A Fast Method for the Optimization of Polarimetric Contrast Enhancement in Partially Polarized Condition

LEI Shi-wen ZHAO Zhi-qin NIE Zai-ping and LIU Qing-huo2

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A Fast Method for the Optimization of Polarimetric Contrast Enhancement in Partially Polarized Condition

LEI Shi-wen1, ZHAO Zhi-qin1, NIE Zai-ping1,and LIU Qing-huo1,2

(1. School of Electronic Engineering, University of Electronic Science and Technology of China Chengdu 610054; 2. Department of Electrical and Computer Engineering, Duke University Durham NC 27708, USA)

A general signal to clutter plus noise ratio (SCNR) model containing the partially polarized condition is created. Based on this SCNR model, a fast method for the optimization of the polarimetric contrast enhancement (OPCE) problem with constrained transmitted and received polarization is proposed. The method proves the OPCE problem equivalent to the maximization of a linear cost function. The solving of the maximization of the function is simpler than that of the OPCE problem. Hence, the faster performance searching is achieved. The method is theoretically deduced. The numerical experiments demonstrate the effectiveness of this method. Compared with the conventional global search method (GSM) based on three-step method, the proposed method costs less than 5% of the calculation time.

Kennaugh matrix; optimization of polarimetric contrast enhancement (OPCE); polarization ratio; polarization state

The problem of optimally selecting polarization states of the transmitted waveform is extensively studied as it can enhance the performance in target detection, tracking and identification[1-3]. Scattering properties of targets and clutter are polarization- sensitive; hence, the benign applications of polarization can enhance the polarimetric power contrast. They are known as the optimization polarimetric contrast enhancement (OPCE) problem[4-6].

In partially polarized condition, Sinclair matrix cannot provide the whole polarization information. To cope with the OPCE problem in this condition, Kennaugh matrix which provides the whole polarization information is applied. Usually, there are not analytic solutions to the OPCE problem, and numerical methods are applied. Among those methods, the global search method (GSM), which searches the overall two-dimensional polarization space, is the most common used one[7-8]. This kind of method is time- consuming; especially when fast real-time signal processing is required such as in an accurate tracking of high maneuvering target scenario. Aiming to expedite the process of enhancing the desired targets versus the clutter and noise, various fast methods are proposed. Ref.[9-10] proposed iterative numerical methods for the completely polarized condition 0 and the partially polarized condition 0. Those methods are faster than the GSM. However, the method in 0 can only be used in the condition in which the relationship between the transmitted and received polarization states is not constrained. Ref.[11-12] proposed methods based on polarization ellipse parameters. Those methods require for the information of the angle consisting of the target and the clutter on the Poincare sphere frame as well as the sphere center.

In the paper, assuming the Kennaugh matrices is measured, we emphasize on how to fast solve the OPCE problem in partially polarized condition with constrained transmitted and received polarization. We first introduce a general signal to clutter plus noise ratio (SCNR) model for the partially polarized condition. Then an OPCE problem is deduced and a fast method for the OPCE problem is proposed. The method always converges to the optimal result. It expedites the computation of the OPCE problem by converting the problem into an equivalent maximization linear function.

1 Signal to Clutter Plus Noise Ratio Model

In this section, a model of signal to clutter plus noise ratio (SCNR) defined by the Stokes vector and Kennaugh matrix is created. Stokes vector is defined as

where,

(2)

(3)

whereis the received wavelength,andare the spherical coordinates of the antennapointing direction, η is the free space impedance,andare the antenna gain and received electric field strength, respectively.

The received power includes the power of the targets, the clutter and the noise, expressed separately as,and. SCNR can be defined as

(6)

The ratio of the completely polarized power to the total power is defined as the polarization ratio

(8)

In the detection period, the polarization direction of the antenna to the target, clutter, and noise are the same. Hence, SCNR defined in (5) can be rewritten as

where SCR denotes signal to clutter ratio,-{CNP} is the sum of the clutter Kennaugh matrix and the noise Kennaugh matrix.anddenote the received and the transmitted polarization states, respectively. The OPCE problem is to select the optimal polarization states to maximize the SCNR in (10).

2 Polarization Optimization Method

In this section, the OPCE problem based on the SCNR is created. Then a polarization states optimization method is proposed.

