# Multi-Object Deep-Field Digital Holographic Imaging Based on Inverse Cross-Correlation
## 2 Principle  
### Digital Holographic Reconstruction  
Digital holography can be divided into two major steps: interference recording and diffraction reconstruction. In the diffraction reconstruction process, a reconstructed image is reproduced in digital form through numerical calculations that are carried out on computers [2], as described in Equation (1). Here, \(U_{0}(x_{0},y_{0})\) represents the field distribution function of an object before diffraction, \(U(x,y)\) represents the light field distribution function of the reconstructed image, and \(z\) is the diffraction reconstruction distance. The symbols \(F\) and \(F^{-1}\) denote the two-dimensional fast Fourier and inverse transforms, respectively. \(H\) is the transfer function in the frequency domain, which is expressed in Equation (2); \(u\) and \(v\) are used to represent the frequency domain values in the vertical and horizontal directions, respectively; \(\lambda\) denotes the wavelength; and \(k\) represents the wave number. \[U(x,y)=F^{-1}\{F\{U_{0}(x_{0},y_{0})\}\cdot H\} \tag{1}\]  
\[H=e^{i\lambda z}\bigg{[}1-\frac{\lambda^{2}}{2}\bigg{(}u^{2}+v^{2}\bigg{)} \bigg{]} \tag{2}\]  
In the case where \(U_{0}(x_{0},y_{0})\) is known, the key factor for reconstructing the light field distribution of an image \(U(x,y)\) is to find the appropriate diffraction distance \(z\). Figure 1 shows the difference in the reconstructed image at different reconstruction distances. The quality of the reconstructed images varies across different reconstruction distances. When the numerical diffraction distance is equal to the measured length from the object to the photodetector, the reconstructed image of the object is the clearest, while the rest of the positions are blurred. Through numerical simulations, the diffraction reconstruction process can be realized on a computer. Depending on the properties of an object and the diffraction reconstruction distance, we can obtain different reconstructed images. As such, in digital holography,  
Figure 1: The influence of different reconstruction distances on the image. a suitable diffraction reconstruction distance is essential for achieving high-quality and clear reconstructed images. In the case of single-plane objects, determining the appropriate diffraction reconstruction distance \(z\) can often be achieved by employing a focusing evaluation function, which is used to explore a suitable range of measurement distances for the experimental subject. However, when dealing with multiplanar samples, the presence of out-of-focus information from different planes results in the smudging of indicators within the evaluation function, which significantly deteriorates the effectiveness of the search process. However, when researching the reconstruction of multiplanar objects, we noticed that the edge diffraction fringes of the reconstructed objects varied significantly. Based on this observation, we can adopt a new method through which to determine the edge information of an object, thereby locking the position of the plane. Specifically, it is possible to search for alterations in diffraction fringes between neighboring reconstruction frames, which helps to accurately capture the position of an object's borders. Once the boundary information of the planes has been ascertained, it becomes feasible to allocate distinct diffraction reconstruction distances to each plane, thus facilitating the achievement of multiplanar digital holographic imaging with a large depth of field. With this approach, instead of relying on the traditional focusing evaluation function to determine the diffraction reconstruction distance, the edge information of an object is used as an anchor point to accurately locate the positions of diverse planes. This method is able to surmount the challenges posed by multiplanar samples during the reconstruction process, thus allowing each plane to be reconstructed at an appropriate distance and resulting in the heightened accuracy and clearer large-depth-of-field digital holographic imaging of multiplanar objects.