Optimal Polarization for Contrast Enhancement
in Polarimetric SAR Using Genetic Algorithm
2.6 Mutation
In order to maintain a more diverse population among the new generations, the mutation that provides a means for exploring new portions of the solution space is needed. Probability of bit mutation should be kept low enough so that the highly fit chromosomes are not destroyed. Typically, on the order of pmutation =0.01 of the total bits mutate per generation.
Chromosome before
mutation |
 |
0 1 1 |
|
1 1 1 |
|
0 1 0 |
Chromosome after
mutation |
|
0 1 1 |
|
1 0 1 |
|
0 1 0 |
3. Polarimetric Matched Filter
In this paper, the optimal polarization filter algorithm developed by Swartz et al. [1] was used to select the optimal polarizations between two different classes. A brief discussion of the polarimetric SAR and optimal filter is given below.
For a give set of antenna polarizations, the scattered power can be expressed in terms of the scattering matrix for the target
 | (2) |
Where a
m denotes antenna polarizations in the transmitting and receiving mode, i.e.,
 | (3)
|
m = t(transmitted), r (received), and
 | (4) |
Equation (2) can be rewritten according to the definition given in [1] as
 | (5) |
In case of backscattering, S
hv = S
vh, then
Equation (5) may be expressed in the form P =F+
?CF, where C is the covariance matrix of the terrain cover assumed to be statistically distributed, defined by [1]
where ̑ ̑means ensemble average operation and * the complex conjugate.
In [1], the contrast ratio between class A and B is defined as
To maximize r
A/B above, the Lagrange multiplier method is used.
Making use of Lagrange multiplier concept leads to the eigen-equation
Since F represents the optimum matched filter, we now want to calculate the corresponding transmitting and receiving antenna polarizations. In[1], each a
m was solved but involves tedious algebraic operations. Indeed, Chen et al. [2] derived an analytical equations to get antenna polarization angles for transmitter and receiver directly and effectively. it can be readily shown that what is needed are the following ratios
where
At this point, it should be remembered that in general a polarization vector can be transformed into a normalized Stokes vector and related to orientation y and ellipticity ? angles. It turns out that we can simply find the polarization angles in terms of Q
t and Q
r
4. Implementation and results
Unlike tedious algebraic operations used in Lagrange method, GA is trying to find parameters directly. Referring to elements in eigenvector found through Lagrange method, equation (6) can be rewritten to
where R
xx and I
xx are real numbers representing real part and imagery part, respectively.
It can be seen that in this problem there are six parameters to be found to meet the criteria that the contrast ratio between any two classes is maximum.
In this paper, we compare optimal matched filter
F's that are found from the Lagrange method and the GA respectively. The test data is an L-band four-look fully polarimetric image over the San Francisco Bay area required by the JPL airsar system. This image has been widely used in the past as a testing image. The original image is shown in Figure 2. Three classes may be identified by visualization ̑ open water, park area, and urban city.
For comparison, optimal polarizations determined by these two methods and linear polarizations were summarized in Table 1. Obviously, it can be shown that optimal polarization has higher contrast ratios than linear polarization, as expected, and GA provides an alternative to solving optimal polarization problem. The fitness function to be maximized in this case is
fitness = r
Urban/Ocean or r
Park/Ocean or
r
Urban/Park (14)
In addition to get an optimal polarization to discriminate any two classes in an image at one time, GA can also determine an optimal polarization characterizing the best discrimination for any two among all possible classes simultaneously. The fitness function to be maximized in this case is
fitness = r
Urban/Ocean+ r
Park/Ocean+ r
Urban/Park (15)
It should be pointed out that although GA slightly degrades the overall performance when compared to those obtained individually and sequentially, it is much more computationally efficient, as observed in Table 1. This should be regarded as a more practical approach particularly when there are many classes of interest in an image. Table 2 listed polarization angles and corresponding final chromosome for the case.