2.1 Problem Formulation

Considering the radar system receives the echoes of the targets embedded in clutter and noise background, the maximization of the SCNR is choosing the optimization criterion to design antenna. For simplicity, we assume a co-polar condition, i.e., the transmitted Stokes vector is the same to the received Stokes vector (other transmitted and received polarization relationship can be realized by matrix rotation). Hence,

The fundamental principle of optimal reception is to adaptively adjust the polarization states to maximize the SCNR. The OPCE problem is then converted to be the optimization problem,

(12)

Constituting (9) and (10) into (12), the optimization problem is transformed to be,

2.2 A Fast Polarization Optimization Method

In this subsection, we recast (13) in a linear function with two variables which are polarization stateand supplementary parameter. Then we solve the problem numerically. Let us first define the following function,

Define the maximum value of functionwith respect toas

(15)

(17)

Assuming the polarization state corresponding tois; then, the maximum SCNR is

According to the forward analyses, the following Theorem 1 can be summarized.

Theorem 1 The maximum SCNR equals to thewhenand the corresponding polarization state is.

(20)

Let the derivation of the cost function with respect toequal to zero, we obtain

According to matrix decomposition theory, there is a unitary matrixand a diagonal matrixsuch that. Let

(22)

The domain of Lagrange multiplierthat maximizes (19) is 0,

(24)

where,

and,

(26)

“eigs” denotes the eigenvalues of matrix.

Substituting (21) into (19), there is:

(28)

Hence, the function (19) is monotone increasing with respect to the variable. The conclusion can be established by the following theorem 2.

Theorem 2 The maximization of (19) can be calculated by the Lagrange method in (20). The function is a monotone increasing one with respect to the Lagrange multiplierin (20).

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Given the maximum value of the numerator and the minimum value of the denominator of (10) areand, respectively, the upper bound ofis:

Given the minimum value of the numerator and the maximum value of the denominator of (10) areand, respectively, the low bound ofis:

(30)

Hence the search intervals ofis:

The procedures for the proposed method are:

4) else go to 2).

3 Numerical Experiments

Experiments are accomplished by Matlab 2010 code running on a 32-bit computer with CPU AMD Athlon 3.0GHz, RAM 4G. Monte Carlo simulation time is 100. Let us consider the following Kennaugh matrix: the target Kennaugh matrixin 0 and the clutter Kennaugh matrixin 0.

(33)

Some assumptions are: 1) the distances of the target and the clutter to the antenna are the same, i.e.,. 2) the antenna gains are the same, i.e.,. Hence,and. Equation (10) shows the SCR does not affect the selection of the optimal polarization state, we choose SCR = 10 dB. Three different clutter to noise ratios(CNRs), i.e., CNR = 10 dB, 0 dB and ?10 dB are tested to validate the proposed method.

Considering the target is completely polarized, i.e.,. To test the performance of proposed method in the partially polarized condition, three conditions are operated: 1) Low polarization ratio; 2) Middle polarization ratio; 3) High polarization ratio. The results obtained by the GSM in a small search-step, i.e.,, is considered to be the real ones.

Experiment I: Low Polarization Ratio

The maximum SCNRs corresponding to CNR=10 dB, 0 dB and ?10 dB are22.333 9 dB, 19.824 3 dB and 12.509 3 respectively. Their corresponding polarization states are (?0.754 4, 0.633 0, 0.173 6), (?0.771 5, 0.608 2, 0.186 5) and (?0.771 5, 0.608 2, 0.186 5), respectively.

The average time consumed by the proposed method is about 5% of that consumed by the GSM with the similar calculation accuracy. The optimal polarization sates to different CNRs are similar to each other.

TABLE I the Proposed Method V.S. the GSM withp=0.01

MethodCNR/dBPolarization StateSCNR/dBTime/s GSM10(-0.754 4, 0.633 0, 0.173 6)22.332 40.040 6 0(-0.754 4, 0.633 0, 0.173 6)19.823 00.040 2 ?10(-0.754 4, 0.633 0, 0.173 6)12.508 10.040 3 Proposed Method10(-0.773 6, 0.605 7, 0.185 7)22.333 60.002 4 0(-0.772 3, 0.607 6, 0.185 6)19.824 40.002 3 ?10(-0.770 8, 0.609 3, 0.185 5)12.509 30.001 9

Experiment II: Middle Polarization Ratio

The maximum SCNRs corresponding to CNR = 10 dB, 0 dB and-10 dB are 22.2646 dB, 19.7747 dB and 12.497 3, respectively. Their corresponding polarization states are (-0.674 8, 0.730 0, 0.108 9), (-0.709 3, 0.691 0, 0.139 2), and (-0.756 6, 0.629 2, 0.177 9) respectively.

