SUPER-RESOLUTION OF POLARIMETRIC SAR IMAGES FOR SHIP DETECTION Chen Jiong

Yang Jian

Tsinghua University Email:[email protected]

Tsinghua University Email: [email protected]

Abstract—Polarimetric SAR images are used for aerial and space imagery applications, such as target detection, tracking, and resource exploration. However, spatial resolution is limited due to the signal bandwidth and the antenna dimension. Superresolution reconstruction is the process to reconstruct a highresolution image from multiple low-resolution images. Polarimetric SAR provides multiple images of the same scene at the different channels (HH, VV, HV and VH). In this paper, the POCS method is applied to extract the information from LR images in different channels to generate the HR image. The multiplicative characteristics of noises in SAR images are utilized to construct the convex sets in POCS method. Ship targets are selected as detection object for super-resolution. Results from simulated and real data have validated the effectiveness of proposed method.

I. I NTRODUCTION Synthetic aperture radars (SAR) have been found useful for making visual observations of Earth from space to obtain useful information. Spatial resolution is one of the most important parameters in a SAR imaging system. Images with high resolution (HR) are desired and often required in most applications such as target detection, tracking, recognition, etc., for an HR image can offer more details. However, spatial resolution is also extremely difficult to improve due to the bandwidth of signal and dimension of antenna. Polarimetric SAR provides multiple images of the same scene at the four channels (HH, VV, HV and VH), which present different scattering characteristic to the corresponding polarimetric state. Super-resolution is a technique to combine information in multiple images together to construct a high resolution image. The basic problem of high-resolution image reconstruction using multiple images was first addressed by Tsai and Huang [1]. The POCS formulation was first suggested by Stark and Oskoui [5]. Their method was extended by Tekalp to include observation noise [6]. Ships are important targets in both military and civil applications. Pastina used the super-resolution techniques for ship detection in polarimetric SAR image by introducing in parametric spectral estimators [7]. However, as a spectral domain method, the characteristics of Polarimetric SAR images are not fully extracted. POCS method has been validated theoretically and practically to be an effective super-resolution process to preserve the prior knowledge of multiple image while reduce the interference of noise. In this paper, we apply POCS method in polarimetric SAR image for ship targets.

This paper is organized as follows. In Section 2, we give a description of the multi-channel images in Polarimetric SAR system. In Section 3, we review the POCS method. Section 4 constructs the convex sets in POCS method by utilizing the multiplicative characteristics of noises and details the proposed algorithm for ship target in polarimetric SAR images, and Section 5 provides experimental results of both simulated and real data. Conclusions are given in Section 6. II. MULTI-CHANNEL POLARMETRIC SAR IMAGE A polarimetric SAR measures the complete scattering matrix S of a medium. The scattering matrix with given complex elements is given by: ! Shh Shv S= (1) Svh Svv To obtain an accurate knowledge of a target of interest in the SAR scene it is important to characterise it via its complete polarimetric scattering matrix. In most situation, elements in scattering matrix are complex so we should normalize each amplitude to (0, 255) to get the gray images in each channel. Outliers with extremely large values have been eliminated and set to 255 during the normalization process. Each channel provides a single LR image, thus four images from respective channels make up the LR image set for super-resolution, denoted as sk (m1 , m2 ). In the reciprocal backscattering case which is widely accepted in practice, we have Shv = Svh , so in most application, we have 3 different images. The polarimetric information can also be represented by the covariance matrix   m11 m12 m13 m14    m21 m22 m23 m24    M = (2)   m31 m32 m33 m34  m41 m42 m43 m44 where mi,j = mj.i in the reciprocal case, and m11 = m12 = m22 =

(|Shh |2 +2|Shv |2 +|Svv |2 ) 2

(|Shh |2 −|Svv |2 ) 2

(|Shh |2 −2|Shv |2 +|Svv |2 ) 2

(3)

onto the convex sets Ci and defining the relaxed ith projection ∆ operator Ti = (1 − λi ) I + λi Ci , 0 ≤ λ ≤ 2, the central theorem of POCS states that the recursion

