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Abstract
With single-photon sensitivity and picosecond resolution, single-photon imaging technology is an ideal solution for extreme conditions and ultra-long distance imaging. However, the current single-photon imaging technology has the problem of slow imaging speed and poor quality caused by the quantum shot noise and the fluctuation of background noise. In this work, an efficient single-photon compressed sensing imaging scheme is proposed, in which a new mask is designed by the Principal Component Analysis algorithm and the Bit-plane Decomposition algorithm. By considering the effects of quantum shot noise, dark count on imaging, the number of masks is optimized to ensure high-quality single-photon compressed sensing imaging with different average photon counts. The imaging speed and quality are greatly improved compared with the commonly used Hadamard scheme. In the experiment, a 64 × 64 pixels’ image is obtained with only 50 masks, the sampling compression rate reaches 1.22%, and the sampling speed increases by 81 times. The simulation and experimental results demonstrated that the proposed scheme will effectively promote the application of single-photon imaging in practical scenarios.
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1. Introduction
Single-photon imaging (SPI) technology can realize single-photon extremely low-light imaging, which has broad application prospects in bioluminescence imaging [1–3], astronomical observation, military reconnaissance, special operations missions and other fields [4–8]. Traditional SPI is mainly divided into point-by-point scanning imaging and direct imaging with a single-photon detector array. Point-by-point scanning imaging scans the measured target pixel by pixel through an optical scanning element and restores the image by recording the photon count of each pixel. Its imaging speed is slow, which makes it difficult to meet the requirements for imaging dynamic targets [9]. The single-photon detector array imaging is a potential new technology. However, it is technically difficult and costly to implement [10,11], and not applicable at the present.
Compressed sensing (CS) can reconstruct the sparse original signal of the transform domain by sub-Nyquist sampling, which provides a new idea for efficient SPI [12–14]. Single-photon compressed sensing imaging (SP-CSI) modulates the target light field by loading a series of different masks on the spatial light modulator (SLM), recording the photon count corresponding to each mask by one single-photon detector, and recovering the measured target image by the correlation between the mask and the photon counting. The imaging pixel is determined by the pixel size of the SLM, the imaging speed and the quality mainly depend on the number and structure of the masks. Therefore, restoring high-quality images with fewer masks by designing a reasonable mask structure is one of the most important research topics in SP-CSI.
The random speckle mask has the characteristics of simple structure and easy implementation, which is one of the earliest masks used [15,16]. However, it needs a large number of masks to recover the image, which is inefficient, and it is sensitive to signal fluctuation noise, which is not suitable for SP-CSI. The orthogonal transform masks are widely used in SP-CSI, which effectively improved the imaging speed, for instance, Hadamard transform masks [17], Fourier transform masks [18], and cosine transform masks [19]. Among them, the Hadamard transform mask is usually used since its binary mask structure is easy to implement and robust to noise. Moreover, the number of masks required for imaging could be reduced by reordering the Hadamard masks while ensuring excellent imaging quality, such as Russian Dolls ordering [20], Cake Cutting ordering [21](CC), Wavelet Decomposition Coefficient ordering [22], Total Variable Score ordering [23], and Dataset Learning ordering [24]. Further, dictionary learning mask design schemes based on Singular Value Decomposition or Principal Component Analysis (PCA) construct masks by extracting the main features of the measured target [25–27] to recover high-quality images with a smaller number of masks. However, the existing SP-CSI does not fully consider quantum shot noise and dark count, the mask design of SP-CSI has not been optimized.
In this paper, an efficient SP-CSI scheme based on the PCA algorithm and Bit-plane Decomposition (BD) [28] is proposed, which improved the imaging speed and quality significantly by constructing Best Choice (BC) masks. By using the PCA algorithm to extract the main features of the target image as gray masks, the correlation between the mask and the target image is enhanced. Then the grayscale mask is decomposed into eight binary masks by the BD algorithm, and the binary mask that best represented the grayscale mask is selected as the BC mask, which simplified the experimental operation and improved the imaging quality and speed. The scheme was optimized by studying the relationship among quantum shot noise, the number of masks and image quality. Experimental and simulation results show that the imaging quality of SP-CSI does not always improve with the increase of the number of masks because of the effect of quantum shot noise, there is an optimal number of masks for different photon counts to ensure high-quality imaging. Therefore, the imaging quality can be improved by rationally choosing the number of masks. Finally, the images are further optimized by the classical image optimization algorithm.
