Ship Detection Using Polarization Cross-Entropy Yilun Chen, Jiong Chen, Jian Yang, Weijie Zhang Department of EE, Tsinghua University Beijing, 100084, China
[email protected]
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 extensive experimental results, it is found that the distribution of the polarization cross-entropy in ocean region can be well approximated by a generalized exponential distribution, yielding a CFAR ship detection method using the proposed polarization cross-entropy. Experimental results demonstrate the effectiveness of the new method in ship detection.
Introduction A polarimetric synthesis aperture 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. Extracting appropriate parameters from full polarimetric data is also very essential for target detection in polarimetric SAR images. Cameron proposed the coherent target composition method [3] in 1990, which was then applied to ship detection by Robert Ringrose in 1999 [6], using SIR-C single-look complex image. In 1996, Cloude and Pottier took a review of target decomposition theory and proposed the concept of polarization entropy [4]. 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 clutters. 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 exponential distribution. Then we employ a constant false-alarm 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.
Ship detection using polarization cross-entropy 1．
Eigen-decomposition of Polarimetric Coherence matrix
For a reciprocal case, the scattering vector (Pauli basis) of a target is 1 T (1) k shh svv , shh svv ,2shv , 2 where h and v denotes horizontal polarization basis and vertical polarization basis, respectively. The coherence matrix is then defined as (2) T k kH , which can be decomposed as T U U H . (3) The diagonal matrix contains three eigenvalues of the coherence matrix T , expressed by
1 2
, 3
(4)
where 1 2 3 0 . The unitary matrix U contains eigenvectors of T
U u1 u2
u3 ,
(5)
where the vectors ui , i 1..3 can be formulated as
ui cos i
sin i cos i e ji
T
sin i sin i e ji .
(6)
Based on the eigen-decomposition results shown above, Cloude and Pottier [4] defined three important parameters for the representation of target information, which are the polarization entropy H , alpha angle , and anisotropy A . H , and A are defined as follows:
i
3
H pi log 3 pi , pi i 1
3
i 1
,
(7)
i
3
pii ,
(8)
2 3 . 2 3
(9)
i 1
and A
These parameters are proved to be effective in terrain classification [12]. For the ocean and ships have different scattering characteristics, it is reported that polarization entropy H can be useful to detect ships in oceans [8], [9], even ship wakes [5].
2．
Polarization cross-entropy
Most current available polarimetric parameters are descriptive. As mentioned in the above section, for example, the polarization entropy describes the randomness of the scattering mechanism of targets, angle describes the physics behind the scattering process (e.g., when is close to 0, the scattering corresponds to single-bounce scattering produced by a rough surface). 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 covariance 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,
1 Tt U t 2 1 Tc U c
2
U H t 3
(10)
U H . c 3
(11)
The polarization cross-entropy is then defined as 3
PCE pi log i 1
pi , qi
where pi and qi are the normalized eigenvalues of Tt and Tc , respectively,
(12)
pi
i 3
, qi
i
i 1
i
(13)
3
i 1
i
The polarization cross-entropy is easily proved to have the following properties: (1) PCE 0 for any polarimetric coherency matrices Tt and Tc (2) H t H c if PCE 0 , where H t and H c 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 Figure 1. By estimating Tt and Tc from the target cell and clutter cell, respectively, the pixel’s PCE can be obtained from eq.(12).
Figure 1 Structured cell window for calculating polarization cross-entropy. The guard area ensures no target cells are included in the clutter statistics estimation.
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 eq.(12) and eq.(7), we have 3
PCE 1 pi log3 pi 1 H
(14)
i 1
where H is the polarization entropy. From this point of view, the polarization entropy can be regarded as the target behaves “unlike” white-Gaussian clutter, where such “unlike” is measured by the proposed polarization cross-entropy. 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. Through these results we can see that the new parameter is very useful for target detection in an ocean area. For the ocean itself, the value of polarization cross-entropy is close to zero; for the pixels containing potential targets, the value of polarization cross-entropy rises significantly.
(a)
(a) Figure 2 Span images (a) and their corresponding PCE images (b) of polarimetric data from NASA/JPL AirSAR. Targets in oceans can be clearly seen from the PCE images.
3．
The distribution of polarization cross-entropy over ocean regions
In this section, we discuss the statistical distribution of polarization cross-entropy over ocean regions, which would be used in our detection algorithm. The theoretical closed form of the polarization cross-entropy may be quite complex given the assumption of a Whishart distribution of 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:
f ( x) Ke (u / ) , x 0 .
