# Multi-Object Deep-Field Digital Holographic Imaging Based on Inverse Cross-Correlation
## 3 Numerical Calculation Results  
### Viability Verification  
Figure 3 illustrates the numerical calculation process of the large-depth-of-field digital holography of multiplanar objects when subjected to inverse cross-correlation. Figure 3a shows the proposed optical path system: the coherent light from the laser is collimated by a spatial filter (\(SF\)) and a lens (\(L\)), which is then separated into an object path and a reference path using a beam splitter, \(BS_{1}\). After encountering reflector \(M_{2}\), the object's light path interacts with the three-plane object, as shown in Figure 3b, which then incidents to the beam-combining mirror \(BS_{2}\), and this is then received by the photodetector. The reference light is reflected from reflector \(M_{1}\) to the beam-combining mirror \(BS_{2}\). The angle of \(BS_{2}\) is fine-tuned to separate the reference light from the object light. After the two lights interact, the off-axis hologram can be observed in the photodetector. Assuming that the length and width of the image output by the photodetector are \(W\) and \(H\) (respectively), after the specific off-axis digital holographic filter processing, the diffraction-reconstructed image under different values of reconstruction distance, \(z\), is calculated on a computer using the angular spectrum diffraction method. Two adjacent reconstruction distances, \(z_{1}\) and \(z_{2}\), are selected. The color bars' values indicate the normalized amplitudes. And a cross-correlation operation is performed. Since the reconstructed images have the same size, a cross-correlation coefficient matrix with a size of \((2W-1)*(2H-1)\) can be obtained, as shown in Figure 3c. The pixel coordinates are found under a specific threshold \(\zeta\) in the matrix, thereby denoting the edge coordinates of the object to be measured. In addition, the neighboring coordinates can be used as vertices to filter out the planes of different measured objects, as shown in Figure 3d. In the simulation, the size of the object and the photosensitive surface of the CCD were the same, the diffraction distance from the farthest object to the CCD was 0.93 m, and the three measured objects were uniformly distributed in the z-direction within the range of the depth of field--namely 0.62 m. The sample size of the CCD was 512 \(\times\) 512 pixels, and the whole sensitive area was 5 \(\times\) 5 mm. Image reconstruction was performed in steps of 0.062 m. The actual position of the reconstructed image should be at the coordinates with step index values of 50, 55, and 60. The simulations are performed by MATLAB R2023a. With the initial condition that the different measured objects were located in different planes, the exact location of each object in the plane could be found by integrating the depth estimation operator, as shown in Figure 4. Figure 4a presents the intercepted pixel ranges of different objects, which are respectively distinguished by color into four different planes: blue, red, gray, and orange. Figure 4b illustrates the normalized reconstruction distance coefficients, which are calculated using the depth estimation operator in the plane range corresponding to Figure 4a. The results are the same as those presented in the simulation data, thus verifying the feasibility of the proposed algorithm. Figure 4c shows the reconstructed images of each object at their optimal reconstruction distances. After fusing the images of the extreme value points, the results shown in Figure 5 were obtained. The color bar values indicate the normalized amplitude. It is evident that each measured object is clearly displayed without blurring. The depth of field under this simulation can reach 2.497 m according to our calculations, thus highlighting the significant effect of multiplane depth-of-field expansion. Figure 4: Intercept plane reconstruction results. (**a**) Different intercept planes. (**b**) Normalized reconstruction distance coefficients. (**c**) The reconstructed image at the maximum point.