On the Polarization Entropy Jian Yang Yilun Chen Yingning Peng Dept. of Electronic Engineering Tsinghua University Beijing, China [email protected] Abstract—The polarization entropy is an important parameter and it has been used for target classification, target detection and so on. For calculating the polarization entropy, we have to obtain the eigenvalues of a covariance matrix and then use logarithm. Therefore, some calculation cost is necessary to get all the entropy values of adjacent n by n windows for all pixels in a polarimetric SAR image. In this paper, the authors propose a new method to calculate the polarization entropy, based on the least square method. Using a polarimetric SAR image, we validate the effectiveness of the proposed method.

λ 3 + a2 λ 2 + a1λ + a0 = 0 , where

a0 = c11c23 c32 + c13 c22 c31 + c12 c21c33 − c11c22 c33 − c12 c23 c31 − c13c21c32 a1 = c11c22 + c22 c33 + c33c11 − c12 c21 − c23c32 − c31c13

Keywords-polarization; Synthetic Aperture Radar (SAR); remote sensing

I.

INTRODUCTION

In polarimetric radar remote sensing, the polarization entropy is one of important parameters for target classification, target detection and so on [1-6]. For calculating the polarization entropy, however, one has to obtain the eigenvalues of a covariance matrix and then use logarithm. Consequently, some computation cost is necessary when we need get all the entropy values of adjacent n by n windows of every pixel in a polarimetric SAR image. In general, there may be more than one million pixels in a SAR image. So it is important to propose a simple method to calculate the polarization entropy. This paper will solve this problem. II.

3×3

.

(1)

[C ] = ( c ) ij

3×3

(3b) (3c)

. Then the polarization entropy is defined by 3

H = −∑ ki log 3 ki , ki =

∑

i =1

λi

.

3

λ j =1 j

(4)

For target classification or target detection in a polarimetric SAR image, we need obtain more than one million entropy values in general. Consequently, some computation cost is necessary. So it is important to propose a simple method to calculate the polarization entropy. THE NEW FORMULA 3

Consider that let p ( x) = b0 + b1 x + b2 x + b3 x + b4 x

4

approximate − x log 3 x . Since − x log 3 x equals zero when

x = 0 or x = 1 , we assume that p (0) = 0 and p (1) = 0 . It leads to b0 = 0 and b4 = −b1 − b2 − b3 . So it is reasonable to 3

let

AH = b1 + b2 ∑ ki + b3 2

i =1

(2a)

3

∑ k − (b + b 3

i

i =1

1

2

+ b3 )

3

∑k

4 i

i =1

3

approximate H = −

or equivalently

∑ k log i

3

ki . Use the least square method

i =1

This paper is devoted to last Dr. Ernst Lueneburg. This work was supported by the National Natural Science Foundation of China (40271077), by the National Important Fundamental Research Plan of China (2001CB309401), by the Science Foundation of National Defence of China, by the Research Fund for the Doctoral Program of Higher Education of China, by the Aerospace Technology Foundation of China, and by the Fundamental Research Foundation of Tsinghua University.

0-7803-9050-4/05/$20.00 ©2005 IEEE.

,

2

Its eigenvalues are obtained by

[C ] x = λ x ,

(3a)

Let λi (i = 1, 2, 3) be the three eigenvalues of the matrix

III.

The entropy is originally used to describe the chaos extent in nature. Later, it was introduced to radar polarimetry [1-3]. Consider the following polarimetric covariance matrix ij

,

a2 = −c11 − c22 − c33 .

THE POLARIZATION ENTROPY

[C ] = ( c )

(2b)

2012

min

∫∫ ( H − AH )

2

dk1dk2 ,

(5)

Ω

Ω is the area determined by k1 − k2 ≥ 0 , k1 + 2k2 − 1 ≥ 0 and 1 − k1 − k2 ≥ 0 , shown in Figure 1.

where

2012

Letting

∂ ∂bi

∫∫ ( H − AH )

cost is reduced. Figure 3 shows the procedure for calculating the polarimetric entropy. 2

dk1dk2 = 0 ,

(6)

Ω

we obtain 3

∑k

AH = 2.3506 − 5.7613

3

2 i

∑k

+ 6.0611

i =1

Note that ki =

∑

λi

i

− 2.6504

i =1

∑k

4

.(7)

i

i =1

is derived from the eigenvalues of the

3

λ j =1 j

covariance matrix

3

3

[C ] = (c

)

i j 3× 3

, i.e., the roots of (2b).

According to Vieta’s Theorem, the following formula can be derived from (7)

AH = 3.9408

a1 a22

+ 7.5818

a0 a23

− 5.3008

a12 a24

,

Figure 1. The area Ω for the least square method.

(8)

where a0 , a1 and a2 are given by (3a-3c). Obviously, one may directly obtain the polarization entropy from (8) without solving an eigenvalue equation. On average, the corresponding absolute error is 0.006 in theory, i.e.,

∫ H − AH dk dk 1

2

= 0.006 .

(9)

Ω

This demonstrates a good approximation by the proposed formula (8). However, if the maximum eigenvalue of the covariance matrix [C ] is approximately equal to 1, the polarization entropy is very small and the relative error by (8) is a little large although the absolute error is very small. When k1 ≥ γ = 0.95 , we can prove that a0 ≤ 0.0006 a23 and

Figure 2. The area Ω1 for the least square method.

a1 ≤ 0.0481 a22 . For this case, the following formula of the polarization entropy is derived, based on the least square method over a small area Ω1 as shown in Figure 2:

AH = 5.2819 if

a1 a

2 2

a1 a22

≤ 0.0481 and

+ 54.8584

a0 a23

a0 a23

− 35.6980

a12 a24

,

(10)

≤ 0.0006 .

When we calculate the polarization entropy by the proposed formula, it is better to get t1 =

a1 a

2 2

and t2 =

a0 a23

first. After comparing t1 and t 2 with 0.0481 and 0.0006, respectively, we submit t1 =

a1 a22

, t2 =

a0 a23

2

, and t1 =

a12 a24

into Figure 3. Procedure for calculating the polarimetric entropy.

(8) or (10) to obtain the entropy. In this way, the computation

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2013

IV.

CALCULATION RESULT AND CONCLUSION

To validate the effectiveness of the proposed formula, we used a NASA/JPL AIRSAR L-band image of San Francisco. As shown in Figure 4, we selected three areas consisting of a sea area, a park area and an urban area. For each pixel, the entropy of an adjacent 3 by 3 window was calculated by (4) and (8) or (10). The calculation results are given in Table 1, validating a good approximation by the proposed formula. In addition, we calculated the polarization entropy of the whole area by C programming language. Except the computation time on covariance matrices, the running time for obtaining the entropy by the proposed formula was only 5% of that by (4), demonstrating the efficiency of the proposed formula. TABLE I.

sea area park area urban area whole area

COMPARISON OF CALCULATION RESULTS BY TWO FORMULAE

Av. value of H 0.058 0.607 0.459 0.360

Av. value of AH 0.055 0.607 0.473 0.365

Av. absolute error 0.003 0.006 0.016 0.008

Figure 4. The span image of San Francisco.

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