## 2 Method
### The Super-Resolution BIFT Imaging by Analytic Extrapolation  
As BIFT's uniqueness is guaranteed, the light field can be accurately reconstructed to the theoretical limit of the Cramer-Rao bound [26]. However, due to the cut-off effect of the objective lens, high spatial frequencies cannot be recovered, but only low spatial frequencies within the finite aperture \(P(f_{x}^{\prime},f_{y}^{\prime})\). In the BIFT imaging method, an object \(O\) (\(x_{0},y_{0}\)) is projected into the image plane by the band-limited imaging system. The wave field \(U\big{(}f_{x},f_{y}\big{)}\) is the Fourier transform of the bounded object field with the finite FOV, and thus introducing additional constraints of analyticity. The analyticity indicates three properties as follows:  
1. The optical field \(U\big{(}f_{x},f_{y}\big{)}\) exhibits continuous differentiability and can be extended indefinitely in the Fourier plane;
2. Any point of the optical field \(U\big{(}f_{x},f_{y}\big{)}\) can be expressed as the sum of each other points in the field. In other words, any point of the optical field \(U\big{(}f_{x},f_{y}\big{)}\) is not independent from others;
3. The entire function can be uniquely determined by its behavior within any arbitrarily small region. The unknown portion of the analytic function can be inferred from its analyticity in that small region. For a more comprehensive description of analyticity in the BIFT system, we provide the following mathematical analyses. Using the Equations (1) and (2), the Fourier transform of \(f(x,y)\) is given by  
\[U(f_{x},f_{y})=P(f_{x},f_{y})\iint h_{f}(f_{x}-f_{x_{0}},f_{y}-f_{y_{0}})U_{0 }(f_{x_{0}},f_{x_{0}})d_{f_{x_{0}}}d_{f_{y_{0}}}, \tag{6}\]where \(U_{0}(f_{x_{0}},f_{x_{0}})\) is the Fourier transform of \(O(x_{0},y_{0})\) and \(h_{f}(f_{x_{0}},f_{y_{0}})\) is the Fourier transform of the finite FOV of \(S(x_{0},y_{0})\). Considering \(U_{0}(f_{x_{0}},f_{y_{0}})\) with the discrete sampling with sampling point \((n_{0},m_{0})\). The Equation (7) can be rewritten as  
\[U(f_{x},f_{y})=P(f_{x},f_{y}){\sum\limits_{n_{0}}}{h_{f}(f_{x}-n_{0},f_{y}-m_{0} )U_{0}(n_{0},m_{0})}. \tag{7}\]  
As we can see in Equation (7), although the function \(U_{0}(n_{0},m_{0})\) is discrete, the Fourier transform of the finite-aperture FOV is continuously differentiable. Therefore, the convolved light field \({\sum\limits_{n}}{h_{f}(f_{x}-n_{0},f_{y}-m_{0})U_{0}(n_{0},m_{0})}\) is also continuously differentiable. Such continuous differentiability of this complex function ensures that it satisfies the Cauchy-Riemann equations, thereby exhibiting analyticity. However, due to the cut-off effect of the objective lens \(P(f_{x},f_{y})\), we can only obtain partial information about the analytic function \({\sum\limits_{n}}{h_{f}(f_{x}-n_{0},f_{y}-m_{0})U_{0}(n_{0},m_{0})}\) within the finite aperture, which is \(U(f_{x},f_{y})\). As a result, property 3 can be used to infer the whole object's spatial frequency from the known part of \(U(f_{x},f_{y})\). To better understand the principle of analytic continuation, the Taylor series expansion serves as a simple example. By using the Taylor equation, a one-dimensional case of \(U_{0}(f_{x_{0}},f_{y_{0}})\) can be expressed as  
\[U_{0}(f_{x_{0}})=U(n)+U^{\prime}(n)(f_{x_{0}}-n)+U^{n}(n)(f_{x_{0}}-n)^{2}+\dots, \tag{8}\]  
In Equation (8), due to the continuous derivatives of the analytic function, the function outside the finite aperture can be approximated by constructing a polynomial series with coefficients from multiple order derivatives at the point within the band-limited region. However, in practice, the spatial frequencies within the aperture are also sampled by the camera's pixels. Calculating the derivative of a point using discrete difference operations introduces significant errors. Therefore, we use matrix-based methods to analyze the analytic extrapolation.