2. Theory and methods
2.1 Theoretical model of single-photon compressed sensing imaging
The theoretical model of SP-CSI is shown in Fig. 1, the light source illuminates the measured target, and the reflected light signal is projected to the SLM, such as a Digital Micro-mirror device (DMD). The SLM carries out two-dimensional spatial modulation of the light field of the measured target by loading the mask. The modulated light is focused on the single-photon detector, and recovers the target image using the correlation between the mask and the photon counts. The basic principle can be expressed by the following equation:
where X is the measured target; A is a sensing matrix of size M × N2, consisting of M mask matrices of size N × N; B is the photon count; and e is the noise photon count. The target image can be restored by multiplying both sides of Eq. (1) with the inverse matrix of A at the same time. Unlike recovering the image through an iterative algorithm, only inverse transformation of a matrix is needed here to restore the image which greatly shortens the time required for image recovery. But not all sensing matrices have an inverse matrix. When the sensing matrix does not have an inverse matrix, the pseudo-inverse matrix of the sensing matrix can replace the inverse matrix.
Fig. 1. Single-photon compressed sensing imaging schematic. DMD: Micro-mirror device; SPD: Single-photon detector.
2.2 Efficient single-photon compressed sensing imaging
In this work, an efficient SP-CSI scheme based on the PCA and BD algorithms is proposed, which significantly improved the imaging speed and quality at visually acceptable pixel sizes. Firstly, an image dictionary is constructed by the same kinds of images. Assume that M images of size N × N are collected, and each image is then converted into a one-dimensional vector of size N2. These one-dimensional vectors are combined to form an image dictionary C of size M × N2. The PCA algorithm is used to remove the redundant image features in C while retaining the main image features as masks. For which, a new image dictionary D is obtained by centralizing C, the covariance matrix DDT of D is calculated, and the eigenvalue decomposition of DDT is done. Then, the eigenvectors are arranged in descending order according to the eigenvalues and the eigenvectors are taken as the eigenmatrix mask. Since only a small number of the eigenmatrix mask contains the main features of the target, the number of masks for SP-CSI will be greatly reduced, thus improving the imaging speed. The eigenmatrix masks are usually grayscale masks, whereas binary masks have simpler experimental operations, faster imaging speed, and a broader range of applicability than grayscale masks. If the grayscale masks can be characterized by binary masks, the whole imaging scheme will be further optimized. The pixel values of grayscale masks are usually decimal numbers from 0-255, so 8-bit binary numbers can be used to represent each pixel value of the masks. Therefore, the BD algorithm is used to decompose a gray mask into eight binary masks (eight bit-plane masks) by combining the pixels in the same bit plane. Among the eight binary subgraphs, the eighth subgraph (the eighth bit-plane subgraph) is the most representative of the original grayscale mask because it contains the most low-frequency information. So, the eighth subgraph is selected as the BC mask, which makes it easy to operate and also improves the imaging speed while ensuring the imaging quality. Meanwhile, the classical image processing algorithm is adopted in this work, which further improves the imaging quality. Image contrast is enhanced by setting a reasonable threshold for pixel truncation, and part of the image noise is removed by median filtering.
The existing SP-CSI experiments do not fully consider the effect of quantum noise fluctuation on the performance of the imaging system. In this work, we studied the effect of photon fluctuations caused by quantum shot noise on imaging quality in an extremely weak light environment. At the same time, the dark count is considered in the simulation and experiment. The dark count is mainly caused by the instrument itself, which has little impact on the image quality of SP-CSI. The dark count per unit of time is evenly distributed on each mask, and the influence of dark count on imaging quality will be further reduced. Therefore, this paper focuses on analyzing the effect of quantum shot noise on imaging quality.