Other elements in Mueller matrix can be calculated from scattering matrix as well. When polarimetric information is pre2 2 sented in this form, the LR images consisted of |Shh | , |Shv | , 2 and |Svv | could be obtained by solving the equation 3, Assuming each LR image represents scattering feature of terrain or specified object for different polarization, fusing all the features into a higher resolution image is a promising way to extract details in image and provide more convenience for the application of detection, recognition, etc.. Speckle caused by coherence greatly reduces the quality of SAR images. For polarimetric SAR, the speckle in the image has the characteristics of multiplicative noise, in the sense that speckle noise level is proportional to the scene signal level. This model has been derived and verified for singlelook and multilook processed SAR images []. Let zi denote the amplitude level of an arbitrary pixel in the image, si and vi denote the signal and the multiplicative noise, respectively. The pixel level may be written as zi = si vi

x(l+1) = Tm Tm−1 · · · T1 x(l) ,

l = 0, 1 . . .

converges weakly to a feasible solution that lies in Cs . We define following closed, convex constraint set for each pixel of each channel images sk (m1 , m2 ), where r(x) denotes the residual associated with an arbitrary member yk (n1 , n2 ) of the constraint set: The quantity δk (m1 , m2 ) reflects the statistical confidence with which the region of high-resolution image is a member of the set Ck . In SAR images, the statistics of δk (m1 , m2 ) are identical with those of the noise nk which is proportional to scene signal power. The projection Tm (l) of x(l) (n1 , n2 ) onto Ck (m1 , m2 ) can be defined as where Wk (m1 , m2 ; n1 , n2 ) denotes the weight from pixel (m1 , m2 ) in kth LR image to pixel (n1 , n2 ) in HR image. p, q are coordinates in the SR grid for which Wk (m1 , m2 ; n1 , n2 ) is not zero. In addition to the principal constraint set, we use also the amplitude constraint set

(4)

where vi has unity mean and standard deviation σi . It is assumed that vi is statistically independent from si and vi (i 6= j). vi is regarded as a measure of the speckle strength.

CA = {x (n1 , n2 ) : 0 ≤ x (n1 , n2 ) ≤ 255}

The POCS method describes an iterative approach to incorporating prior knowledge about the solution into the reconstruction process. It projects an initial estimate onto constraint sets iteratively to obtain a solution that is consistent with all the constraints. In the POCS method the unknown signal f is assumed to be an element of the appropriate Hilbert space H. It is assumed that m properties (constraints) p1 , p2 ...pm of the signal f are known a priori. For each prior information pi , there is attributed a set Ci , which is the set of all the signals having the propriety πi . Often these sets are closed, meaning that they contain their limit points, and convex, meaning that if s1 , s2 ∈ Ci , then µs1 + (1 − µ) s2 ∈ Ci for any 0 ≤ µ ≤ 1. Clearly, the estimate of f belongs to the solution ∆ set Cs = ∩m i=1 Ci , assuming this intersection is not empty. Denoting by Pi a projection operator of an arbitrary signal

IV. SUPER RESOLUTION ALGORITHM FOR POLARRMETRIC SAR IMAGES When applying POCS method to polarimetric SAR images, two key points to be solved is: 1. How to construct the convex set and 2. How to obtain the weight matrix W . A. Constructing the Convex Set The convex set Ci is the set in which all the signals have the same propriety πi . In POCS method, we define a convex

n o (l) (l) Ck (m1 , m2 ) = x(l) (n1 , n2 ) : rk (m1 , m2 ) ≤ δk (m1 , m2 )

(l)

(9)

that bounds the estimated image gray levels in the range (0, 255). POCS is a simple and effective method for solving ill-posed problem such as super-resolution reconstruction, where the number of LR images is insufficient and blur operators are illconditioned. It also allows a convenient inclusion of a priori information. However, these methods have the disadvantages of nonuniqueness of solution depending on the initial value in iteration.

III. POCS METHOD

rk (m1 , m2 ) = sk (m1 , m2 ) −

(5)

X

x(l) (n1 , n2 ) Wk (m1 , m2 ; n1 , n2 )

(6) (7)

n1 ,n2

Tm x(l) (n1 , n2 ) = x(l) (n1 , n2 ) + λ

            

(x)




rk (m1 ,m2 )−δk (m1 ,m2 ) Wk (m1 ,m2 ;n1 ,n2 )

P

p,q

(x)

Wk2 (m1 ,m2 ;n1 ,n2 )

0




rk (m1 ,m2 )+δk (m1 ,m2 ) Wk (m1 ,m2 ;n1 ,n2 )

P

p,q

Wk2 (m1 ,m2 ;n1 ,n2 )

(x)

rk (m1 , m2 ) > δk (m1 , m2 ) (x) rk (m1 , m2 ) ≤ δk (m1 , m2 ) (x)

rk (m1 , m2 ) < −δk (m1 , m2 )

