Single-pixel correlated imaging (SPCI), which represents significant research within computational imaging, exhibits remarkable imaging capabilities in complex scenarios^[1–3]. This method leverages light field modulation and compressive sensing technology to reconstruct the target image using a single-pixel detector, thereby addressing the limitations of traditional multipixel detectors in specific optical environments^[4–6]. In recent years, SPCI has found extensive application in various domains, including biological imaging, remote sensing detection, and imaging in low signal-to-noise ratio (SNR) environments^[7,8]. However, despite the superior imaging capabilities of SPCI demonstrated in numerous scenarios, it still faces challenges such as high computational complexity, limited imaging efficiency, and reduced imaging quality under low-light conditions^[9,10]. These limitations have significantly impacted the practical implementation of SPCI in high-real-time and high-precision imaging tasks, thereby creating an urgent need to develop more efficient SPCI techniques to improve its overall performance.

In recent years, researchers have carried out in-depth investigations into the core challenges of SPCI and proposed a range of optimization strategies^[11,12]. The quality of SPCI and computational efficiency have consistently been mutually constraining factors. The proposal of optical intensity autocorrelation has realized high-order SPCI^[13]. The results show that the visibility of the imaging increases with the increase of the order, but a longer integration time is required. The integration time required for imaging is determined by the intensity distribution ratio between the two detectors and the complexity of the object being measured. In addition, employing a vortex beam with wavefront shaping can enhance the resolution of SPCI^[14]. However, this approach does not reduce computational time and leads to an increase in system complexity. Owing to the lack of correlation between the random phase and the signal, it is often challenging for system noise to facilitate SPCI in low-light conditions. Therefore, enhancing imaging efficiency by reducing computational complexity and ensuring imaging quality under low-light conditions through algorithmic and system optimization holds great significance.

In this paper, we propose an innovative SPCI method based on symmetrically related random phases. Unlike the conventional SPCI that relies heavily on a large number of independent scattering media, this method employs symmetric correlations with the imaging target to construct an optimized random phase sequence. This approach not only reduces the number of required random phases but also enhances the coherence characteristics of the light field, thereby effectively improving the signal extraction capability. This method not only decreases the volume of computational data but also mitigates system noise interference, thereby enhancing the efficiency and stability of the imaging process. Regardless of the complexity of the image, the use of 2000 scattering media results in an SNR exceeding 20 dB and a structural similarity index measure (SSIM) greater than 0.5 for the reconstructed image. SPCI can be completed within 2 min. The dynamic regulation of multiple random phase loads was accomplished through a spatial light modulator (SLM). Additionally, a dual-path thermal-optical SPCI system and a single-path computational SPCI system were established. High-quality imaging under low-light conditions was successfully achieved. It introduces a novel approach to advanced imaging technology and holds great significance for the implementation and application of imaging in complex environments.

The SPCI technique is a statistical-based method that utilizes the second- or higher-order correlation characteristics of the light field to reconstruct the target image. The single-pixel detector only captures the distribution of target and scattered light field intensities, without acquiring complete phase information. However, in SPCI, the target image is reconstructed by utilizing intensity signals acquired under multiple independent modulation states of the light field.

The incident light field is temporally modulated by utilizing symmetrically related random phases as scattering media, and the signal is subsequently detected through the imaging target using a single-pixel detector for correlation measurement. The symmetry correlation can be leveraged with the imaging target to design an optimized random phase sequence. Information redundancy compression is implemented by reducing the number of random phases. The original process, which demands a large number of independent random phases, can be approximately replaced with fewer yet more effective phases, thereby achieving a reduction in computational complexity. The ith association measurement utilizes a random phase ψi of size L×L to spatially modulate the light source, resulting in structured light that corresponds to the spatial distribution, as depicted in Fig. 1(a). The structured light generated through spatial encoding subsequently illuminates the imaging target, carrying its spatial information. Finally, the intensity of this structured light is detected and recorded by a single-pixel detector lacking spatial resolution capability.