3. Results and discussion
3.1 Simulation results
To evaluate the effectiveness of the proposed scheme, simulation experiments are performed. This paper takes MNIST handwritten Arabic numerals as the measured target and compares the imaging results of the BC scheme with the widely used CC scheme. The CC scheme is the Cake Cutting ordering scheme mentioned in the introduction.
To enhance the imaging resolution, the 28 × 28 pixels scale images in the MNIST handwritten dataset are transformed to 64 × 64 pixels scale, so the number of CC masks is 4096 for the full sampling. Since non-integer magnification images will have some distortion, the Bicubic interpolation method provided by the image class in Python can be used in the process of magnifying the image to obtain a magnification effect closer to high-resolution images. At the same time, considering the simplicity of the MNIST handwritten dataset, the magnified image basically retains all the features of the original image. The BC scheme and CC scheme mainly use low-frequency information to preserve the main features of the object, so some distortion will not affect the mask construction and image recovery. The 4096 handwritten Arabic numerals were selected to construct the image dictionary, and a computer experiment shows that 600 eigenmatrix masks contain 99% of the features of the image dictionary. Therefore, the number of BC masks optimized based on the eigenmatrix mask was set as 600. The computer experiment explains why the number of BC masks is 600. A series of eigenmatrices and eigenvalues can be obtained by using the PCA algorithm for the image dictionary/dataset. The eigenvalue represents the contribution of the eigenmatrix to the features contained in the image dictionary. After the eigenmatrices are arranged in ascending order according to the eigenvalues, the eigenmatrices with greater contribution are used preferentially. The contribution rate of each eigenmatrix is calculated and accumulated. The cumulative result shows that the first 600 eigenmatrices already contain 99% features of the image dictionary, so there is no need to use more eigenmatrices as masks. Because the eigenmatrix mask is the grayscale mask, the BD algorithm is used to decompose the 600 eigenmatrix masks into 8 × 600 binary masks. Each eigenmatrix mask corresponds to eight binary masks. Select one of the eight binary masks obtained from the decomposition of each eigenmatrix mask as the BC mask. Among the eight binary masks, the eighth binary mask containing the most low-frequency information is selected as the BC mask. Therefore, the BC mask is finally composed of 600 binary masks. To facilitate the evaluation of image quality, the pixel values of imaging results were mapped to the range of 0-255. To better simulate the practical situation, quantum shot noise and dark count were introduced into the simulation process. The qualitative comparison of image quality is done from a visual point, and the quantitative comparison is done by using two image quality evaluation criteria: peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) [29].
There are three steps for simulation experiment. Step I: Construct the BC mask based on the MNIST handwritten dataset. The selected 4096 handwritten Arabic numeral images in the MNIST handwritten dataset to constitute an image dictionary. The PCA algorithm was used to obtain the eigenmatrix mask of the image dictionary, and the BD algorithm was used to obtain the BC mask of the eigenmatrix mask, as shown in Fig. 2.
Step II: Comparison of the imaging quality using the BC scheme and the CC scheme. Taking the handwritten digit “3” as the target. The imaging quality of the BC scheme and the CC scheme were compared qualitatively and quantitatively, as shown in Fig. 3. It shows that with the increase in the number of used masks, the imaging quality of the BC scheme and CC scheme gradually increases. When the number of masks is small, the BC scheme outperforms the CC scheme. When the number of masks is large, the CC scheme outperforms the BC scheme, but there is no significant difference in the imaging quality between the two schemes from the visual point. The BC scheme inevitably discards the high-frequency information and some low-frequency information of the object while pursuing the ultimate mask compression rate. For example, use the BD algorithm to decompose a grayscale image into eight binary subgraphs, and select one of them as a BC mask. When obtaining a BC mask, seven subgraphs will be discarded. Because the BC scheme restores the object image by retaining the lower-frequency information of the object, the quality of the image will not be greatly improved as the number of BC masks increases. However, the CC scheme retains low-frequency information but does not discard high-frequency information. Therefore, with the increase of the number of masks, the CC scheme will gradually introduce some high-frequency information to improve the quality of the image. By optimizing the imaging results with the classical image optimization algorithm, that is, the pixel truncation and median filtering, the BC scheme can recover higher-quality images with fewer masks, and the BC scheme performs as well as the CC scheme when more masks are used, as shown in Fig. 4. Therefore, on the premise of ensuring imaging quality, the BC scheme has a faster imaging speed.