(15)
The parameter is associated with the decreasing rate of the peak and is with the variance. The constant K is adjusted in order to have
K
1
f x dx 1 . That is, 0
,
(16)
where
eu u 1du . 0
(17)
Eq.(15) includes a distribution family with different parameters and , 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 1 , eq.(15) 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 generalized Laplacian distribution [11].
Figure 3 Generalized exponential distribution family.
The coefficients and in eq.(15) can be computed by the first and second moment from the distribution of the polarization cross-entropy
m1 xf x dx 0
(18)
and
m2 x 2 f x dx . 0
(19)
Substitute eq.(15) into both the integrals, we have
m1 K
2 2
(20)
m2 K
3 3 .
(21)
and
Therefore,
m12 m2
F 1
(22)
and
1 m2 , 3
(23)
x . F x 3 1 x x
(24)
where
2
2
The function F 1 x is shown in Fig. 4. In practice, the F 1 function can be computed through a look-up table with interpolation correction. And m1 , m2 can be calculated directly by m1 x
and m1 x 2 , where
denotes the operator for empirical mean value.
Figure 4 Plot of
F 1 ( x) by eq.(24).
Fig.5 gives an example where the PCE histogram of a ocean region is well fitted by distribution in eq. (15), with 8.128 103 and 0.987 estimated by eq.(22) and eq.(23), respectively.
0. 12 dat a his t ogram fit dis t ribut ion 0. 1
0. 08
0. 06
0. 04
0. 02
0
0
0. 01 0. 02 0. 03 0. 04 0. 05 0. 06 0. 07 0. 08 0. 09
Figure 5 The histogram of PCE over ocean regions and the fit exponential distribution.
4．
PCE based CFAR detection algorithm
Give distribution model of ocean clutter as shown in eq.(15), it can be proved that the false alarm Pfa and the detection threshold t have the following relationship, t 1 , 1 Pfa ,
(25)
where , is the incomplete Gamma function, defined as [10]
x, a eu u a 1du . x
0
(26)
Therefore, the detection threshold t can be calculated by 1
1 t 1 1 Pfa , ,
(27)
where 1 , is the inverse incomplete Gamma function. One may refer to [10] for its solution. Specifically, when 1 , eq.(27) turns out to be
t ln Pfa . The procedure of the proposed CFAR detection algorithm can be summarized as Table 1.
(28)
Table 1 Procedure of the proposed detection algorithm (1) Calculating
the
polarization
cross-entropy
over
the
whole
polarimetric SAR image; (2) For each pixel
i, j ,
given a false alarm rate Pfa
t (a) Calculate target’s PCE xi , j from the target cell around pixel
i, j
(Fig.1) (b) Estimate the distribution parameter
i, j
, i , j from the clutter
cell by eq.(22) and eq.(23). (c) Calculate the local detection threshold ti , j with Pfa and
i, j
, i , j
by eq.(27). t (d) If xi , j ti , j then mark the pixel
i, j
as target, else mark as
non-target.
Experimental results The proposed detection algorithm is validated through polarimetric SAR data measured by NASA/JPL AirSAR. The image was selected from the Sydney coast, Australia. Fig.6 shows a set of images from original data to final detection result. The false alarm rate is set to be 0.5%. For comparison, we also calculated the polarization entropy (Fig.6(c)). It can be seen that the polarization entropy can reflect descriptive scattering characteristics of an ocean area, e.g., one can even observe patterns of ocean current from the entropy image. However, the target points such as ships are difficult to recognize, whereas from the polarization cross-entropy image (Fig.6(b)), 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 (Fig.6(d)).
(a) Span image
(b) PCE image
(c) Polarization entropy image
(d) Detected result
Figure 6 Experimental results for CFAR detection algorithm. The polarimetric data is captured by NASA/JPL AirSAR over Sydney coast, Australia.
Conclusion The polarization cross-entropy has been proposed in this paper as a discriminative parameter to detect ships in ocean areas. The new parameter is based on eigen-decomposition of two polarimetric coherency matrices and could well reflect the difference of scattering characteristics of a target and local clutter. The polarization cross-entropy is validated to be very effective to discriminate ships 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 by NASA/JPL AirSAR data. It should be pointed that in the proposed detection algorithm we only utilize the PCE parameter. It is expected to achieve better result if we add other polarimetric information, such as power intensity and other parameters in our detection algorithm.
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