The imaging target (continuous image) can be assumed to be discretized in space, resulting in a digital image τm of size L×L. Due to the use of symmetrically related random phases, each pixel of this matrix corresponds one-to-one with the units of the random phase in space. Therefore, the detected light intensity can be represented as χi=∑aL∑bLψi(a,b)·τm(a,b),(1)where variables a and b represent the row number and column number of the random phase unit (image matrix pixel), respectively. By rearranging the matrices ψi and τm into vectors ψi and τ with a length of N=L×L according to rows or columns, Eq. (1) can be conveniently expressed as an inner product between two vectors: χi=∑jLψi(j)·τ(j)=⟨ψi,τ⟩,(2)where j denotes the index of an element within the vector.

A square matrix of size N×N, denoted as the random phase matrix ψ, is formed by arranging N linearly independent random phase row vectors ψ1,ψ2,…,ψN in the column direction. So, as shown in Fig. 1(b), the entire process of correlation measurement for imaging targets can be represented as χ=|⟨ψ1,τ⟩⟨ψ2,τ⟩⋮⟨ψN,τ⟩|=ψτ,(3)where vector χ, depicted in Fig. 1(c), consists of N measured values arranged as a column. The N random phase vectors must be linearly independent in order for the random phase matrix ψ to have an inverse matrix ψ−1. Based on this requirement, we can reconstruct the target image using the following equation: ψ−1χ=ψ−1ψτ=τ.(4)

Traditional compressive sensing methods typically depend on computationally intensive iterative reconstruction algorithms, whereas the Hadamard-based method necessitates a large number of orthogonal mode sets. In contrast, our proposed method, by exploiting the intrinsic phase symmetry, enables efficient sharing of computational results across different stages of the process, thereby substantially minimizing redundant calculations. Utilizing symmetrically related random phases not only reduces the number of phases required for experiments but also diminishes the computational complexity of matrix operations at the algorithmic level. Consequently, by evaluating the correlation between the imaging target and the multiple intensity distributions obtained under symmetrically related random phase modulation, the structural information of the imaging target can be reconstructed with high fidelity. Given that the random phase demonstrates matrix symmetry correlation with the target image, this approach effectively mitigates redundancy in random phase utilization, minimizes the volume of computational data, and ultimately enhances imaging efficiency.

The quality of imaging in the SPCI system is directly influenced by the number of random phases. The random phase of the scattering media is utilized to induce diverse wavefront disturbances, thereby enhancing the system’s SNR and imaging resolution through multiple modulations. The scattering pattern of the light field is modulated by each random phase, as illustrated in Fig. 2(a), resulting in distinct variations. In conventional SPCI, over 8000 scattering media are typically employed to achieve effective image reconstruction^[15,16]. Our system conducts image reconstruction in the computation and simulation experiment, considering various conditions with different numbers of random phases. The number of pixels for the reconstructed image of the target object is set to 200×200. The imaging results obtained using varying numbers of random phases are presented in Figs. 2(b)–2(d). The imaging target used in Fig. 2(b) is a simple letter “F.” We conducted simulations to observe the reconstruction effects with varying numbers of random phases, specifically N=100, N=500, N=1000, N=2000, and N=5000. The letter “F” can be displayed when N=500, but the image exhibits blurred edges and significant noise. The overall contour lacks clarity. The noise gradually diminishes, and the image edges become sharper with the progressive increase in the number of random phases. When N=1000, the shape of letter “F” becomes significantly more distinct, exhibiting near-complete suppression of noise and a refined edge structure, thereby enabling the faithful reproduction of intricate details. This suggests that a moderate increase in the number of random phases can significantly enhance image quality. However, once the random phase quantity reaches a certain threshold, further increases have limited impact on improving image quality.

