## 2 Principle of Tomographic Laser Absorption
### Principles of the MIARO Algorithm  
Using the proposed algorithm MIARO, the global optimal solution can be found in a shorter time because of the realization of the adaptive iteration step size and the regular optimization of process control, which takes advantage of different iteration algorithms. Firstly, global optimization is sought by the Landweber algorithm, and the initial iteration value is provided based on MAART by iterating a small number of times. The Landweber algorithm is a synchronous iterative reconstruction algorithm or simultaneous iterative reconstruction technique (SIRT), where each iteration involves all rows of the system of equations, which can realize fast global optimization. The specific iteration form is  
\[\mathbf{\alpha}^{(k+1)}=\mathbf{\alpha}^{(k)}-\lambda_{k}\mathbf{L}^{T}\Big{(}\mathbf{L}\mathbf{ \alpha}^{(k)}-\mathbf{A}\Big{)} \tag{4}\]  
where \(\mathbf{\alpha}\) is the reconstruction parameter, and \(\mathbf{A}\) is the integral projection value. \(\mathbf{L}\) is the sensitivity matrix, and \(\lambda_{k}\) is the \(k\)-th iteration step. The specific value of \(\lambda_{k}\) can be found in refs. [26; 27], and its expression is as follows:  
\[\lambda_{k}=\frac{\left\langle\mathbf{r}^{(k)},\mathbf{r}^{(k)}\right\rangle}{\left\| \mathbf{L}^{T}\mathbf{r}^{(k)}\right\|_{2}^{2}},\mathbf{r}^{(k)}=\mathbf{A}-\mathbf{L}\mathbf{\alpha} ^{(k)} \tag{5}\]  
where \(<,>\) denotes the inner product. In MAART, the adaptive change in the iterative step can be implemented, and reconstruction parameters can be introduced into the iterative relationship. The higher the contribution of a reconstruction parameter to a projection value, the larger the possible step size of an iteration. But the MAART iterative algorithm is sensitive to the initial value, and it cannot be set to 0. The initial parameter is provided via a global search of optimization using the Landweber algorithm so as to realize a faster iterative convergence. The iterative form of the MAART algorithm is expressed as [28]  
\[\alpha^{(k+1,i)}=\alpha^{(k,i)}-\lambda\Big{(}L_{i}\alpha^{(k,i)}-A_{i}\Big{)} \tag{6}\]  
\[\lambda=\beta\times\frac{\alpha_{j}^{(k)}L_{ij}}{\sum\limits_{j=1}^{N}\alpha_ {j}^{(k)}L_{ij}} \tag{7}\]  
The reconstruction process is often a kind of pathological or ill-posed inverse problem. In order to solve such a problem, it is necessary to regularize the constraints on the reconstructed and true distributions with the aid of some additional a priori information to suppress the noise and artifacts introduced by data incompleteness. Regularization can effectively improve the instability of the laminar inverse problem. Therefore, the establishment of suitable and accurate regularization terms is key to the success of a regularization method. One of the frequently adopted regularization models is Tikhonov regularization, which is usually used to attenuate the smoothness of the mutation response of neighboring points in a reconstruction region, and a typical function is defined as follows:  
\[\alpha^{(k)}_{(i,j)}=(1-\theta)\alpha^{(k-1)}_{(i,j)}+\theta(\sum\limits_{y=j-1 }^{j+1}\sum\limits_{x=i-1}^{i+1}\alpha^{(k-1)}_{(x,y)})/8 \tag{8}\]  
where \(i,j\) is the rank order of a grid. The Adaptive Regular Optimization (ARO) algorithm proposed here is used to realize the change in the adaptive regularization parameters and to introduce the reconstruction parameters into the iterative process. In ARO, the traditional fixed smoothing coefficient \(\theta\) cannot accurately reflect the actual distribution characteristics, and the specific choice of the smoothing slip coefficient will also be directly related to the reconstruction quality. When \(\theta\) takes a larger value, the reconstruction parameters will be over-smoothed. On the other hand, a smaller value of \(\theta\) can result in the reconstruction parameters falling into a local optimal solution, and hence, the reconstructed distribution tends to become steeper.