# A Surrogate Model Based Multi-Objective Optimization Method for Optical Imaging System
## 2 Methodology
### Kriging Surrogate Model  
The Kriging surrogate model originated in the areas of mining and geostatistics, which involve temporally and spatially correlated data. The unique characteristic of Kriging stems from its ability to combine global and local modeling. The Kriging surrogate model is one of the unbiased models with the smallest estimation variance, which could provide efficient and reliable prediction. Extensive reviews of the Kriging model used in simulation, sensitivity analysis, and optimization in the design process can be found in [139]. Due to its high accuracy and good performance for complex nonlinear problems, the Kriging surrogate model was chosen as the surrogate model to provide prediction for the imaging quality of the optical imaging system in this paper. The Kriging model consists of two parts, the regression model and the stochastic process. The regression model represents the global tendency of the analyzed function, and the stochastic process represents the spatial correlations in the design space of interest [140]. \[y=\sum\limits_{i=1}^{k}\beta_{i}f_{i}(\mathbf{x})+Z(\mathbf{x}) \tag{1}\]  
Figure 2: Experimental design of the LHS method in 2D (**left**) and 3D (**right**). where \(\mathbf{x}\) is \(n\)-dimensional vector, \(y\) is the unknown function of \(\beta_{i}f_{i}\) (regression model) and \(Z(\mathbf{x})\) (stochastic process). The regression models with polynomials of orders 0, 1 and 2 were adopted here and detailed in Table 1.  
\(Z(\mathbf{x})\) represents a local deviation from the regression model and is the realization of a stationary, normally distributed Gauss random process with zero mean, variance, and non-zero covariance. The covariance matrix of \(Z(\mathbf{x})\) is given by:  
\[cov\Big{[}Z\Big{(}\mathbf{x}^{(i)}\Big{)},Z\Big{(}\mathbf{x}^{(j)}\Big{)} \Big{]}=\sigma^{2}R\Big{(}\mathbf{x}^{(i)},\mathbf{x}^{(j)}\Big{)}\;i,j=1, \ldots,ns \tag{2}\]  
where \(\sigma^{2}\) is the process variance, and \(R\) is an \(ns\times ns\) symmetric correlation matrix. In addition, \(R\Big{(}\mathbf{x}^{(i)},\mathbf{x}^{(j)}\Big{)}\) is the spatial correlation function between any two points \(\mathbf{x}^{(i)}\) and \(\mathbf{x}^{(j)}\) of \(ns\) sample points. A popular Gaussian correlation function is used here, and the function can be expressed:  
\[R\Big{(}x^{(i)},x^{(j)}\Big{)}=\exp\left[-\sum_{k=1}^{m}\theta_{k}\Big{|}x^{(i )}-x^{(j)}\Big{|}^{2}\right] \tag{3}\]  
where \(\theta_{k}\) is the \(k_{\text{th}}\) element of the correlation vector parameter \(\theta\). The regression term \(\sum_{i=1}^{k}\beta_{i}f_{i}(x)\) can choose a constant value, a linear model, or a quadratic model. In this paper, the quadratic model was adopted here. In the implementations, \(\mathbf{x}\) is normalized by subtracting the mean from each variable, and then dividing the values of each variable by its standard deviation:  
\[x_{\text{norm}}\;=\;\frac{x-x_{\text{mean}}}{x_{\text{std}}} \tag{4}\]  
The Kriging predictor is:  
\[\hat{y}(x)=\hat{\mu}+r(x)^{T}R^{-1}(Y-F\hat{\mu}) \tag{5}\]  
where \(F\) is a matrix that can be written as:  
\[F= \begin{bmatrix}f_{1}(\mathbf{x}_{1})&\cdots&f_{k}(\mathbf{x}_{1})\\ \vdots&\ddots&\vdots\\ f_{1}(\mathbf{x}_{ns})&\cdots&f_{k}(\mathbf{x}_{ns})\end{bmatrix} \]  
When the order of the regression models is 0, \(F\) is a column vector of length ns filled with ones. \(Y\) the column vector with responses of sample points, and \(r(x)\) is the correlation vector, which can be written as:  
\[r(x)=\Big{[}R\Big{(}x,x^{1}\Big{)},R\Big{(}x,x^{2}\Big{)},I,R(x,x^{n_{s}}) \Big{]}^{T} \tag{6}\]  
For a give parameter \(\theta\), \(\hat{\mu}\) and \(\hat{\sigma}^{2}\) can be calculated as:  
\[\hat{\mu}=\Big{(}F^{T}R^{-1}F\Big{)}^{-1}F^{T}R^{-1}Y \tag{7}\]