The imaging target chosen for simulation calculation is a slightly more intricate “smiley face.” Figure 2(c) illustrates the imaging results under various conditions of random phase quantities. The “smiley face,” in comparison to the letter “F,” encompasses more intricate details such as the eye and mouth structures. When N=500, the details of the image are not clearly visible, and noise becomes more prominent. As the number of random phases increases to N=1000, the overall image structure gradually becomes more distinct, particularly with improved restoration of fine details in the eyes and mouth region of the smiley face. The image quality approaches its optimal state, and further increasing the number of random phases only yields marginal enhancements in the imaging performance. The target image for simulation calculations was a highly intricate “dragon,” which allowed us to evaluate the impact of random phase quantity on complex image imaging. Figure 2(d) illustrates the impact of varying random phase numbers on the reconstruction outcome of the “dragon” image. The “dragon” features an abundance of intricate curves and details, including the intricately designed dragon scales and meticulously crafted body contours. As the number of random phases increases to N=1000, the dragon’s overall outline becomes more distinct, revealing gradual emergence of intricate details such as the curved structure of its head and body. The effective suppression of noise in the image and the accurate reconstruction of fine structures, such as dragon scales and claws, were achieved only when N=2000 and N=5000. The quantity of random phases has a significant impact on the imaging quality, particularly in the initial stages. The improvement in image quality tends to stabilize when the number of random phases reaches a certain level, and further increasing the number of random phases has limited impact. We observed that the imaging time was 0.1 min when 100 random phases were employed, 0.4 min for 500 random phases, 0.7 min for 1000 random phases, 1 min for 2000 random phases, and 1.8 min for 5000 random phases. By leveraging the symmetric relation with the imaging target to design optimized random phases, the imaging time has been significantly reduced.

The curve depicted in Fig. 3(a) illustrates the variation of imaging quality for the letter “F” when different numbers of random phases are employed. The image includes inset figures depicting reconstructed images for N=1 and N=2000. The reconstructed image exhibits a higher level of noise and weaker signal intensity when the number of random phases is limited. The increase in the number of random phases to N=700 leads to a rapid improvement in SNR, resulting in 19.4 dB. The image quality reaches a stable state when N=800. The SSIM gradually improves with an increasing number of random phases. The SSIM of 0.517 at N=800 indicates a high level of similarity between the image structure and the original target. The variation in image quality of a complex “smiley face” under different numbers of random phase modulation is illustrated in Fig. 3(b). The inset images, respectively, depict the reconstructed images at N=1 and N=2000. The SNR of the “smiley face” increases as the number of random phases increases. The SNR and SSIM values are significantly lower when N is 800 or below, indicating the presence of noticeable noise. The SNR reaches 20.01 dB, and the SSIM reaches 0.52 when N attains a value of 1000, indicating an enhanced visibility of the image’s detailed structure. As the number of random phases increases, the enhancement in image quality becomes increasingly marginal. The variation in imaging quality of the most complex “dragon” under different numbers of random phases is illustrated in Fig. 3(c). The inset images illustrate the reconstruction effect at N=1 and N=2000. The SNR and SSIM remain low in the case of N=1000, indicating that the reconstructed image still exhibits a significant level of noise. The increase in the number of random phases to N=2000 leads to significant improvements in both SNR and SSIM, resulting in a substantial reduction in noise levels and enhanced image clarity. The reconstructed image achieves satisfactory quality with the SNR greater than 20 dB and the SSIM greater than 0.5, regardless of its complexity when N reaches 2000. Therefore, the proposed method enables effective control over the number of random phases utilized, thereby enhancing imaging efficiency.

The dual-path thermal-optical SPCI system was constructed, wherein a scattering medium based on 2000 symmetrically related random phases was employed to experimentally validate the imaging quality. The signal light intensity passing through the target object (measured by the single-pixel detector) is correlated with the reference light field without passing through the object [detected by the charge-coupled device (CCD)], thereby reconstructing the image of the object. The technology achieves the separation of detection and imaging, employing a nonlocal imaging method known as off-object imaging. The schematic diagram of the system is illustrated in Fig. 4(a). The modulated optical field is obtained by sequentially loading different random phases using a transmissive phase modulation SLM (pixel number: 1920×1080, pixel size: 8 µm). The laser emitted from the laser device, after passing through the beam expander, is directed onto the SLM loaded with random phases. Subsequently, the light is focused onto the target object via a lens L1. Lens L2, positioned behind the target object, converges the light onto the single-pixel detector (KY-APRM-100M-S-FS-1 mm). By utilizing the beam splitter (BS), the symmetric optical path of the scattered beam can be achieved, allowing the CCD (MV-GEF401GC-T-CL) to detect the spatial distribution of the light field through lens L3. Lens L3 focuses the spatial distribution of the light field onto the photosensitive region of the CCD. To evaluate the anti-interference performance of the proposed method, an experiment was conducted using a laser power set at 15 mW under low-light conditions.

