## 3 Simulation Result
### Improving the Recovery from Noise by Analytic Extrapolation  
Although the analytic extrapolation method performs well under noise-free conditions, it is sensitive to noise, which often leads to the failure of reconstructing high-resolution phase images. In this section, we analyze the impact of noise on analytic extrapolation using eigenvalue decomposition methods and address the noise issue through regularization to achieve high-resolution image reconstruction. When exiting the noise in a finite spatial frequency area, the Equation (10) can be rewritten as,  
\[\mathbf{y}\mathbf{+}\mathbf{n}\mathbf{=}\mathbf{A}(\mathbf{x}\mathbf{+}\mathbf{n}_{\mathbf{x}}) \tag{15}\]  
where \(\mathbf{n}\) is the vector of noise and \(\mathbf{n}_{\mathbf{x}}\) is the analytic extrapolation error introduced by the noise. The simple relation between the \(\mathbf{n}\) and \(\mathbf{n}_{\mathbf{x}}\) can be given by  
\[\mathbf{n}\mathbf{=}\mathbf{A}\mathbf{n}_{\mathbf{x}} \tag{16}\]  
For a square matrix \(\mathbf{A}\), we can decompose it into eigenvalues \(\lambda_{k}\) and eigenvectors \(\mathbf{\varphi}_{\mathbf{k}}\), which are also called eigenmodes, as follows (for non-square matrices, we can use the singular value decomposition):  
\[\mathbf{A}=\sum_{k=1}^{M}\lambda_{k}\mathbf{\varphi}_{\mathbf{k}}\mathbf{\varphi}_{\mathbf{k}}^{T} \tag{17}\]  
Combining Equation (17), the Equation (16) can be rewritten as  
\[\lambda_{k}{}^{-1}\mathbf{\varphi}_{\mathbf{k}}\mathbf{n}=\mathbf{\varphi}_{\mathbf{k}}\mathbf{n}_{ \mathbf{x}}k=1,2,\ldots N \tag{18}\]  
Considering that Gaussian white noise is uniformly distributed in whole spatial frequency, and the eigenvectors are orthogonally normalized, the \(\mathbf{\varphi}_{\mathbf{k}}\mathbf{n}\) will have a similar value to \(\mathbf{n}\). As a result, the key factor to affect the error of analytic extrapolation error is the eigenvalue \(\lambda_{k}\). When the \(\lambda_{k}\) larger than 1, the \(\mathbf{\varphi}_{\mathbf{k}}\mathbf{n}_{\mathbf{x}}\) will have a smaller value than \(\mathbf{\varphi}_{\mathbf{k}}\mathbf{n}\), which means the noise can't be amplificated by the analytic extrapolation. In contrast,  
Figure 5: The comparison of the amplitude and phase image among the analytic extrapolation, diffraction-limited, and ground truth. (**a1–a3**) The dashed line in Figure 4 with different ERRs. (**b1–b3**) The phase image of diffraction limitation, analytic extrapolation, and ground truth at ERR of 2.  
when the \(\lambda_{k}\) much smaller than 1, the \(\varphi_{k}n_{x}\) will have an especially larger value than \(\varphi_{k}n\), and the noise will dominate the analytic extrapolation process. Unfortunately, in the BIFT system, the matrix \(A^{\prime}\) often has an eigenvalue much smaller than 1, as shown in Figure 6.  
In Figure 6, the eigenvalues decrease rapidly as the number of k increases, indicating that higher-order eigenmodes passing through the band-limited system experience severe attenuation. This attenuation is exponential with increasing mode number, as demonstrated in Figure 6b. When higher-order patterns collected by the BIFT system are submerged in noise, they cannot be effectively recovered by the analytic extrapolation method. Consequently, the range of spatial frequencies that can be recovered through analytic extrapolation ultimately depends on the noise level. To reduce the noise effect in the analytic extrapolation process, we added a small positive value \(\varepsilon\) to the small \(\lambda_{k}\) ensuring the noise is not amplificated too large in the recovery process. The selection of the value of the \(\varepsilon\) is depend on the signal to noise ratio. By the regulation of the \(\varepsilon\), the super-resolution image can be recovered by analytic extrapolation under the noise corruption. Figure 7a illustrates \(e,e_{rec}\) versus the varied (SNR) from 0 to 80 dB at the condition of ERR = 2. In Figure 7a, the \(e\), \(e_{rec}\) was decreased with the increase of SNR.