i) PCE image 3

The polarization cross-entropy is then defined as P CE =

3 X

pi log3

i=1

pi qi

(10)

where pi and qi are the normalized eigenvalues of Tt and Tc , respectively. It should be better if pi and qi are corresponding to the similar scattering mechanism for the same i. However, for the worst case, they can be sorted decreasingly. pi =

λi 3 P

,

qi =

λi

i=1

µi 3 P

(11)

µi

i=1

The polarization cross-entropy is easily proved to have the following properties: (1) P CE ≥ 0 for any polarimetric coherency matrices Tt and Tc ; (2) Ht = Hc , if P CE = 0, where Ht and Hc are the polarization entropies derived from matrices Tt and Tc . For a given pixel of a polarimetric SAR image, its polarization cross-entropy can be calculated from the structured cell window shown in Fig.1. The target cells can be selected to be 3 × 3 pixels and the clutter cells are selected to be the 1 ∼ 2 pixel-margin of a 9×9 pixel-window empirically. In the clutter cells, the pixels with large span value could be trimmed off to make sure that the clutter cells are independent and identically distributed. By estimating Tt and Tc from the target cell and clutter cell, respectively, the pixel’s PCE can be obtained from (10). Specifically, we let (q1 , q2 , q3 ) = (1/3, 1/3, 1/3), which means the clutter is supposed to be white-Gaussian distributed, then with (10) and (7), we have P CE = 1 +

3 X i=1

pi log3 pi = 1 − H

where H is the polarization entropy.

(12)

Fig. 2. Span, HV and PCE images with different frequency. Image1 : L-band, image2 and image3 : C-band.

Extensive experimental results have shown polarization cross-entropy is efficient in extracting discriminative features to enlarge the contrast between target and clutters, especially for ocean target. Fig.2 demonstrates its effectiveness through evaluating a series of full polarimetric data collected by NASA/JPL AIRSAR in different regions. For ship targets are always presenting high response in cross-polarization channels, we give HV images for comparison. C. The distribution of polarization cross-entropy over ocean regions In this section, we discuss the statistical distribution of polarization cross-entropy over ocean regions. The theoretical closed form of the polarization cross-entropy may be quite complex given the assumption of a Wishart distribution of

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Fig. 3.

Generalized exponential distribution family

Fig. 4. The histogram of PCE over ocean regions and the fit exponential distribution.

polarimetric coherency matrix. Rather than deriving a theoretical solution, we aim at observing the result from practical data. With the polarimetric SAR data over several different areas, we have found empirically that the ocean’s polarization cross-entropy histograms is well modeled with the following distribution: β f (x) = Ke−(u/α) , x ≥ 0 (13) The parameter β is associated with the decreasing rate of the peak and α is with R ∞the variance. The constant K is adjusted in order to have 0 f (x) dx = 1. That is, β   K= αΓ 1/β

where

Γ (β) =

Z

(14)

∞

e−u uβ−1 du

(15)

0

(13) includes a distribution family with different parameters α and β, as shown in Fig.3. Sparse distributions, i.e., a pdf with a peak at zero and decreasing tail, can be well approximated by the adopted distribution. When β = 1 , (13) yields the exponential distribution. Therefore, the presented distribution family is named as generalized exponential distribution, which can be regarded as a single-edge version of the generalized Laplacian distribution [13]. The coefficients α and β in (13) can be computed by the first and second moment from the distribution of the polarization cross-entropy R∞ R∞ m1 = 0 xf (x) dx m2 = 0 x2 f (x) dx (16) Substituting (13) into both the integrals, we have     2 3 m1 = K αβ Γ β2 m2 = K αβ Γ β3 β = F −1 where



m21 m2



α=

1 m2 Γ( β ) 3 Γ( β )

 2 Γ 2/x F (x) =     Γ 3/x Γ 1/x

III. PCE BASED CFAR DETECTION ALGORITHM Given the distribution model of ocean clutter as shown in eq.(13), it can be proved that the false alarm Pf a and the detection threshold t have the following relationship, !  β 1 t Γ , = 1 − Pf a (20) α β where Γ (·, ·) is the incomplete Gamma function, defined as [10] Z x Γ (x, a) = e−u ua−1 du (21) 0

Therefore, the detection threshold t can be calculated by    β1 1 −1 t=α Γ 1 − Pf a , (22) β where Γ−1 (·, ·) is the inverse incomplete Gamma function. One may refer to [10] for its solution. Specifically, when β = 1, (22) turns out to be t = −α ln Pf a

(23)

The procedure of the proposed CFAR detection algorithm is summarized in Algorithm 1.

(17) IV. EXPERIMENTAL RESULTS

Therefore,

r

In practice, the F −1 (·) function can be computed through a look-up table with interpolation correction. can be

And  calculated directly by m1 = hxi and m1 = x2 , where h·i denotes the operator for empirical mean value. Fig.4 illustrates an example where the PCE histogram of an ocean region is well fitted by distribution in (13), with α = 8.128 × 10−3 and β = 0.987 estimated by (18).

