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Abstract
Compressive ghost imaging (CGI) can effectively reduce the number of measurements required for ghost imaging reconstruction. In most cases, however, when using illumination patterns as measurement matrices, CGI has not demonstrated the ability to reconstruct high-quality images at an ultra-low sampling rate as perfect as claimed by compressive sensing theory. According to our analysis, the reason is that the non-negative nature of light intensity causes measurement matrix in compressive ghost imaging to be inconsistent with the essential requirements of good measurement matrix in compressive sensing theory, leading to low reconstruction quality. Aiming at this point, we propose a bipolar compressive ghost imaging method to improve the reconstruction quality of ghost imaging. The validity of the proposed method is proven by simulations and experiments.
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1. Introduction
Ghost imaging (GI) is an indirect computational imaging method which enables detecting and imaging as separate parts. Compared with traditional direct imaging techniques, ghost imaging uses a single-pixel detector and takes the advantages of anti-turbulence perturbation and lensless imaging. The idea of GI originated from the photon entanglement of spontaneous parametric down-conversion, and the first GI experiment was completed in 1995 [1]. In 2002, Bennink et al. successfully completed a ghost imaging experiment based on a random scattered light field from a classical light source [2]. Later, the computational ghost imaging schemes was proposed and requirements for GI experiments reduced greatly [3]. In 2010, Ferry et al. [4] proposed differential ghost imaging (DGI), which adds a new differential bucket detector to record a reference light, and performs differential calculation on measurements, which can dramatically enhance the signal-to-noise ratio (SNR) and improve the image reconstruction quality. At present, GI has been one of the frontiers and hot spots in the field of optical imaging and has wide application in the fields of optical computing [5,6], lidar imaging [7], life science [8,9], secure communication [10,11] and so on.
Meanwhile, a variety of imaging algorithms of GI have been proposed to improve imaging quality and reduce imaging time. Among them, non-iterative imaging algorithms include normalized ghost imaging [12], time-correspondence ghost imaging [13,14] and pseudo-inverse ghost imaging [15], etc, which are fast reconstruction methods but suffer from problems such as high number of measurements and low reconstruction quality. In order to dramatically improve the quality and efficiency of ghost images, in 2009, Katz et al. [16] experimentally demonstrated that compressive ghost imaging (CGI) by utilizing compressive sensing (CS) [17,18], an iterative algorithm, could get a better performance with far fewer measurements than conventional GI. After that, the researchers combined normalized ghost imaging, time-correspondence ghost imaging and compressive ghost imaging together to propose the correspondence normalized ghost imaging based on compressive sensing [19] to further improve the reconstruction quality. In 2015, researchers proposed complementary CGI [20,21] to improve CGI quality, which uses the complementary 0/1 binary speckle pattern pairs consisting of the speckle pattern and its inverse speckle pattern to illuminate the object. And then the differential signals and differential speckle pattern are used to reconstruct images. By the way, + 1/-1 binary measurement matrix can be formed by these differential speckle patterns and the +1/-1 binary measurement matrix has been demonstrated to have better performance than the 0/1 binary measurement matrix. However, this also means that one valid measurement requires twice measurements to be completed in the experiment. In 2022, Cheng et al. propose a singular value decomposition compressive ghost imaging (SVDCGI) [22], a post-processing method, in which singular value decomposition is used to transform the measurement matrix into an orthogonal matrix before compressive reconstruction and then the quality of reconstructed images has improved significantly because the orthogonal matrix is also considered to be an excellent measurement matrix in CS theory. From above, we are aware that the unipolarity constraint may limit the imaging quality of CGI and that it is feasible to remove the unipolarity limitation in experiments by some post-processing algorithms such singular value decomposition. But, singular value decomposition is a computation of high complexity and poor robustness, which means SVDCGI, the current optimal ghost imaging method to the best of our knowledge, may be inefficient for practical imaging and motivates us to propose a new post-processing method to improve imaging quality.
In optical systems, unipolarity caused by the non-negativity of light intensity is prevalent and may have some negative effects. Optical Scanning Holography (OSH) [23] is also a single-pixel imaging technology in which the unipolarity constraints result in increased background intensity and decreased contrast of images. To eliminate this effect, OSH introduces a DC-Block in the detection circuit to break the unipolarity constraints. Inspired by this approach, we propose a post-processing algorithm to equivalent to a DC-Block. We call it bipolar compressive ghost imaging (BCGI) in which the measurement signal in the experiment is subtracted by a constant to break unipolarity constraints, and correspondingly, the measurement matrix becomes another equivalent matrix with bipolar. Numerical simulations demonstrate the better robustness and efficiency of our method compared to SVDCGI. And effectiveness of our method is demonstrated in the experiment.
