The single-pixel imaging (SPI) system requires only a single-pixel detector to measure the total light intensity and is not sensitive to phase, which makes it suitable for imaging objects in complex environments. If an object is obstructed by an opaque obstacle, and the obstacle is sparsely distributed, the obstacle’s signal and the object’s signal can be separated over time using the principle of light time of flight, which allows for image reconstruction of the object. For large-area occlusions, the current effective method combines the self-reconstruction characteristics of Bessel beams to achieve the SPI of the object. However, existing studies have only demonstrated the feasibility of this approach without deeply analyzing the underlying factors such as the shape and position of obstacles and the conditions for complete imaging. We tackle this issue by exploring the influence of different types of obstacles on Bessel SPI, as well as the conditions for effective imaging, thus providing a reference for applying Bessel beams in SPI.

The Bessel beam is generated by projecting an annular slit onto the DMD and combining it with a Fourier lens. By shifting the annular slit in a specific sequence, we can scan the Bessel beam across the object’s surface accordingly. Bessel-SPI leverages this scanning Bessel beam as the illumination mode in SPI, combined with a compressed sensing algorithm. In this study, we analyze how the shape and position of obstacles affect the Bessel beam, the beam’s self-reconstruction after occlusion, and the resulting SPI imaging. We also define the conditions needed for complete imaging based on the Bessel beam’s non-diffraction and self-reconstruction distances. When these conditions are met, Bessel-SPI can produce a full image of the object. Comparing the transmission and SPI imaging results of the Bessel beam and Hadamard mode under identical occlusion conditions highlights the advantages of Bessel-SPI for imaging occluded objects.

First, the field of view changes of the object in different positions are compared in the absence of obstruction, as shown in Fig. 4, along with the image quality under different sampling times. This demonstrates that Bessel-SPI can achieve image quality comparable to Hadamard-SPI at the same sampling rate, as shown in Fig. 5, thus verifying the feasibility of Bessel-SPI. Secondly, obstacles are classified into central occlusion type and peripheral occlusion type. For central occlusion type obstacles, simulations are conducted for the Bessel intensity distribution on the object surface and the corresponding Bessel-SPI results. It is proved that when the distance between the object and the obstacle satisfies z₂>fa/d (where f represents focal length of lens,a represents size of obstacle in x direction,and d represents ring diameter), and the distance between the lens and the object satisfies z₁+z₂<2Rf/d (where R is the radius of the lens), the central spot of the Bessel beam can self-reconstruct after passing through the obstacle. Thus, Bessel-SPI can image the object completely. For peripheral occlusion type obstacles, the non-diffraction distance of the Bessel beam will be shortened. The object can be fully imaged only when the distance between the object and the lens is less than or equal to z₁+z₂<z₁+fa/d, where a represents the size of the central transmissive region of the obstacle in the x direction. Finally, comparison of the imaging results from Bessel-SPI and Hadamard-SPI under the same occlusion conditions shows that the experimental results align with the theoretical and simulation results.

Based on the self-healing characteristics of Bessel beams, a scheme is proposed using scanning Bessel light as the illumination mode for SPI experiments. The self-healing characteristics of Bessel beams and the imaging results of Bessel-SPI are analyzed under different shapes and positions of obstacles. The simulation results show that Bessel-SPI can achieve image quality similar to Hadamard-SPI when there are no obstacles. In the presence of obstructions, the commonly used Hadamard-SPI lacks obstruction resistance. However, when combined with the compressed sensing algorithm, Bessel-SPI can perform SPI for obstructed objects at a low sampling rate and exhibits greater obstruction resistance. The simulation results also demonstrate that for central occlusion type obstacles, when z₂>Z_min and z₁+z₂<Zmax, Bessel-SPI can capture the complete structure of the object, where Z_min=fa/d. For peripheral occlusion type obstacles, the non-diffraction distance of the Bessel beam becomes Zmax'=z₁+fa/d. Bessel-SPI can only obtain the complete structure of the object when the object’s position satisfies z<Zmax'. In this paper, we analyze the applicability of SPI for the special case of opaque obstacle occlusion and improve the imaging research of SPI under such conditions. This work can be extended to dynamic and 3D objects, enabling SPI for occluded moving objects and occluded 3D objects. In addition, leveraging the inherent characteristics of SPI, Bessel-SPI can maintain imaging capabilities in more complex environments, such as line-of-sight imaging under partial occlusion.