Fig. 3. Imaging results using the BC scheme and CC scheme. (a) The target to be measured; (b) Imaging results using different numbers of masks. The number at the top of the figure represents the number of used masks, and the notation on the left represents different schemes; (c) PSNR of BC and CC schemes; (d) SSIM of BC and CC schemes.
Step III: Study the relationship among quantum shot noise, the number of masks and image quality. To further study their relationship, by changing the mean photon count measured per mask, the handwritten digit “3” was taken as the target, and the BC scheme was used to conduct the simulation experiment. The simulation results are shown in Fig. 5. The results show that when the mean photon count rate is large, the effect of quantum shot noise on the imaging quality is limited. The image quality first increases and then gradually becomes stable with the increase of the number of masks used in image restoration. When the mean photon count rate is low, the effect of quantum shot noise on the imaging quality is gradually obvious. The image quality first increases and then decreases with the increase of the number of masks used in image restoration.
Fig. 5. Simulation results of the PSNR and SSIM. (a) PSNR with different photon count; (b) SSIM with different photon count.
The study found that with the increase of the number of masks, additional noise will be introduced, leading to the deterioration of imaging quality. Because the effect of quantum shot noise is different for different masks. The masks which contain high-frequency information will not contribute to the improvement of imaging quality but introduce additional noise caused by the presence of quantum shot noise. Therefore, in the SP-CSI, when the mean photon count rate is low, the mask containing low-frequency information should be preferentially used. Here, the number of masks corresponding to the inflection point of image quality is the optimal number of masks. The imaging speed and quality can be further improved by selecting the optimal number of masks.
3.2 Experimental results
An SP-CSI experimental system was built to verify the above simulation experiments. The illumination light source is a red laser diode (Letesos LD-650-10 mw). A DMD (Texas Instruments Discovery V7000) with pixel scale 1024 × 768 is used for generating masks. The DMD consists of many tiny mirrors, one for each pixel. The mirror is rotated at different angles to represent the 0/1 state, so a binary mask can be easily generated by DMD. The BC mask and CC mask used in the experiment are both binary masks. In the experiment, when using all the pixels on DMD, in order to reconstruct the 64 × 64 pixels image, every 16 × 12 array of the DMD is operated as a single pixel. In terms of photon detection, a single-photon detector (Siminics SPD500) is used to detect reflected light modulated through masks. In terms of photon record, a time-correlated single-photon counter (Siminics FT100) was used to record the photon count corresponding to each mask. The imaging target used in this paper is a handwritten digital model printed by a laser printer on a black acrylic plate.
To verify the Step II results of the simulation, the visual qualitative and numerical quantitative comparison of the imaging quality using the BC scheme and the CC scheme with the handwritten digit “3” in Fig. 3(a) as the target is shown in Fig. 6.
Fig. 6. Comparison of the BC scheme and the CC scheme for different numbers of masks. (a) Take the Handwriting digit “3” as an example, and make a qualitative comparison from the perspective of visuals. CC1 and BC1 are the normalized results, CC2 and BC2 are the images of optimized by the classical image optimization algorithm; (b) PSNR comparison of CC1 and BC1 imaging results; (c) SSIM comparison of CC1 and BC1 imaging results.