(8)

set Ci for every pixel (m1 , m2 ) in each LR image sk , denoted as Ck (m1 , m2 ). For the POCS is an iterative approach, the convex set would be adjusted adaptively in each iteration. When applied to the polarimetric SAR images, the convex set should be constructed considering the characteristics of multiplicative noise, which means, in the region with high intensity, the noise will be larger proportionally. So the constraint of convex set in these regions should be looser. In our method, we simply modify the traditional definition of convex set in XX by defining the delta as δk (m1 , m2 ) = µsk (m1 , m2 )

(10)

where sk (m1 , m2 ) is the pixel intensity in LR image sk and µ is the multiplicative factor of intensity to noise.

Fig. 1.

Orignial HR Image

B. Obtaining Weight Matrix where Wk (m1 , m2 ; n1 , n2 ) denotes the weight from pixel (m1 , m2 ) in kth LR image to pixel (n1 , n2 ) in HR image. An acceptable assumption is that Wk (m1 , m2 ; n1 , n2 ) has a form of Gaussian distribution as Wk (m1 , m2 ; n1 , n2 ) = e

(n1 −nx )2 +(n1 −ny )2 σ2

(11)

where nx and ny denote the corresponding central pixel in HR image to pixel (m1 , m2 ) in kth LR image, and σ controls the standard variation. This assumption is simple but effective and normally accepted. However, the estimation of Wk (m1 , m2 ; n1 , n2 ) could be refined by some other advanced methods such as statistical and learning approaches. C. Full Description of Algorithm The algorithm of reconstruct high-resolution polarimetric SAR image using POCS method can be described fully as follows: Algorithm 1 POCS ALGORITHM FOR POLSAR IMAGES procedure POCS(matrix S or M , IterM ax, W ) Get LR images sk from S or M i←1 while i < IterM ax do for every LR image do for every pixel in this LR image do (l) Get Ck (m1 , m2 ) by 6 and 10 (l) Calculate rk (m1 , m2 ) by 7 (l) (l) Project x (n1 , n2 ) to Ck (m1 , m2 ) by 8 end for end for Calculate the average suqare error end while end procedure

(a) HH Fig. 2.

(b) HV

(c) VV

Simulated LR Images in Different Channels

in September 1993 has been used, which is characterized by the presence some ships representing the target to be superresolved. Firstly we validate the proposed method by using the simulated LR images in different channels deteriorated from a high-resolution span image with respective W and effected by multiplicative noise. Figure 2 shows the simulated LR images which are of half the resolution of original HR image shown in figure 1. The W of each channel is assumed as random 2 × 2 matrix with the sum of all the elements to be 1. The super resolution result is shown in figure 3, and the curve of average square error with iteration shown in figure 4 illustrates the convergence of proposed method. Then we use the data of the three polarimetric channels HH-VV-HV at the L frequency band as LR images. Figure 5 shows the 15 × 24 samples section of the data, at HH, VV and HV channels respectively, chosen for testing the proposed techniques of Section 4. With the multiplicative noise and point-target model W assumption, we apply the POCS method to increase the resolution from one pixel to a 4 × 4 region. Figure 6 shows the result from which we can find that the shape of a ship is depicted more precisely and the details such as the bow and poop can be recognized more easily, which will increase the accuracy of detection effectively. VI. CONCLUSION

V. EXPERIMENTAL RESULTS To validate the POCS method described previously, we apply them to live recorded polarimetric JPL AIRSAR L-band data, . Particularly the data set of the Sydney Sea collected

In this paper we have derived the super-resolution approach based on POCS method to polarimetric SAR images, aiming at improving the geometric resolution. fusing the information at the different polarimetric channels inside super-resolution

of other application. ACKNOWLEDGMENT This work was supported in part by the Foundation of Doctoral of Tsinghua University and in part by the Excellent Young Teacher Program of Ministry of China. R EFERENCES

Fig. 3.

Reconstructed HR Image

0.35

Square Error

0.3

0.25

0.2

0.15

0.1

0

Fig. 4.

5

15

20

Average Square Error with Iteration

(a) HH Fig. 5.

10 Iteration Time

(b) HV

(c) VV

Real LR image from Different Channels

Fig. 6.

Reconstructed Result

techniques. The multiplicative property of speckle noise in SAR image is utilized to construct the convex sets in POCS method. Experiments with both simulated and real data have validated the effectiveness of proposed method. Besides for ship detection, the method is readily adopted to the application

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