2. Theory of bipolar compressive ghost imaging
A schematic diagram of computational GI is shown in Fig. 1. A group of variational illumination patterns are successively loaded into a computer-controlled spatial light modulator, and the corresponding speckle patterns are projected onto the object beyond a certain distance. Then all the transmissive or reflective photons from the object are collected onto a bucket detector by a lens.
2.1 Compressive ghost imaging
According to the principle of Fourier optics, the intensity recorded by the single-pixel detector can be represented as the sum of the image with the illumination speckle pattern after point-to-point multiplying. Thus, GI can be considered as a linear detection process, whose acquisition process can be described mathematically by matrix multiplication. To simplify the calculation, a grayscale image is usually represented by a one-dimensional matrix $\mathbf{p} = {[{p_1},{p_2}, \cdots ,{p_N}]^T}$, $N$ is the total pixel number, T represents transpose. The variational illumination patterns form the measurement matrix
2.2 Bipolar compressive ghost imaging
From Eq. (2), it is not difficult to know that the measurement sequence ${\mathbf{B}_{M \times 1}}$ is non-negative. Refer to the detection circuit of OSH, DC-Block makes AC electrical signals exhibit bipolarity by removing the DC component and then causes the output signal to fluctuate at zero mean. Similarly, to make ${\mathbf{B}_{M \times 1}}$ being bipolar, the easiest way is to subtract a constant, say, the mean value of LIS. Therefore, when a mean value is subtracted from the LIS, a new measurement sequence ${\tilde{\mathbf{B}}_{M \times 1}}$ with bipolarity is formed and fluctuates at zero mean:
3. Simulation
To test the effectiveness and robustness of our method, three types of noise environments were considered, namely noiseless, Gaussian white noise and constant noise. The results of the noiseless environments can be considered as the optimal solutions to provide a reference for the results of constant noise and Gaussian white noise. In this simulation, constant noise means the noise item $\mathbf{n}$ in Eq. (2) is defined as a constant sequence and we set the constant equaling to the maximum value of LIS. The purpose of considering constant noise is to reveal the effect of a high-power background light source on imaging quality. Gaussian white noise means the noise item $\mathbf{n}$ in Eq. (2) is defined as a Gaussian sequence, and we set detection SNR equaling to 20 dB. The detection SNR of LIS is defined as $10 \times {\log _{10}}({{\raise0.7ex\hbox{${R({WP} )}$} \!\mathord{\left/ {\vphantom {{R({WP} )} \sigma }} \right.}\!\lower0.7ex\hbox{$\sigma $}}} )$, where $R({\cdot} )$ means the range of LIS and $\sigma $ is the standard deviation of the Gaussian sequence. In many GI systems investigated in previous literatures, DMD is the most common spatial light modulator for measurement collection. For DMD, binary projections can reduce systematical error and faster detection speed compared to using a gray value projections [25]. Thus, we just demonstrate these results using binary measurement matrix under the condition that all of the elements are 0 or 1 with an equal probability of 50%. In order to more objectively verify the robustness of the two methods, we introduce structural similarity (SSIM) [26] as the evaluation index. SSIM is a number between 0 and 1. A larger SSIM means the difference between the output image and the original image is smaller, i.e. the image quality is better.
In this numerical simulation, a binary image and another grayscale image with 128 × 128 pixels are selected as the target objects. Sampling rate γ is defined as the number of measurements divided by the number of image pixels. BCGI is compared with CGI and SVDCGI. Figure 2 shows the reconstruction results with different methods at 20% sampling rate. For the binary object, the letters “SCU” is visible in the images reconstructed by all three methods regardless of the noise environment. But there are also some significant differences among these images. Firstly, for noiseless environments, two virtually lossless binary images are reconstructed by SVDCGI and BCGI while the image reconstructed by CGI has severe noise. When Gaussian white noise is added to LIS, the white part of these images reconstructed by SVDCGI and BCGI becomes blurred. Secondly, constant noise not only severely blurs the white part of the image in SVDCGI and CGI but also makes the black part appear salt-and-pepper noise in SVDCGI. However, BCGI reconstructs the virtually lossless image and is not affected by this constant noise. The above results show that compressive sensing does give ghost imaging the ability to reconstruct meaningful objects at under-sampling, but its imaging quality is poor. This phenomenon is more obvious for complex grayscale objects. For the grayscale object, CGI almost fails. But both SVDCGI and BCGI are able to reconstruct detail-rich images for the grayscale object in the noiseless environment, which proves breaking the unipolarity constraint does help to improve the imaging performance for complex objects. It is worth noting that the clarity of BCGI reconstructed images is higher than that of SVDCGI in both noiseless and Gaussian noise environments. Furthermore, for the environment of constant noise, a significant but predictable difference between the results of SVDCGI and BCGI can be found that constant noise severely degrades the reconstruction results of SVDCGI, while BCGI is almost unaffected, because the constant noise is apparently eliminated for the BCGI in Eq. (3). So both effectiveness and robustness of BCGI are better than that of SVDCGI.