(18)

(19)

The proposed detection algorithm is validated through the NASA/JPL AIRSAR full polarimetric SAR data in the Sydney coast, Australia. The data is in C-band, with incident angles spanning from 21.8o ∼ 71.9o . Fig.5 shows a set of images from original data to final detection result. The false alarm rate is set to be 0.5%. Fig.5a and Fig.5b are span and HV images as reference, on which some target points are

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Algorithm 1 PCE based CFAR detection algorithm procedure • 1. Calculate the polarization cross-entropy by Fig.1 over the whole polarimetric SAR image; • 2. For each pixel (i, j) , given a false alarm rate Pf a 1) Get target’s PCE xti,j and clutter’s PCE by Fig.1, which have been calculated in Step 1 2) Estimate the distribution parameters (αi,j , βi,j ) from the clutter cells by eq.(18). 3) Calculate the local detection threshold ti,j with Pf a and (αi,j , βi,j ) by eq.(22). 4) If xti,j > ti,j then mark the pixel (i, j) as target, else mark as non-target. end procedure

from ocean background through practical polarimetric SAR data. Furthermore, we proposed a new CFAR ship detection algorithm based on polarization cross-entropy. The detection algorithm is evaluated to be efficient mainly by C-band NASA/JPL AIRSAR data. The universality and robustness of the proposed method in lower frequency image or windy conditions need to be further studied and validated. It should also be pointed out that in the proposed detection algorithm we only utilize the PCE parameter, ignoring any differences in the scattering mechanisms. It is expected to enhance the detection, if we’ll add other polarimetric information, such as similarity parameter of diplane [14] because ships often induce dihedral scattering, which is often not present on the ocean surface. ACKNOWLEDGMENT This work was supported in part by the National Natural Science Foundation of China (40871157). R EFERENCES

(a) Span image

(b) HV image

(c) Polarization entropy image

(d) Detected result

Fig. 5. Experimental results for the CFAR detection algorithm. Points 1 ∼ 6 in (d) indicate there are potential targets. However, in span image (a), potential targets 1, 2, 6 are not so obvious and in HV image (b), potential targets 3, 4, 6 give very little response, which are all pointed out by dashed arrows.

difficult to recognize. From the polarization cross-entropy image shown in Fig.5c, it is very clear to discriminate targets from ocean background, and the cross-entropy of ocean pixel itself provides a very little response. This is propitious to the following detection step which is also demonstrated by the detection result shown in Fig.5d. In the detection result, points 1 ∼ 6 indicate there are potential targets, while some are not so obvious in either span or HV images. V. CONCLUSION The polarization cross-entropy has been proposed in this paper as a discriminative parameter to detect ships in ocean areas. It is validated to be very effective to discriminate ships

[1] L.M Novak and M.C. Burl, Optimal Speckle Reduction In POL-SAR Imagery and its Effect on Target Detection SPIE Conference, Orlando, FL, February, 1989 [2] R. D. Chaney, M.C. Bud and L. M. Novak. On the Performance of Polarimetric Target Detection Algorithms IEEE AES Magazine, pp: 1015. November 1990 [3] K. Conradsen, A. A. Nielsen, J. Schou and H. Skriver. A test statistic in the complex Wishart distribution and its application to change detection in polarimetric SAR data, IEEE Trans. Geosci. Remote Sensing, Vol. 41(1), pp. 4 C 19, 2003 [4] J. Schou, H. Skriver, A. A. Nielsen, K. Conradsen. CFAR edge detector for polarimetric SAR images, IEEE Trans. Geosci. Remote Sensing, Vol. 41(1), pp. 20 C 32, 2003 [5] W. L. Cameron, N. Youssef, and L. K. Leyng, Simulated polarimetric signatures of primitive geometrical shapes IEEE Trans. Geosciences Remote Sensing, 34(3): 793-803, 1996 [6] R. Ringrose, N. Harris, Ship Detection Using Polarimetric SAR Data, In Proc. of the CEOS SAR workshop. ESA SP-450, October 1999 [7] S. R. Cloude, E. Pottier, A Review of Target Decomposition theorems in Radar Polarimetry IEEE Transactions on Geoscience Remote Sensing, 34(2):498-518, March 1996 [8] S. R. Cloude, et al. An Entropy Based Classification Scheme for Land Applications of Polarimetric SAR. IEEE Trans. Geosci. Remote Sensing, 1997, 35(1): 68-78. [9] R. Touzi, F. Charbonneau, R.H Hawkins, Ship-Sea contrast Optimization When using polarimetric SAR, IGARSS.01, Sydney, Australia, 2001 [10] R. Touzi, F. Charbonneau, Characterization of symmetric scattering using polarimetric SAR, IGARSS’02, TORONTO, CANADA, 2002 [11] E. Pottier, W.M. Boerner, D.L. Schuler, Polarimetric detection and estimation of ship wakes, IGARSS’99, Germany, 1999 [12] J. Cody, An Overview of Software Development for Special Functions, Lecture Notes in Mathematics, 506, Numerical Analysis Dundee, G. A. Watson (ed.), Springer Verlag, Berlin, 1976. [13] S. G. Mallat, A theory for multi-resolution signal decomposition: The wavelet representation, IEEE Pat. Anal. Mach. Intell., vol. 11, pp. 674693, July 1989. [14] J. Yang, Y. N. Peng, S. M. Lin, Similarity between two scattering matrices, Electronics Letters vol. 37(3), pp. 193-194, 2001.