## 4 Experimental Validation  
The in-line interferometer with the DSB filter shown in Figure 10 was implemented to experimentally validate the theoretical analysis of the transverse and longitudinal magnifications of the reconstructed images.  
The microscopic objects were illuminated using an expanded and collimated laser beam, with a wavelength \(\lambda~{}=~{}632.8\) nm, manufactured by Lumentum. The lens L2, with a focal length \(f\) = 85 mm, creates an image of the object plane (\(s_{o}~{}=~{}113\) mm) at the image plane (\(s_{i}~{}=~{}347\) mm), where a CCD (model Blackfly BFS-U3-31S4M-C manufactured by Flir) was positioned. The intensity recorded by the CCD was proportional to the square of the complex amplitude in the image plane, resulting in the loss of phase information. Consequently, during the reconstruction of the focused image, the conjugated defocused image was superimposed. To reconstruct the focused image without the influence of the conjugated image, it was essential to have complex amplitude data, which included both magnitude and phase information.  
\begin{table}
\begin{tabular}{c c|c|c|c|c|c|c} \hline  & & \multicolumn{3}{c}{**USAF 1951**} & \multicolumn{3}{c}{**Microsphere**} \\ \cline{3-8}  & & **Ent** & **TC** & **GI** & **Ent** & **TC** & **GI** \\ \hline \multirow{2}{*}{\(\mathrm{M_{L}}\)} & Ave. & \(-9.4841\) & \(-9.4786\) & \(-9.4511\) & \(-9.4686\) & \(-9.4756\) & \(-9.5697\) \\  & S.D. & \(0.0089\) & \(0.0058\) & \(0.0256\) & \(0.0138\) & \(0.0524\) & \(0.0693\) \\ \multirow{2}{*}{\(\mathrm{M_{T}}\)} & Ave. & \(-3.0767\) & \(-3.076\) & \(-3.0837\) & \(-3.0770\) & \(-3.0738\) & \(-3.0864\) \\  & S.D. & \(0.0125\) & \(0.0120\) & \(0.0128\) & \(0.0176\) & \(0.0103\) & \(0.0572\) \\ \hline \end{tabular}
\end{table}
Table 1: Summary of the average (Ave.) and the standard deviation (S.D.) values of \(\mathrm{M_{L}}\) and \(\mathrm{M_{T}}\) for the two simulated microscopic objects for the three sparsity image criteria: Tamura coefficient (TC), Gini Index (GI), and entropy (Ent).  
Figure 9: (a) \(\mathrm{M_{L}}\) as a function of the microsphere displacement \(\Delta z_{o}\) and (b) \(\mathrm{M_{T}}\) as a function of \(\Delta z_{o}\).  
The DSB filter was employed in the in-line interferometer to obtain magnitude and phase information (digital hologram). The experimental scheme used to implement the DSB followed the proposal given in [25]. A description of the experimental arrangement used to implement the DSB filter is provided in the Supplementary Materials.  
The transverse and longitudinal magnifications of the reconstructed images were calculated for the following microscopic objects: (a) a resolution test USAF 1951 and (b) glass microspheres (diameter: 14.5 \(\upmu\)m \(\pm\) 1 \(\upmu\)m), as shown in Figure 11. In agreement with the simulations section, an extra example, a microscope reticle, was also experimentally studied, and the obtained results are provided in the Supplementary Materials. The microscopic objects were placed on a linear translation stage equipped with a vernier micrometer having a resolution of 10 \(\upmu\)m. The objects were shifted in steps of 1 mm in the axial direction, covering a range from \(\Delta z_{o}~{}=~{}-10\) mm to \(\Delta z_{o}~{}=~{}+10\) mm. In each displacement, the focused images were reconstructed.