## 3 Uncertainty Evaluation
The uncertainty of \(e_{Z}(x,y)\) at the pixel position of (\(x_{i},y_{j}\)) is then evaluated by \[u[e_{Z}(x_{i},y_{j})]=\frac{\overline{e_{Z}}(x_{i},y_{j})-e_{Z}(x_{i},y_{j})}{2}. \tag{29}\]  
In addition, according to Equation (18), the least-squares estimation \(A^{\prime}_{pq}\) of elements in matrix A\({}^{\prime}\) can be calculated, as follows:  
\[\hat{A}^{\prime}_{pq}=\frac{1}{\alpha^{p}}\hat{A}_{pq}. \tag{30}\]  
It follows that the uncertainty in \(A^{\prime}_{pq}\) can be evaluated by the following equation:  
\[u^{2}\Big{(}\hat{A}^{\prime}_{pq}\Big{)}=\Big{(}\frac{p}{\alpha^{p+1}}\hat{A} _{pq}\Big{)}^{2}u^{2}(\alpha)+\bigg{(}\frac{1}{\alpha^{p}}\bigg{)}^{2}u^{2} \big{(}\hat{A}_{pq}\big{)}. \tag{31}\]  
With \(A^{\prime}_{pq}\) and \(u(A_{pq})\), the values of \(e_{Z}(x/\alpha,y)\) and \(\partial e_{Z}(x/\alpha,y)/\partial\alpha\) can be obtained at any pixel position of \((x_{i},y_{j})\). The uncertainty of \(e_{Z}(x/\alpha,y)\) at any pixel position \((x_{i},y_{j})\) can be also evaluated in a similar manner to Equation (28). Likewise, the value of \(e_{X}(x,y)\) and its uncertainty \(u[e_{X}(x,y)]\) at any pixel position \((x_{i},y_{j})\) can be calculated as well. The PV value of \(e_{Z}(x,y)\) in the region \(\Omega\) of interest can be evaluated by  
\[\mathrm{PV}[e_{Z}(x,y)]=\max[e_{Z}(x,y)]-\min[e_{Z}(x,y)],\quad(x,y)\in\Omega, \tag{32}\]  
where \(\max(\cdot)\) and \(\min(\cdot)\) represent the maximum and minimum of a function. The peak and valley values of \(e_{Z}(x,y)\), \(\max[e_{Z}(x,y)]\), and \(\min[e_{Z}(x,y)]\) can be readily calculated according to the obtained expression of \(e_{Z}(x,y)\). The uncertainty in the calculated PV of \(e_{Z}(x,y)\) can then be evaluated, as follows:  
\[u^{2}\{\mathrm{PV}[e_{Z}(x,y)]\}=u_{\mathrm{P}}^{2}[e_{Z}(x,y)]+u_{\mathrm{v} }^{2}[e_{Z}(x,y)], \tag{33}\]  
where \(u_{\mathrm{P}}[e_{Z}(x,y)]\) and \(u_{\mathrm{v}}[e_{Z}(x,y)]\) represent the uncertainties of \(e_{Z}(x,y)\) at the peak and valley positions, respectively, which can be directly obtained by substituting the coordinates of the peak and valley positions into Equation (29). Other approaches to evaluate the uncertainty in PV could simply double the maximum value of the uncertainty matrix or applying the Monte Carlo (MC) simulation. However, the former method obviously overestimates the uncertainty in the PV, except coincidence. The MC method could estimate the PV uncertainty by collecting the PV value in each trail assuming that all of the significant errors are included and the probability distributions of each error source is known. Nevertheless, it is burdensome to run the simulation when routine experiment is conducted in a similar condition and the noise could exist explicitly in the evaluated results. Therefore, in this paper, the MC method is not employed, but just the part of error form is analyzed directly to evaluate the uncertainty in the peak and valley separately, and then combine them together. Since the PV of each error form is usually larger than the uncertainty in this experiment, this method is thus a reasonable way to obtain the uncertainty in PV. As can be noted from the above description, the success of the proposed evaluation method lies on the fitting performance of the polynomials fitted to \(e_{Z}(x,y)\) and \(e_{X}(x/\alpha,y)\). There are two measures that are commonly used to evaluate the goodness-of-fit in statistics.