# Infrared and Visible Image Fusion: Methods, Datasets, Applications, and Prospects
## 2 Infrared and Visible Image Fusion Methods
### Method Based on Compressed Sensing
When representing an excessively sparse signal using compressed sensing, the measurement matrix's \(\Phi\) selection must meet the following qualifications for inequality:  
\[1-\varepsilon\leq\frac{\left\|A\alpha\right\|^{2}}{\left\|\alpha\right\|^{2}} \leq 1+\varepsilon,\varepsilon>0 \tag{9}\]  
The RIP property of the measurement matrix is a necessary and sufficient condition for the accurate reconstruction of the original signal by compressed sensing theory. The signal reconstruction algorithm must be implemented during the sampling and compression processes to restore the original signal if the compression matrix satisfies the reconstruction criteria. The original signal can be obtained by solving the underdetermined equations when the number of equations is less than the unknown quantity since the sample data are sparse and the dimension is smaller than that of the original signal. A popular reconstruction methodology for signal reconstruction is the convex optimization method. The original signal \(x\) can only be successfully recovered from the \(M\)-dimensional measurement projection signal \(y\) if the perception matrix \(A\) in Equation (8) satisfies the RIP condition. Firstly, the sparse coefficient \(\alpha=\Psi^{T}x\) is obtained from the inverse problem of the measured value, and the original signal \(x\) is obtained by substituting it into Equation (6). The reconstruction signal \(\alpha\) is an optimization problem for the following \(l_{0}\) norm solution (8). \[\dot{\alpha}=\min_{\alpha}\left\|\alpha\right\|_{l_{0}},s.t.y=\Phi\Psi\alpha \tag{10}\]  
where \(\left\|\alpha\right\|_{l_{0}}\) indicates \(\alpha\)'s zero norm or the number of non-zero elements in it. Because solving \(l_{0}\) is an NP-hard problem, the identical solution can be achieved by swapping \(l_{1}\) minimal norm solving for \(l_{0}\) minimum norm solving. \[\dot{\alpha}=\min_{\alpha}\left\|\alpha\right\|_{l_{1}},s.t.y=\Phi\Psi\alpha \tag{11}\]  
The compressed sensing matrix \(A\) satisfies the K-RIP requirement, and the measurement matrix is frequently chosen as a random Gaussian distribution matrix to obtain the non-correlation of \(\Phi\) and \(\Psi\), allowing the original signal to be reconstructed with high precision. \(l_{1}\) optimization is a convex optimization problem with \(O(N^{3})\) computational complexity. The most frequent algorithms are the basis tracing method, the interior point method, and the gradient projection method. The internal point method produces precise answers, but it is slow to calculate. The gradient projection method is faster than the interior point method, although it is less precise. The lowest \(l_{1}\) norm optimization issue can be turned into an unconstrained optimization problem to improve computational efficiency. \[\dot{\alpha}=\arg\min_{\alpha}\left\|A-y\right\|_{l_{1}}+\lambda\left\|\alpha \right\|_{l_{1}} \tag{12}\]  
A significant advancement in the field of signal processing, compressed sensing will integrate sampling and compression of signals, enable compressed sampling of signals, obtain fewer measurement data, and enable efficient transmission, storage, and process ing [40]. However, it is still necessary to improve the fusion effect of compressed sensing. Consideration can be given to designing efficient analog-to-information conversion front-ends to eliminate the traditional analog-to-digital converters for the compressed sampling of images or designing better reconstruction algorithms to improve the reconstruction of images [41].