## 4 Genetic-OMP Algorithm
Therefore a lower spatial information gain is the superior individual. 2. Spatial information characteristics The OMP algorithm only considers the relationship between the observation matrix and the measurement vectors and does not take into account the effect of \(A_{j}\) on the subsequent support set election. If \(A_{j}\) is chosen in the \(i\)th iteration, the updated residual matrix has a direct impact on the choice of the \(i+1\)th support set. To facilitate the identification of \(\Psi\), the result of the \(i\)th selection should lead to a smaller possible range of support sets. Therefore, we introduce spatial information characteristics as a judgment tool. The spatial information characteristic is numerically equal to the spatial information entropy \(H\big{(}A_{j}\big{)}\) of \(A_{j}\). The spatial information characteristic represents the uncertainty that the choice of \(A_{j}\) brings to the option of the support set in the next iteration. \(\Psi\) is obtained in continuous iteration. As the number of iterations increases, the range of \(\Psi\)'s values should continue to shrink. Accordingly, the spatial information characteristics of each selected \(A_{j}\) should be as small as possible. In summary, spatial information gain and spatial information characteristics constitute the information characterization norm. Based on the norm, lower spatial information gain and smaller spatial information characteristics are superior individuals. So, \(F_{i}\) is satisfied. \[F_{i}=e^{-\lambda_{2}} \tag{31}\]  
\[\lambda_{2}=\frac{Gain\big{(}A_{j},r_{i}\big{)}+H\big{(}A_{j}\big{)}}{2} \tag{32}\]  
After calculating the \(F_{i}\) of each individual, the individual can be chosen. Based on the norm, we use a roulette wheel method of choice. In the \(i\)th iteration, \(G_{i}\) roulette choices are performed, and each choice will produce a single individual. The number of times \(G_{i}\) are made is decided by the outcome of \(G_{i-1}\)'s option. When selection is achieved by using the roulette strategy, there is a probability that an individual with a high \(F_{i}\) will be chosen multiple times. This leads to the fact that after \(G_{i}\) choices, the number of individuals selected is usually smaller than \(G_{i}\). Thus, the number of selections is gradually reduced in each iteration, which is also a reflection of the \"meritocracy\" of the genetic algorithm. The detailed steps of the Genetic-OMP are described in Algorithm 1. ```
0: Observation matrix \(A\), Measurement vector \(V\), Sparsity \(k\)
0: support set \(\Psi\)
1: Initialize: Residual \(r_{0}=V\), iteration counter \(i=0\), preliminary index set \(\Psi=\phi\), roulette selection number \(G_{0}=n\)
2: Calculate \(J_{i}\), and select \(K\) individual according to the size of a as the preliminary index set \(\Theta\). 3: Calculate \(F_{i}\) of each element in \(\Theta\), roulette the elements in \(\Theta\) according to \(F_{i}\), and select the support set index \(\Psi_{i}\) of this iteration. Update \(\Psi_{i}\) into \(\Psi\)
4: Update residuals according to least squares estimates: \(\hat{d}=\big{(}A_{\Psi}^{T}A_{\Psi}\big{)}^{-1}A_{\Psi}^{T}r_{i-1}\), \(r_{i}=y-A_{\Psi}^{T}\hat{d}\)
5:\(i=i+1\). If \(i>K\) stops the iteration
```  
**Algorithm 1** Genetic-OMP Algorithm