The experimental results of Fig. 6 are consistent with the simulation results of Fig. 3. The imaging quality of the BC scheme is better than that of the CC scheme, with PSNR and SSIM maximum enhancement of about 2 dB and 0.15, respectively. The BC mask and CC mask are binary masks, so the number of masks used in the experiment is their real number. When the number of BC masks is 50, the image could be recovered effectively with the BC scheme. The initial imaging PSNR is 13.7 dB, the SSIM is 0.27, and the optimized imaging PSNR is 15 dB, SSIM is 0.73. Compared with the full sampling that requires 4096 masks to recover the image of 64 × 64 pixels, the compression rate of the BC scheme reaches 1.22% and the sampling speed is increased by 81 times. The imaging speed is limited by the mean photon count of the single-photon detector and the modulation speed of the DMD. If the modulation speed of the DMD is set to 1 kHz, and the mean photon count rate is set as 1 kcpm (counts per mask), that is, the mean photon count rate is 1 Mcps (counts per second), the sampling time of 50 BC masks in the BC scheme is only 0.05s.
To verify the Step III simulation results, the handwritten digit “3” is taken as the target, investigated the relationship among quantum shot noise, the number of masks and image quality. The experimental results in Fig. 7 are consistent with the simulation results in Fig. 5.
Fig. 7. Experimental results of the PSNR and SSIM. (a) PSNR with different photon count; (b) SSIM with different photon count.
The influence of quantum shot noise on the imaging quality increases with the decrease of photon count. When the photon count rate is low, the imaging quality first improves and then gets worse with the number of masks increasing. For instance, if we set the mean photon count to 1 kcpm, the PSNR and the SSIM improve by 5 dB and 0.13, respectively, using the optimal number of masks compared to the full sampling. Therefore, using the optimal number of masks in extremely weak light will further improve the imaging speed and quality. However, due to the diversity of mask structure, variability of light intensity and uncertainty of noise, it is difficult to accurately calculate the optimal number of masks, but the optimal number of masks is a common phenomenon in SP-CSI.
To illustrate the universality of our scheme, the handwritten digits “2”, “4”, and “5” were used as the targets, and 50 BC masks were used for the image recovery experiment. The experimental results are shown in Fig. 8. As can be seen from Fig. 8, handwritten digits “2”, “4”, and “5” can all recover the original image well with 50 BC masks after image optimization, which proves the universality of the BC scheme for restoring handwritten digital images. It should be noted that the proposed BC scheme in this paper constructs targeted masks for specific objects, and such masks cannot work effectively on other objects. For a class of objects, one needs to build a dataset for that class of object. For example, the masks used in this paper are designed for the MNIST handwritten dataset, so the mask only works when the target is Arabic numerals.
Fig. 8. Imaging results using 50 BC masks: (a) Original images of handwritten Arabic numerals “2”, “4”, and “5”; (b) Restored optimized images for handwritten Arabic numerals “2”, “4”, and “5”.
4. Conclusion
In this work, an efficient SP-CSI scheme is proposed based on the PCA and BD algorithms. By designing a reasonable mask structure, selecting the optimal number of masks and the most suitable optimization algorithm, the imaging speed and imaging quality are significantly improved. The main features of the image dictionary are extracted and used as grayscale masks by the PCA algorithm, which enhances the correlation between masks and images while compressing the number of masks. The grayscale masks were decomposed by the BD algorithm, and the binary mask that best represented the grayscale mask was selected as the BC mask, which simplified the experimental operation and improved the imaging quality and speed. The scheme was optimized by studying the relationship among quantum shot noise, the number of masks and image quality at the level of single-photon. Experimental and simulation results show that the imaging quality of SP-CSI does not always improve with the increase of the number of masks because of the effect of quantum shot noise, there is an optimal number of masks for different photon counts to ensure high-quality imaging. Finally, the images are further optimized by the classical image optimization algorithm. In the experiment, the mask compression rate achieves 1.22% when PSNR is 15 dB and SSIM is 0.73. The proposed imaging scheme provides a new research method and idea for SP-CSI in extremely weak light, which advances the application of SP-CSI in practical scenarios.
Funding
Applied Basic Research Project of Shanxi Province, China (201901D211342); National Natural Science Foundation of China (62105193).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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