## 2 Materials and Methods
### Analysis Methods
[22] used a denoising method based on singular value decomposition to get a denoised PSF, but it produced small visible artificial ripples to the denoised PSF [19]. The computed PSF, in particular the model of [PERSON] and [PERSON] [23], also takes into account the conditions in which the experiment is carried out, and it has an advantage of being estimated for all of the object space. It is free of noise, flexible, and suitable to any specific condition of the experimentation, unlike experimental PSF, which is not modifiable because it is measured to represent a specific condition in which the images of biological sample must be recorded. The BF-PSF model was reported previously by [PERSON] and [PERSON] [16]. However, they do not assume a real point object in their model. The model of [PERSON] and [PERSON] [23] is more an accurate model for PSF generation that is suitable for the brightfield setting [10], and this was discussed and publicly made available as an ImageJ plug-in by [PERSON] et al. [24]. According to the model, the image of a point object located at \(X_{p}=\left(x_{p},\,y_{p},\,z_{p}\right)\) is defined by Equation (4) [23]:  
\[PSF\left(X,X_{p}\right)=\left|A\int_{0}^{1}e^{i\,\,w\left(\rho,\,X,\,X_{p} \right)}\,\,J_{0}\left(\frac{2\pi}{\lambda}\,r\,NA\,\rho\right)\rho\,d\rho \right|^{2} \tag{4}\]  
where \(X=\left(x,\,y,\,z\right)\) is a point on the image plane, \(NA\) is the numeric aperture of the objective, \(A\) is an amplitude constant, \(\rho\) is the radius of the microscope's limiting aperture in the microscope's back focal plane, \(J_{0}\) denotes the Bessel function of the first kind of order zero, and \(w\left(\rho,\,X,\,X_{p}\right)\) is the phase which enables aberrations to be taken into account in the model, \(r=\sqrt{\left(x-x_{p}\right)^{2}+\left(y-y_{p}\right)^{2}}\). The phase \(w\left(\rho,\,X,\,X_{p}\right)\) is defined by Equation (5), as there are two layers (immersion medium of refractive index \(n_{c}\) and the specimen layer of refractive index \(n_{m}\)) [25]:  
\[ \begin{array}{ll}w\left(\rho,\,X,\,X_{p}\right)&=k[z_{p}\sqrt{n_{m}{}^{2}- NA^{2}\rho^{2}}-WD\sqrt{n_{c^{*}}{}^{2}-NA^{2}\rho^{2}}\\ &+\left(z_{p}-z_{n}+n_{c}\left(\frac{WD}{n_{c^{*}}}-\frac{z_{p}}{n_{m}}\right) \right)\sqrt{n_{c}{}^{2}-NA^{2}\rho^{2}}]\end{array}  \tag{5}\]  
The parameters \(n_{c^{*}}\) and \(WD\) are the design refractive index of the immersion medium and working distance of the objective, respectively, and \(k\) is the wave number. The model of [PERSON] and Lanni [23] allowed for the restoration of our BF images. Defining the wave field traversing the specimen by \(\underline{E}_{z}(x,y)=\sqrt{I_{z}(x,y)}e^{-i\phi_{z}(x,y)}\), where \(I_{z}(x,y)\) is the intensity, \(\phi_{z}(x,y)\) the phase, \(z\) denotes the position along the optical axis, and \((x,y)\) are the two-dimensional coordinates in the plane orthogonal to the optical axis. Therefore, the denotation \(I_{z}(x,y)\) indicates the intensity \(I(x,y,z)\). The TIE is obtained by substituting \(E_{z}(x,y)\) into the paraxial wave equation and taking the imaginary part of the resulting expression; the TIE is defined by [6]:  
\[\frac{\partial I_{z}(x,y)}{\partial z}=-\frac{\lambda}{2\pi}\
abla_{\perp}\! \cdot\!(I_{z}(x,y)\
abla_{\perp}\phi_{z}(x,y)) \tag{6}\]  
where \(\lambda\) is the wavelength of the illumination source and \(\
abla_{\perp}\) denotes the gradient operator in the lateral dimensions \((x,y)\). Due to its non-interferometric nature, the illumination can be quasi-monochromatic and partially-coherent [26].