Single-photon imaging techniques are widely used in the fields of remote sensing^[1–3] and fluorescence imaging^[4,5]. Commonly used single-photon imaging techniques include scanning imaging^[1,2,6], detector array imaging^[7–9], and single-pixel imaging (SPI)^[10–13]. Scanning imaging methods have the disadvantage of low imaging speed and the requirement for mechanical scanning components. Single-photon detector array imaging currently encounters limitations in achieving large-scale pixel imaging due to technical constraints. Additionally, the inconsistency among image elements hinders uniform performance, which in turn limits overall imaging quality^[14]. SPI is a kind of computational imaging method that utilizes a spatial light modulator to modulate the target optical field and reconstructs the image by recording the observed values with a single-pixel detector, which has the advantages of high sensitivity, high resolution, and high photon utilization by combining compressed sensing^[15–18]. The implementation of single-pixel single-photon imaging primarily involves two key steps: modulation of the target optical field and single-pixel detection^[19–23]. Specifically, the modulation of the target optical field typically utilizes digital micromirror devices (DMDs), which are characterized by a large pixel size and a high pattern projection rate (PR)^[22,24,25]. Single-pixel detectors, also known as bucket detectors, typically utilize avalanche photodiodes (APDs) operating in Geiger mode^[26]. The DMD is located in an image plane of the object after a lens, and onto which is programmed a series of patterns, each one probing a different configuration of image intensities and measured by a single-pixel detector. The image is obtained using a reconstruction algorithm. To overcome the interference of the shot noise of photon detection, long integration time is required. As a result, it is difficult to increase the frame rate of SPI, which makes it unsuitable for imaging an object’s dynamic features^[11]. In our prior research, we introduced the theory of quantum compressed sensing and successfully achieved the extraction of high-frequency dynamic features from discrete random photon sequences^[27–29]. This makes it possible for us to perform SPI of target dynamic features at the photon level^[24–26].

In this work, a photon-level dynamic feature SPI method is proposed, capturing the dynamic features of objects in the frequency domain through discrete random photon detection. A compressed measurement system is constructed by discrete random photon detection to capture the object’s frequency domain feature. The amplitudes of the frequency domain feature peaks are utilized as observed values for single-pixel compressed sensing image reconstruction. Experimentally, SPI is performed using a reordered Hadamard pattern according to the structured characteristic of the Hadamard matrix to accelerate the computational process and to simultaneously reduce the memory consumption of the matrix storage^[30]. By recording the photon arrival times during the integration time of each pattern and obtaining the amplitudes of the frequency domain feature peaks using the discrete Fourier transform (DFT) for image reconstruction, we successfully captured 167 and 200 Hz featured frequencies and achieved high-quality image reconstruction with a data compression ratio (CR) of 20%. Our method provides a new detection dimension and effectively expands the application of photon-level SPI in real-world scenarios.

SPI is performed by modulating the target optical field and using a single-pixel detector to record the photon counts corresponding to each pattern and then reconstructing the image based on the correlation between the patterns and the photon counts, as shown in Figs. 1(a) and 1(e). Assuming that the pixel scale of the object image is n×n, the sampling process can be expressed as y=A·x,(1)where x is the vector representation of the measured image with length L=n×n. A is the observation matrix, i.e., the patterns loaded to the DMD, with dimensions m×n×n, implying that m patterns are included. During the sampling, m patterns are sequentially loaded to the DMD and corresponding observed values are recorded, namely photon counts yi, i={1,2,…,m}, as shown in Fig. 1(f). The image reconstruction is realized by inverting the above measurement process, i.e., multiplying both sides of Eq. (1) by the inverse matrix of the observation matrix A at the same time. Reordered Hadamard patterns with optimized coding sequences enable high-quality sampling of specific images. Patterns that elicit strong signal feedback at the detector are prioritized for sampling, whereas those generating weaker feedback are assigned to subsequent positions. This approach significantly reduces the number of samples required while maintaining high-quality image reconstruction. Here, we employ the cake-cutting ordering method of the Hadamard pattern, which prioritizes sampling regions of the image with gradual gray value transitions.

For objects with dynamic features, the radiated optical field is modulated. This is expressed at the photon level as a modulation of the probability of a photon being detected. Although photon detection is in the form of discrete randomization, the dynamic features of the object are still contained in the photon sequence in the form of probability. What we need to do is figure out how to extract dynamic information from discrete photon sequences. This work focuses on extracting the dynamic features of the object in the frequency domain. As shown in Figs. 1(g) and 1(h), the dynamic features of the object can be obtained by applying DFT on photon arrival time {t1,t2,…,tp} for each pattern^[31–33]: F=∑j=1pexp(2πiftj),(2)where F is the amplitude of the feature peak of the dynamic object in the frequency domain, tj is the arrival time of the detected photon, and f is the dynamic frequency of the object.