The physical diagram of the system is depicted in Fig. 4(b). The photograph captured by the laser after imaging the target object is presented in Fig. 4(c), illustrating that the imaged object depicts a letter “F” with a hollow center. The traditional SPCI is constrained by a limited number of random phases and weak light conditions, which fundamentally limit its capability for image reconstruction, as shown in Fig. 4(d). The reconstructed imaging in Fig. 4(e) clearly distinguishes the letter “F” after conducting 2000 symmetrically related random phase correlation tests, exhibiting a distinct overall structure and restored details. The system is also capable of achieving high-quality imaging in unpredictable scattering media. The single-instance imaging process is finalized within 2 min. However, the traditional method requires a significantly longer time, exceeding 40 min. The experimental results verified that the proposed method enables the dual-path thermal-optical SPCI system to not only enhance the efficiency of image reconstruction but also improve imaging quality in low-light conditions.

Furthermore, the development of a single-path computational SPCI system has been accomplished. Since the random phase loaded onto the SLM is artificially designed and known to us, we calculate the scattered light field generated by this known random phase. The complex amplitude of the wavefront generated by the SLM with random phase modulation is subjected to a two-dimensional Fourier transform. The resultant values are then multiplied by the propagation factor, thereby obtaining the phase accumulation for each spatial frequency component during propagation. Subsequently, an inverse Fourier transform is applied, yielding the light field distribution on the object plane. The calculated field, serving as a pattern positioned in front of the target object for the incident beam, is directly correlated with the signal detected by the single-pixel detector. This correlation effectively eliminates the necessity of using a CCD for capture. The system only requires a single optical path, where a single-pixel detector receives the signal light passing through the target object. The system’s schematic diagram is depicted in Fig. 5(a). The laser beam, after passing through a beam expander, enters the SLM loaded with random phases for modulation. The resulting beam is subsequently focused onto the target object by lens L1. After interacting with the object, the light is further relayed and focused by lens L2. Finally, the light is detected by a single-pixel detector that lacks spatial resolution. The system’s physical diagram is depicted in Fig. 5(b). The letter “F” with a hollow center serves as the designated imaging target, as illustrated in Fig. 5(c). The conventional SPCI method fails to achieve efficient image reconstruction when utilizing 2000 random phases and a laser power of 15 mW, as depicted in Fig. 5(d). Using the proposed method, after 2000 iterations of random phase correlations, the image was successfully reconstructed, as depicted in Fig. 5(e). In comparison with images obtained using traditional methods, the reconstruction results exhibit a significant enhancement in imaging quality, thereby facilitating the precise differentiation of components within the letter. The imaging process can be completed in under 1.5 min. Since the random phase to be utilized is directly correlated with the detection signal, the CCD detection process is simplified, thereby further reducing the imaging time. This single-path computational SPCI system not only obviates the necessity for a reference light path but also streamlines the computational workload to a certain extent. Given that SPCI is fundamentally based on the projection sampling principle, its current capability is limited to the effective reconstruction of hollow binary target objects. The exploration of more complex and realistic scenarios will constitute a key focus for future development.

In conclusion, a novel SPCI method based on symmetrically related random phase modulation is proposed to address the issues of high computational complexity, limited imaging efficiency, and reduced imaging quality in low-light conditions that are prevalent in traditional SPCI techniques. This method not only optimizes light field regulation by reducing the number of random phases but also enhances signal extraction capability, thereby significantly improving imaging quality. The experimental results demonstrate that when the number of random scattering media reaches 2000, the SNR of the reconstructed image exceeds 20 dB, the SSIM is greater than 0.5, and a single imaging process can be completed within 2 min. Meanwhile, both the dual-path thermal-optical SPCI system and the single-path computational SPCI system have successfully achieved high-quality imaging under low-light conditions, further validating the advancement and feasibility of this method. This research has not only achieved significant breakthroughs in terms of imaging efficiency, computational complexity, and system stability but also opened up new avenues for the application of SPCI technology in complex environments.