## 4 Results and Discussion
### Suppression of the Twin-Image Effect
#### 4.1.4 Discussion of Results in Section 4.1  
Although the complex field amplitude reconstruction method based on the Gerchberg-Saxton algorithm and the spatial-frequency method have a different physical and algorithmic basis, the results show that the conditions under which their application significantly improves the quality of determination of the particle image boundary coincide: \(z_{R}\geq\frac{2d^{2}}{\lambda}\). Here, d for a particle of arbitrary shape is the particle's circumscribed diameter. This suggests that this condition does not depend on the implemented algorithm but is related to the holographic image quality of the used particle, which is based on determining the sharpness of the image boundary (boundary contrast or boundary intensity jump).  
The fact that \(z_{R}\) is about twice the Rayleigh distance \(\frac{d^{2}}{\lambda}\) defining the conditional boundary between Fresnel and Fraunhofer diffraction [33; 34; 35] can be explained as follows.  
This is explained by the distribution of the wave field corresponding to the virtual and real images in the boundary region of the real particle image, which is marked with a green circle in Figure 3. Figure 12 shows the results of numerical calculations of the diffraction  
Figure 11: Dependence of the particle image quality indicators (opaque circle particle with a diameter of 89.7 \(\upmu\)m (**a**,**c**) and 133.2 \(\upmu\)m (**b**,**d**)) vs. the distance between the particle and the hologram plane at the stage of digital hologram calculation for model particles before (round points) and after (square points) applying the spatial-frequency method. (**a**,**b**) Boundary contrast K; (**c**,**d**) boundary intensity jump \(\Delta\)I.  
pattern using Formula (2) on an opaque circular disk in the boundary region for different distances z.  
The above diagrams show that when z varies from the Rayleigh distance \(\frac{\mathrm{z_{R}}}{2}\) to the Rayleigh double distance \(\mathrm{z_{R}}\), there is an intersection of the first dark fringe of the diffraction pattern from the virtual image and the external area identified along the boundary of the real particle image to determine the quality indicators used. This affects the quality indicators K and \(\Delta\mathrm{I}\), leads to errors in determining their values and, as a result, errors in determining the best image plane. With \(\mathrm{z>z_{R}}\), these contributions are opposite: the Poisson light spot of the diffraction pattern from the virtual image increases the average intensity of the internal area highlighted along the boundary of the particle image, and the first minimum of the diffraction pattern does not actually change the average intensity of the external one. With \(\mathrm{z=z_{R}}\), the contribution of the virtual image to the average intensity of the internal and external areas identified along the boundary of the particle image is almost the same, and the effect on the boundary contrast is maximized. Further, this situation is maintained, but the intensity of the diffraction pattern fringes from the virtual image is substantially reduced by diffraction divergence, so the effect on the boundary contrast is also reduced. Hence, this causes the error in determining the best image plane at \(\mathrm{z>z_{R}}\).  
Thus, the distance \(\mathrm{z_{R}}\) determines the minimum distance to the studied particle, at which the twin image is most suppressed.  
Figure 13 shows a holographic image of a particle obtained by a submersible DHC (Figure 2) during a natural field experiment [36].  
Figure 12: Diffraction patterns on a circular disk in the transition region from Fresnel diffraction to Fraunhofer diffraction. Green line—external area along the boundary of the real particle image, blue line—internal (used to determine quality indicators).