Fig. 2. Reconstructing images of CGI, SVDCGI and BCGI for two different objects, (a) binary image; (b) grayscale image.
Due to the fact that different measurement matrices may produce different reconstruction results in CS even if they have the same distribution property, the simulation was executed 20 times for the grayscale object. Figure 3 (a) - (d) show the average SSIM curves of reconstructed images for noiseless, 30 dB and 25 dB and 20 dB of detection SNR, separately. From the SSIM curves in Fig. 3 (a) (noiseless) and Fig. 3 (b) (30 dB), it can be seen that although the reconstruction quality of both BCGI and SVDGI increases with the sampling rate, the quality of BCGI is higher than that of SVDCGI, which indicates that the effectiveness of our method is better than that of SVDCGI. In addition, although the SSIM of BCGI decreases at the same sampling rate when the SNR of LIS falls down to 25 dB, it is not worse than that of SVDCGI, which indicates that the robustness of BCGI is better than that of SVDCGI. When the detection SNR is further reduced to 20 dB, BCGI can be found that the SSIM curve does not increase at the stage of high sampling rates, but it is worth noting that the image quality of BCGI at the stage of low sampling rates is still substantial as shown in Fig. 2 (b). In addition, we compare the effect of different levels of random Gaussian noise on the reconstruction quality at the same sampling rate. The curves in Fig. 3 (e) show the situation that sampling rate is 10%. We can see that images quality reconstructed by BCGI is always much higher than that of SVDCGI when the detection SNR is higher than 20 dB. Furthermore, we also counted the reconstruction time for these three methods. The average time curves in Fig. 3(f) shows that BCGI requires less refactoring time than SVDCGI. When the sampling rate is 20%, the average reconstruction time is 1.23s for CGI, 7.26s for BCGI, and 23.75s for SVDCGI; if the sampling rate is increased to 90%, the average reconstruction time is 4.76s for CGI, 55.40s for BCGI, and up to 1117.16s for SVDCGI. The reason is that the computation requirements of singular value decomposition increase significantly when the matrix dimension rises, while our method only requires simple processing of the data.
Fig. 3. The average SSIM curves of reconstructed images in Gaussian white noise (a)noiseless; (b) detection SNR =30 dB; (c) detection SNR =25 dB; (d) detection SNR = 20 dB; (e) sampling rate = 20%. And (f) the curves of average reconstructed time at different sampling rate.
4. Experiment
The physical device of ghost imaging is shown in Fig. 4. The object is a 14cm × 14 cm picture printed on a A4 paper. A computer-controlled digital projector (EPSON CB-S03) projects the speckle patterns used for reconstruction onto the surface of the object at the same time interval in turn. The light reflected from the object is collected by a lens and eventually received by the detector (TUSCEN-H674ICE). In addition, two same type filters (PHTODE LB-1) are respectively placed in front of the projector and in front of the detector. The filters only pass the visible light from 460 to 650 nanometers to reduce the effects of environment light. A green light-emitting diode (LED) is used to bring a controlled-level background noise to the experiment. The LED (ADVANCED-ILLUMINATION SL162) is centered at wavelength of 530 nanometers. In the experiment, the measurement matrix is a binary matrix, in which 0 and 255 have an equal probability of 50%.
First, we conducted GI experiments with the LED off. The results of CGI, SVDCGI and BCGI are shown in Fig. 5 (a) at different sampling rates. For binary objects, all methods can reconstruct images at sampling rates as low as 5%. But there is one obvious difference between the images reconstructed by CGI and by two other post-processing methods that although the SSIM of images reconstructed by CGI improves with increasing sampling rate, the noise is still noticeable. For BCGI and SVDCGI, two post-processed methods, the experimental and simulation results are consistent that high quality images can be obtained even at very low sampling rates. However, unlike CGI, which fails for grayscale objects, both BCGI and SVDCGI successfully reconstruct the images, which confirms that eliminating the unipolarity constraint is an effective direction to improve imaging quality. That is to say, both from the subjective visual and objective SSIM values BCGI shows a better performance than SVDCGI.