Fig. 1(g) shows the photon time sequences measured during each pattern load to the DMD, and the operation of Eq. (2) on the time sequences yields a spectrum of dynamic features of the object, as shown in Fig. 1(h). The observed values used for image reconstruction are the amplitudes of the spectrum peaks. For example, the dynamic frequencies of the two objects in our experiment are 200 and 167 Hz, respectively. The arrival time of each measured photon is recorded, and the observed values F1={fa1,fa2,…,fa4096} and F2={fb1,fb2,…,fb4096} are obtained according to Eq. (2). After that, images were obtained according to the traditional SPI reconstruction algorithm, as shown in Figs. 1(c) and 1(d), corresponding to 200 and 167 Hz feature images, respectively. The results show that the interference of other frequency components can be effectively suppressed when imaging a certain characteristic frequency.

The experimental setup is shown in Fig. 2. The imaging object consists of two parts: a four-blade fan and a two-blade fan. The light source is a continuous laser (LD-633, Lightsource) used to illuminate the object area. The light source is attenuated to the single-photon level using a neutral density filter (GCBZ-254C1, Daheng Optics). It is then split equally into two beams by a beam splitter (BAW10R, Thorlabs) to ensure uniform illumination of both objects. The two-beam chopper (GCI-15, Daheng Optics) carries different dynamic frequencies. The two laser beams pass through polymer-engineered diffusers (ED1-C20, Thorlabs), and the laser spots are converted from a Gaussian distribution to a uniform distribution to ensure uniform illumination of the imaged object area. The optical signal reflected from the object is imaged by an imaging Len1 (AC254-100-A, Thorlabs) onto a DMD (Texas Instruments Discovery V7000, pixel scale 1024×768), which sequentially displays patterns to realize two-dimensional spatial modulation of the image. The modulated images are coupled by Len2 (AC254-030-A, Thorlabs) into a single-photon detector (SPD500, Siminics). The photon arrival time sequences are then recorded by a time-to-digital converter (TDC) (FT1040, Siminics). During data acquisition, the TDC is triggered by the DMD to record trigger pattern timestamps, which are used to distinguish the data corresponding to each pattern.

In the following sections, we first demonstrate image reconstruction based on the Hadamard cake-cutting pattern. After that, the feasibility of our work on dynamic feature imaging is demonstrated by performing the frequency domain dynamic feature imaging on objects with different frequency features.

We selected two fans as imaging objects, as shown in Fig. 3(a). They have four and two blades, respectively. The pixel scale of the reconstructed image is 64×64, and the patterns are loaded by setting each 16×12 image element of the DMD as a single pixel. Since the Hadamard cake-cutting patterns rank the features of the original Hadamard patterns, the top-ranked patterns can be directly selected for image reconstruction. The CR is defined here as CR=m/n, where n=4096 is the number of fully sampled patterns. m is the number of patterns used to recover the image in the subsampling case. The average photon count per pattern was set to 20 kcps (counts per second). We first performed photon counting imaging, where the observed values used for image reconstruction are photon counts during each pattern load on the DMD. The imaging results obtained at different data CRs are shown in Fig. 3(a). We use the peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM) to quantitatively characterize the imaging quality, as shown in Figs. 3(b) and 3(c). When the CR is within the range of 0 to 15%, higher CR values lead to better image quality. This improvement can be attributed to the fact that we sorted the features of the Hadamard patterns and utilized more patterns for sampling, thereby acquiring more image structure information. Specifically, when the CR is 15%, both the PSNR and SSIM attain their maximum values. However, as the CR continues to increase, the feature differences between patterns diminish. Consequently, the small variations in photon counts per measurement fail to compensate for the interference caused by quantum shot noise. As a result, additional pattern sampling not only fails to extract more image information but also introduces noise, ultimately degrading the image quality. From a mathematical inverse solution point of view, the larger the CR, the smaller the image PSNR and SSIM. The reason is that when solving equations under under-determined conditions, the greater the difference of each equation, the easier it is to get a solution. This is one of the differences between photon-level SPI and traditional photodetectors (PDs).