TABLE II the Proposed Method V.S. the GSM withp=0.5

MethodCNR/dBPolarization StateSCNR/dBTime /s GSM10(?0.665 5, 0.739 1, 0.104 5)22.264 30.040 3 0(?0.715 4, 0.690 9, 0.104 5)19.771 00.040 1 ?10(?0.754 4, 0.663 0, 0.173 6)12.497 30.040 3 Proposed Method10(?0.674 6, 0.730 3, 0.107 0)22.264 30.002 3 0(?0.708 9, 0.691 6, 0.138 3)19.774 70.002 2 ?10(?0.757 7, 0.628 4, 0.176 3)12.497 30.001 9

The average time consumed by the proposed method is about 5% of that consumed by the GSM with the similar calculation accuracy. The optimal polarization states to different CNRs are different.

Experiment III: High Polarization Ratio

The maximum SCNRs corresponding to CNR = 10 dB, 0 dB and ?10 dB are 22.910 8 dB, 20.045 6 dB, and 12.504 3, respectively. Their corresponding polarization states are (?0.488 6, 0.872 4, ?0.013 1), (?0.536 9, 0.842 7, 0.039 3), and (?0.693 2, 0.705 4, 0.147 8) respectively.

TABLE III the Proposed Method V.S. the GSM withp=0.99

MethodCNR/dBPolarization StateSCNR/dBTime/s GSM10(-0.5, 0.866, 0)22.908 50.069 3 0(-0.543 9, 0.843 2, 0.040 2)20.504 00.070 3 -10(-0.698 4, 0.698 4, 0.156 4)12.462 00.069 3 Proposed Method10(-0.486 7, 0.873 6, -0.014)22.911 90.002 3 0(-0.536 0, 0.843 4, 0.040 2)20.045 60.002 3 -10(-0.694 2 , 0.704 5, 0.147 6)12.50430.001 9

The average time consumed by the proposed method is about 3% of that consumed by the GSM with the same calculation accuracy. The optimal polarization states to different CNRs are greatly different.

The proposed method has been proved to be able to obtain the optimal polarization states for all the partially polarized conditions. Compared with the GSM, the proposed method is less time-consuming and more accurately.

4 Conclusions

In the paper, the OPCE problem with constrained transmitted and received polarization state relationship in partially polarized condition is discussed. A general SCNR model is first created to contain the partially polarized condition. A fast method for the OPCE problem is proposed based on the SCNR model. The method has converted the OPCE problem into the maximization problem of a linear function. Hence, the computational burden is greatly reduced. The numerical experiments have demonstrated the proposed method is better and has higher efficiency than the GSM. This method is easily extended to other polarization states conditions, such as the cross- polarize condition, by matrix rotation.

In the following work, we will research on the fast polarization optimization methods for the OPCE problem with unconstrained relationship between the transmitted polarization state and the received polarization state.

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編 輯 稅 紅

部分極化條件下的極化對比度增強優化的快速方法

雷世文1，趙志欽1，聶在平1，柳清伙1,2

(1. 電子科技大學電子工程學院 成都 610054；2. 杜克大學電氣與計算工程系 美國北卡羅拉州達拉姆 27708)

介紹了一種包含完全極化情形和部分極化情形在內的通用信號雜波噪聲比(SCNR)模型。基于該模型，提出了一種適用于收發極化狀態受約束的極化對比度增強優化(OPCE)的快速方法。該方法證明OPCE問題等價于某類線性代價函數的極值問題，且該類線性代價函數的極值問題的求解比OPCE問題的求解容易。從而構建了快速解決OPCE問題的方法。理論分析和數值實驗驗證了該方法的可靠性和高效性，與基于三步法的全局搜索方法(GSM)相比，該方法僅需要5%的計算時間。

Kennaugh矩陣; 極化對比度增強優化(OPECE); 極化率; 極化狀態

TP202+.1

A

2013-10-16；

2014-10-16

部級基金

2013-10-16；Revised date：2014-10-16

Provincial pre-research fund.

10.3969/j.issn.1001-0548.2015.01.009

Biography：LEI Shi-wen was born in 1985，and his research interests include radar signal and information processing, polarimetric information processing, adaptive signal processing, etc.

雷世文(1985-)，男，博士生，主要從事雷達信號與信息處理、極化信息處理、自適應信號處理等方面的研究.