Fig. 5. Results for the GI experiments, γ is sampling rate and SSIM at bottom right corner, (a) LED off; (b) LED on.
Next, a strong LED is used to simulate the strong background light environment by illuminating the object directly. The results are shown in Fig. 5 (b). It can be seen that the disturbance of background light makes CGI and SVDCGI lose the ability to reconstruct object in high quality. The failure in reconstructing grayscale objects means SVDCGI lacks the ability to resist this strong background light. But for BCGI, the visibility degree of the reconstructed image remains high, and the SSIM of the image is slightly lower than that of the image when LED off. In numerical simulations, we predicted that the negative effects of a high-powered but intensity-stabilized background light source would be almost eliminated by BCGI. However, in the experiments, the turned-on LED degrades the quality of the BCGI-reconstructed image, as shown in Fig. 5 for BCGI when LED on, the SSIM values are only around 0.3 instead of 1. We suspect the reason may be that the high-powered background lights suppress the dynamic response capability of the detector, which leads to the values of LIS can not reach the results of numerical simulation, and consequently the discrepancies between expected values and measured values.
In addition, it is worth noting that although the disturbance of background light makes CGI and SVDCGI lose the ability to reconstruct this grayscale object, the quality of SVDCGI reconstructed images still has a great superiority compared to CGI for binary objects. This may imply that the unipolarity constraint may be the culprit that limits the quality of CGI.
5. Discussion
In CS theory, two conditions must be satisfied in order to ensure that the signal can be accurately recovered after a compressive measurement. One is that the original signal must be sparse in some transform domain, and for a natural image, it can always be considered sparse or compressible on an appropriate sparse basis. The other is that the measurement signal should ensure that most of the original signal information is retained while avoiding the introduction of too much extraneous information, which means the measurement matrix needs to be specially designed.
The Restricted Isometry Property (RIP) [27] is used to design the measurement matrix and RIP means that for a measurement matrix F, if there exists a number $\delta \in (0,1)$ to satisfy Eq. (4),
It can be seen that in BCGI, mean value of each column of the new measurement matrix was reduced to 0, but its random distribution properties have not been changed, therefore, it may satisfy the RIP with a higher probability.
In BCGI model, the experimentally detected light intensity sequence is not directly applied to reconstruct the object, but a constant which is the mean value of the light intensity sequence needs to be subtracted from it. By this way, the new signal sequence can be equated to the target object after coding by a new matrix. However, it is worthy to discuss whether the mean value is the optimized one. A numerical simulation is performed in which the measurement matrix consists of a series of reference probing patterns acquired by the experiment and the directly detected signal is used as the target signal, the constant value to be subtracted has a deviation compared to the mean value, and the process is shown as following
Fig. 6. Results for the BCGI by Eq. (7) when measurement matrices are satisfying a uniform random distribution.
Numerical simulations and experiments have demonstrated the effectiveness of BCGI. In our opinion, the physical meaning of BCGI is how to extract information effectively from the non-negative nature of light intensity. The non-negative nature of light intensity is commonly used in many previous GI schemes, which means the measurement matrices in experiment have to exhibits the property known as unipolarity, that is to say, all elements of the measurement matrix are greater than or equal to zero. According to our analysis, the unipolarity hinders the effective extraction of information to some extent, and BCGI can break the unipolarity constraints. In fact, the goal of complementary CGI can also be regarded as to break unipolarity constraints by converting 0/1 binary measurement matrix to +1/-1 binary measurement matrix, and then to achieve better performance. We think this is a topic worthy of further study, and we will try to explain the effect of the unipolarity constraint on CGI using the restricted isometry property in the near future.
6. Conclusion
In conclusion, without complex optical devices, the bipolar compressive ghost imaging we proposed is just a simple but effective post-processing method to improve the imaging performance of ghost imaging for complex objects. After performing an operation that subtracts the column average value for the light intensity sequences and the measurement matrix, a new compressive sensing process is formed. In the new compressive sensing process, the measurement matrix acquires a bipolar character, which makes it easier to meet the RIP and more likely to give a better reconstruction quality.
Funding
National Natural Science Foundation of China (61177009, 61475104); Natural Science Foundation of Sichuan Province (2022NSFSC0565).
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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