# An Efficient Ground Moving Target Imaging Method for Airborne Circular Stripmap SAR
## 4 Proposed Imaging Method
### Solving the Optimization Problem via GA  
GA starts with encoding the parameters of the search space in the form of a chromosome-like structure. Then, an initial population (a population is constituted by a group of chromosomes) is generated. After that, the fitness value of each individual chromosome of the population is calculated based on a defined fitness function. Subsequently, a selection operation is performed, in which the fittest chromosomes will be chosen with a larger probability. Then, the selected chromosomes are used as parents to produce the next generation by the crossover and mutation operations. This procedure is repeated until the stopping criterion is met.  
In this subsection, GA is utilized to solve the optimization problem presented in (24). The specific steps are explained as follows.  
1. _Step 1--Coding:_ Each individual chromosome of the population is encoded as a binary string with a fixed length. The length of the string depends on the range of the parameters to be searched and the required accuracy. For example, the length of the string for the parameter \(v_{e}\) should be \(2^{l_{\text{rec}}-1}<(v_{e,\text{max}}-v_{e,\text{min}})/\Delta v_{e}<2^{l_{ \text{rec}}}\), where \(l_{\text{rec}}\) is the length of the string for \(v_{e}\), and \(\Delta v_{e}\) is the encoding accuracy for \(v_{e}\).
2. _Step 2--Population initialization:_ The population size \(NP\), the maximum number of generations \(G_{\text{max}}\), and the generation counter \(g\) are initialized. All individuals of the population in the \(0 th\) generation are generated randomly.
3. _Step 3--Calculating the fitness:_ Since the optimization problem is based on maximizing the contrast of the target's image, the fitness value of a chromosome is chosen to be the contrast of the image that is focused with the parameters corresponding to this chromosome. The image is obtained via (26), and the contrast is calculated via (25).
4. _Step 4--Selection:_ The selection operation keeps good chromosomes and eliminates inferior ones. In this paper, to improve the efficiency, the roulette wheel selection rule [33] is adopted, and the selection probability for each chromosome is calculated as follows: \[p\big{(}\text{CP}_{g,n}\big{)}=\frac{f\big{(}\text{CP}_{g,n}\big{)}}{\sum\limits _{n=1}^{N}f\big{(}\text{CP}_{g,n}\big{)}}\] (28)where \(f\left(\cdot\right)\) is the fitness function, and \(\text{CP}_{g,n}\) is the _nth_ chromosome of the population in _gth_ generation. In addition, the optimal individual preservation strategy [32] is used. In other words, the optimal individual of the parent population will directly enter the offspring population.
5. _Step 5--Crossover and mutation:_ In this paper, the single-point crossover operator and simple mutation operator [33] are applied to the selected chromosomes to produce the population of the next generation.
6. _Step 6--Judgment:_ If \(g\) is equal to \(G_{max}\) or the maximum fitness value remains stable, jump to step 7. Otherwise, return to step 3.
7. _Step 7--Output the optimal solution:_ The chromosome that has the largest fitness value is the optimal solution, and the corresponding parameters are the solutions for the optimization problem. To obtain the solutions for the optimization problem, a decoding operation should be performed with two steps (taking \(v_{e}\) for example): (1) convert the binary string to the decimal number \(v_{e}\)'; (2) calculate the actual value of \(v_{e}\) by (29). \[\vartheta_{e}=v_{e,\min}+v_{e}^{\prime}\cdot\frac{v_{e,\max}-v_{e,\min}}{2^{ l_{ee}}-1}\] (29)