# Multi-Object Deep-Field Digital Holographic Imaging Based on Inverse Cross-Correlation
## 1 Introduction  
Digital holography can simultaneously record the amplitude and phase information fields of an object in a photodetector under non-contact conditions [1, 2]. The image of the object can then be numerically reconstructed through interference recording and diffraction reconstruction [3]. Digital holography can offer high sensitivity in detecting subtle morphological changes and deformations due to its ability to capture small phase variations. This sensitivity makes it well suited for applications requiring precise measurements. Furthermore, digital holography is highly suitable for recording multi-dimensional information, thereby enabling the measurement of dynamic behaviors under specific conditions [4, 5]. The advantages of digital holography also extend to its potential for quantitative analysis through the computer-based processing of amplitude and phase information. By integrating image processing and analysis techniques, additional valuable information can be extracted from data. As a result, digital holography has found extensive use in diverse domains, including industrial manufacturing, biology, medicine, and materials science. Its applications encompass three-dimensional topography measurement [6, 7], particle field characterization [8], deformation analysis [9], stress distribution mapping, and other areas [10]. Many of its key issues are also commonly discussed [11]. In digital holography, diffraction reconstruction is the key aspect allowing one to realize the reconstruction of three-dimensional scenes. In this process, the selection of the appropriate diffraction distance has a critical influence on ensuring the quality of the three-dimensional reconstruction. In the case of a single-plane image of the object's digital reconstruction, the assumption of a well-defined focal plane often holds true. Consequently,a range of actual measurement distances can be explored, and image quality can be evaluated at various diffraction reconstruction distances to identify the optimal one [12; 13; 14]. However, the situation becomes complicated when the measured object is a multiplanar object; this is because the image quality is affected by multiple planes, and it is also at which point the single-focal plane assumption no longer applies. Therefore, optimizing the selection of the diffraction distance is a crucial technique for realizing multiplanar digital holographic imaging with a large depth of field. In order to accomplish the holographic reconstruction of multiplanar objects, different structures and algorithms have recently been proposed. [PERSON] et al. proposed a spectral multiplexing method under low-coherent-light sources [15]. This method uses the principle that the different angles of the reference light are separated in the spectrum; as such, objects at different locations are recorded in the interference fringes of different directions in order to image multiple depths in a single acquisition. However, this method has a complex and bulky optical path structure, and it requires a precise mechanical displacement stage to adjust the axial distance between different reference optical paths. In addition, in order to achieve spectral separation, the holographic optical path must be set up as an off-axis optical path, thereby losing more spatial bandwidth; meanwhile, at the same time, the relative angle of each reflector has a strict mutual exclusion requirement. [PERSON] et al. proposed an autofocusing algorithm [16] that is based on the area criterion applied to the monitoring of microscopic plankton. By dividing the hologram into regions and focusing them sequentially, the best reconstruction distance for the object can be identified as the distance where the reconstructed object occupies the smallest area, which is determined according to the principle of minimizing the dispersion spot when the object is focused. A limitation of this method is that there are many a priori conditions for determining the existence of objects in a region, and the algorithm is not very widely applicable.