Frequency domain dynamic feature SPI can be achieved by replacing the photon-count sequences in the image reconstruction with the amplitudes of the frequency domain feature peaks of the object. First, a four- and two-blade fan illumination source was loaded with an intensity modulation of 200 and 167 Hz for simulating the dynamic features of the object, respectively. Since the rotational speed of the chopper blades is determined and the inner and outer rings have different numbers of slots, the laser is modulated at different frequencies when it passes through the inner and outer rings of the chopper, respectively. The average number of photons sampled by each pattern was set to 20 kcps. The photon arrival times during the integration time corresponding to each pattern were recorded using a TDC, and the amplitude of the feature peaks was obtained by performing a DFT via Eq. (2). Image reconstruction is performed based on the imaging process shown in Figs. 1(g) and 1(h). The sampling of the object using different patterns exhibits distinct characteristics in the frequency domain. For example, pattern #1 captures all information about the object image area, leading to two prominent peaks in the spectra at 200 and 167 Hz. The 200 Hz peak is higher than the 167 Hz peak because the four blades have a greater surface area compared to the two blades. In contrast, pattern #2 only has two blades, resulting in a single peak at 167 Hz, while the 200 Hz peak is absent. The dynamic feature imaging results in the frequency domain at 200 and 167 Hz were reconstructed using two sets of peak amplitudes, F1={fa1,fa2,…,fa4096} and F2={fb1,fb2,…,fb4096} for all patterns, as shown in Fig. 4. The trend in the image quality change aligns with the previous analysis: the PSNRs for 200 and 167 Hz imaging reach their maximum at CRs of 25% and 20%, respectively, while the SSIMs stabilize. A reduced CR leads to a decrease in the PSNR. Here, the differences in the optimal CR of the two fan blades is due to the different shape profile. The underlying principle of reordered patterns is that the optimized sequence can be sampled with high quality for a particular image. The cake-cutting pattern is suitable for target sampling and low CR reconstruction of regions with slow gray value transformations. The grayscale transformed regions of the two-fan blades have differences, leading to a difference in the optimal CR. The left boundary of the four-blade fan image is cropped when the CR is 10%. This is due to the low CR, which results in insufficient sampling of the contour detail information of the fan blade. This lack of detail in the left boundary of the fan blade can be observed in the original image and becomes even more pronounced after background optimization filters out irrelevant information.

The calculation process of the PSNR and SSIM indicates the difference between the reconstructed image and the original image pixels. So, when the pixel scale of the image is fixed, an increase in the number of pixels occupied by the object being imaged leads to a greater overall difference between the PSNR and SSIM. After image reconstruction, we optimized the images using the same classical image processing methods. The sole distinction is that the object in Fig. 3 occupies more pixels than that in Fig. 4, resulting in higher PSNR and SSIM values for Fig. 3. We conclude that the PSNR and SSIM are relative indices for evaluating image quality and can only reflect image quality when applied to the same imaging object.

In principle, our method can realize object imaging with arbitrary dynamic frequency. However, the implementation is limited by the DMD pattern PR. When the PR is high, not enough dynamic information about the object can be obtained within each pattern integration time, and the dynamic feature image of the object cannot be reconstructed. The effect of the PR on imaging quality was investigated by gradually increasing the PR from 1 to 50 Hz, while setting the average photon count corresponding to each pattern at 20 kcps and maintaining a CR of 50%. The dynamic feature imaging results at 200 Hz with different PRs are shown in Figs. 5(a)–5(c). When the PR was 1 Hz, the PSNR and SSIM were 20.63 dB and 0.953, respectively. As the PR was increased to 20 Hz, the PSNR and SSIM were reduced to 16.05 dB and 0.8549. Furthermore, when the PR is further increased to 50 Hz, there is no effective reconstructed image regardless of the data CR setting. Figure 5(d) shows the decrease of the PSNR and SSIM with PR. Therefore, the selection of the PR must ensure that a sufficient amount of information can be captured within each pattern for image reconstruction.

In this work, a photon-level dynamic feature SPI method is proposed, capturing the dynamic features of objects in the frequency domain through discrete random photon detection. Direct frequency domain SPI is achieved by utilizing the amplitudes of the frequency domain feature peaks as observed values for single-pixel compressed sensing image reconstruction. SPI is performed using a reordered Hadamard pattern according to the structured characteristic of the Hadamard matrix. Experimentally, we successfully captured 167 and 200 Hz featured frequencies and achieved high-quality image reconstruction with a data CR of 20%. The parameters affecting the imaging quality, such as the CR and pattern PR, were quantitatively analyzed. Our method provides a new detection dimension and effectively expands the application of photon-level SPI in real-world scenarios.

Our method is applicable not only to photon-level imaging but also to other imaging experiments under normal light conditions. If it is working in strong light mode, the detector can be a PD or an APD, and the dynamic features of the target can be collected by an oscilloscope or a data acquisition card. We chose the photon level for our experimental demonstration because it is more promising for a wider range of applications. This is because conventional photon-counting requires long integration time to accumulate enough photons to overcome shot noise, resulting in its inability to capture the target’s fast dynamical features. Our work takes the advantage of a high-speed dynamical feature capture method based on sparse photon detection, which realizes target dynamical feature capture at the photon level, thus enabling compressed sensing imaging of target dynamical features. The application scenarios of this method include non-contact dynamic target measurements such as aero-engine rotation imaging or extreme weather drone detection (improving the detection signal-to-noise ratio by measuring the dynamic features of the rotor blades of a drone). Our method can improve the imaging signal-to-noise ratio by capturing the dynamic features of the target to overcome the background noise interference. The highest frequency bandwidth that can be measured is up to 2 GHz.