Ruey-Long SU1, K.S.CHEN1, Jiancheng SHI2
1Institute of Space Science National Central University
Chung-Li, Taiwan 2ICESS, University of California Santa Barbara, CA, USA
The optimal use of polarimetric scattering information provided by fully polarimetric synthetic aperture radar (SAR) is useful for discriminating between different terrain covers. Conventionally, eigenanalysis is used to synthesize a polarimetric matched filter (PMF) image that maximizes the contrast between the features of interest, but it involves many algebraic operations and can get one contrast ratio for any two classes at one time. This paper describes the application of genetic algorithm (GA) to the contrast enhancement. It obtains all contrast ratios characterizing the best discrimination for any two among all possible classes simultaneously. After using an L-band fully polarimetric SAR image as a test data, it shows that the proposed method is very effective and promising.
Remote sensing imagery data are known for their high degree of complexity and irregularity, especially for SAR image. Moreover, there are usually many natural and man-made targets resulting in extremely difficult in the task of segmentation, classification, and detection. The problem of optimization of polarimetric contrast enhancement has been attracting attention in recent years [1-7]. The main purpose is to choose the polarization states that enhances the desired target versus the undesired target (clutter). What we can control is the polarization states of the transmitter and the receiver. For example, Teti et al.  used polarimetric matched filter (PMF) technique based on eigenanalysis to enhance image contrast in selected regions to improve detection and classification of flooded regions. Yang et al.  developed a numerical method for solving the optimal problem of contrast enhancement. However, the polarimetric techniques reported abovementioned for target discrimination have been limited to the special case of two targets because of the complex nature of the optimization problem. In reality, there are more than two possible targets of interest involving the discrimination process. This motivates us to use stochastic-base search method to handle problems with targets more than two so as to enhance the operation efficiency and yet remain effective.
Two stochastic-base optimization procedures that can deal a large number of discrete parameters are simulated annealing and genetic algorithm (GA). In recent years, applications of GA to a variety of optimization problems in electromagnetics have been successfully demonstrated [7-11]. It’s optimization is global in the sense that it has random components that test for solutions outside the current minimum, while the algorithm converges. In particular, GA is much better at dealing with solution spaces having discontinuities, constrained parameters, and/or a large number of dimensions with many local maxima.
Traditional optimization algorithms search for the best solution, using gradients and/or random guesses. Gradient methods have the disadvantages of getting stuck in local minimum, requiring gradient calculations, working on only continuous parameters, and being limited to optimizing a few parameters. Random-search methods do not require gradient calculations, but being a blind-search and tend to be slow. In addition, GA is easy to program and is able avoid the mathematical rigor of traditional optimization methods.
In this paper, we utilize GA to search for a pair of optimal polarimetric states so that at these polarization states the contrast ratio between two classes in SAR image is maximmum. Besides, it can get all contrast ratios for any two classes simultaneously subject to the sum of all individual contrast ratios being maximum. The results also provide an excellent preprocess for subsequent image classification, if desired.
2. Genetic Algorithm
This section begins with a brief overview of GA, followed by a description of a step-by-step implementation. More details on GA can be found in  and references cited there. The goal of the GA is essentially to find a set of parameters that maximize (or minimize) the output of a function (or process). A typical flow diagram of a simple GA optimizer is presented in Figure 1.
2.1 Parameters Encoding
Encoding is a transformation from the parameter space (also called phenotype) to the gene space (also called genotype). Genes in the chromosome represent the coded parameters. Usually, a binary coding is utilized. For example, if the chromosome has three parameters (said var1, var2, var3) to be optimized and use three bits for each parameter, then the chromosome might be written as
Chromosome 1 1 0 0 0 1 1 0 0
Parameters var1 var2 var3
2.2 Population Initialization
If there are M populations in a generation, 1 or 0 is randomly selected for each bit to form a chromosome and consequently construct M chromosomes that represent possible solution of var1, var2, and var3. This randomness guarantees the diversity of population of possible solutions.
Chromosome 1 1 1 0 0 0 1 1 0 0
Chromosome 2 0 1 1 1 1 1 0 1 0
Chromosome M 1 0 0 0 1 1 1 0 1
2.3 Fitness Calculation
Each chromosome has a cost value, founded by evaluating a fitness function, or object function. Fitness evaluation involves decoding of the chromosomes to produce the parameters that are associated with the individual. If a GA tool is available, the good fitness function is the only portion that a user must provide and in general the most difficult part of the whole processes. The fitness value in this example can be expressed as
fitness = f(var1, var2, var3) (1)
Reproduction, also called selection, shows the influence of the fitness value to GA. Although, fitness is the merit of goodness of an individual, reproduction cannot be executed only by selecting the best individual, because the best individual may not be very close to the optimal solution, especially in the very early generation. In order to avoid the problem of premature convergence, some individuals with relative low fitness value must be preserved to ensure that genes possessed by these individuals are not lost prematurely. The selection operator can either be stochastic or deterministic.
The most productive step in GA before convergence is crossover. It stems from the idea that by exchanging information between two chromosomes may produce a new better chromosome. One-point and two-point crossover methods are the most popular.The probability of individuals chosen to attend crossover during each generation is specified by the parameter pcrossover(typically 0.6—1.0). After crossover, the total number of chromosomes remains constant.
|parent 1||1 1 0 0 0||1 1 0 0|
|parent 2||0 1 1 1 1||1 0 1 0|
|offspring 1||1 1 0 0 0||1 0 1 0|
|offspring 2||0 1 1 1 1||1 1 0 0|
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.
|0 1 1||1 1 1||0 1 0|
|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.  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
Where am denotes antenna polarizations in the transmitting and receiving mode, i.e.,
m = t(transmitted), r (received), and
Equation (2) can be rewritten according to the definition given in  as
In case of backscattering, Shv = Svh, 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 
where ̑ ̑means ensemble average operation and * the complex conjugate.
In , the contrast ratio between class A and B is defined as
To maximize rA/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, each am was solved but involves tedious algebraic operations. Indeed, Chen et al.  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
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 Qt and Qr
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 Rxx and Ixx 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 = rUrban/Ocean or rPark/Ocean or rUrban/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 = rUrban/Ocean+ rPark/Ocean+ rUrban/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.
Assuming that the polarimetric responses of all the classes in the image are known a priori, determination of polarization states of the transmitter and the receiver for maximizing the contrast ratios between any two classes was presented in this paper. As compared to conventional method, GA shows the ability to provide another and even convenient way to solve optimal polarization problem. It can get all contrast ratios characterizing the best discrimination for any two among all possible classes individually and simultaneously.
It must be emphasized at this point that GA may not always be the best method to deal with all optimal problems in hand, but it has obvious advantage that it seems to find globally optimal solutions without requiring a great deal of information about the solution space. In some sense, it searches for a solution like an engineer learns from experiments and trial-and-error. That makes GA useful for solving problems that were solved experimentally and empirically in the past.
Figure 1.Flow diagram of GA
Figure 2. L-band SAR image of the San Francisco
|Optimal- Lagrange||28.5990 dB||19.1451 dB||9.5674 dB|
|Optimal-GA||28.5988 dB||19.1449 dB||9.5674 dB|
|Linear-HH||15.1013 dB||7.3075 dB||7.7938 dB|
|Linear-HV||20.3604 dB||17.2547 dB||3.1057 dB|
|Linear-VV||9.4340 dB||4.2362 dB||5.1978 dB|
|GA(simultaneously)||28.5988 dB||19.0547 dB||9.5441 dB|
Table 1. contrast ratios for different polarizations.
Table 2. Polarization angles and solution chromosome for simultaneous case
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