# Infrared and Visible Image Fusion: Methods, Datasets, Applications, and Prospects
## 2 Infrared and Visible Image Fusion Methods
### Method Based on Compressed Sensing  
By using a low-signal-sample compression technique based on the signal's sparsity under specific transformations, compressed sensing (CS) can effectively lower the computing complexity and increase the operation rate of the image processing scheme. The sparsity of the signal is the foundation of compressed sensing. The three primary components of the compressed sensing theory are the reconstruction method, coding measurement, and signal sparse representation [39]. The obtained transformation vectors are said to be sparse because they are projected onto the orthogonal transformation basis, and the absolute values of the majority of the transformation coefficients are often tiny. Let \(x\) be an \(N\)-length discrete real-valued signal that can be represented as a concatenation of orthogonal bases. In the time domain, element x is \(x(n),n\in[j,2,...,N]\). \(\Psi^{T}=[\psi_{1},\psi_{2},...,\psi_{m},...,\psi_{M}]\) is the orthogonal basis, and \(\psi_{m}\) is the \(N*1\) column vector in the expression. A linear combination of orthogonal bases can be used to express \(x\):  
\[x=\sum_{k=1}^{N}\psi_{k}a_{k}=\Psi\alpha \tag{6}\]  
In this situation, \(\Psi^{T}\) is the transpose of \(\Psi\), \(xa_{k}=<x,\psi_{k}>=<\psi_{i}^{T},x>\), \(x\), the N*1 matrix, and \(\Psi\), the N*N matrix. The signal \(x\) is represented equivalently by. The signal \(x\) is referred to as sparse of order \(K\) if it has only \(K\ll N\) non-zero coefficients on the orthogonal basis \(\Psi\). \(K\) denotes the signal's sparsity, and \(\Psi\) is the signal's sparse basis. The sparsity of a compressible signal is the number of nonzero sparsity coefficients \(aK\) with which the signal is represented in the corresponding orthogonal basis \(\Psi\). The original signal is reconstructed from the sparse transform domain using the inverse transform \(x=\Psi^{H}a\) because \(\Psi\) is an orthogonal basis and there are hence \(\Psi\Psi^{H}=\Psi^{H}\Psi=I\) and \(\Psi\in C^{N\times N}\). The sparse signal x is not directly measured in the CS coding measurement model; instead, the sparse signal x is linearly mapped using the measurement matrix \(\Phi=\left[|\varphi_{1}|,|\varphi_{2}|,...,|\varphi_{n}|,...,|\varphi_{M}|\right]\), yielding a set of random measurement values \(y_{m}=<x,\varphi_{m}>\) in the form of a matrix:  
\[y=\Phi x \tag{7}\]  
where \(\Phi\) is the measurement matrix for \(M*N\), and y is the linear projection of \(x\) under \(M*1\)'s measurement matrix. When you add it to Formula (7), you obtain:  
\[y=\Phi x=\Phi\Psi\alpha=A\alpha \tag{8}\]  
Figure 6: Flow chart of image fusion algorithm based on sparse representation. where \(A=\Phi\Psi\) is the matrix, also known as the perception matrix. Equation (7) converts the N-dimensional signal \(x\) to the M-dimensional observed signal y. The dimension of y is substantially lower than the dimension of \(x\), and the reconstruction of \(\alpha\) from y has an infinite number of sets of solutions, a pathological problem. The initial signal in the Formula (8) is K-order sparse, which means it has only K non-zero coefficients and \(K<M\ll N\). When the original signal's sparse representation coefficient \(\alpha\) is known, the original signal \(A\) of Equation (8) is solved inversely, yielding the reconstructed original signal. Solving the nonetheless definite equations by the convex optimization method solves the required inverse problem and yields the sparse coefficients \(\alpha\), thus obtaining the original signal \(X\). The measurement number M (that is, the dimension of y) must meet \(M=O(K\lg(N))\), and the measurement matrix \(\Phi\) in the measurement Formula (8) must meet the restricted isometric RIP (restricted isometry property) to ensure that the original signal is reconstructed without distortion.