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Abstract
We present a technique called single-pixel imaging camera based on complementary detection and optimized encoded modulation (CSPI camera), which can significantly reduce the influence of the disturbance light to single-pixel imaging (SPI). The experiments demonstrates that when the probability of the value “1” for each binary encoded pattern is P=0.5, CSPI camera is still disturbance-free even if the intensity fluctuation of the disturbance light is much larger than the signal’s intensity. The reconstruction results of both traditional SPI and differential SPI are also compared. This technique of CSPI camera can dramatically promote real application of single-pixel imaging Lidar.
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1. Introduction
Single-pixel imaging (SPI), as a computational imaging technique, has attracted a lot of attention recently [1–10]. At present, there are two typical schematics for SPI [9,10]. One is ghost imaging (GI), where the target is illuminated by a series of speckle patterns and the photons reflected from the target are collected onto a single-pixel detector [1–6]. The other is single-pixel camera (SPC), where the target is usually imaged onto a spatial modulation device and the modulated signals are received by a single-pixel detector [7,8]. In comparison with SPC, more investigations were focused on GI because of its potential applications in remote sensing [11–13], three-dimensional imaging [14–16], phase imaging [17–20], optical encryption [21–24], and microwave imaging [25]. Recently, some works have demonstrated that SPC has more advantages than GI in the Lidar application [9,10,26–28]. Firstly, the detection range of GI Lidar is limited by the damage threshold of the modulation device, whereas there is no need for SPC to be considered because the reflection signal is usually weak [26]. Secondly, the quality of SPC is better than GI in the same light disturbance environment [27]. Thirdly, the detection process of SPC usually satisfies the standard modeling of compressive sensing, whereas the case of GI is deviated because only the low-frequency information reflected from the target is detected by the single-pixel detector in long-distance imaging [10,11,15,28]. In addition, the structure’s size of SPC is usually smaller than that of GI. Therefore, the schematic of SPC may be more suitable than that of GI in the area of Lidar remote sensing.
On the other hand, digital micro-mirror device (DMD), as a high-speed light modulator, is widely used for SPC system [8–10]. By controlling the micro-mirrors of DMD, we can obtain a series of random binary encoded patterns with different statistical distribution and the property of encoded patterns has a great effect on the quality of SPC [10,29,30]. For example, Hadamard encoded patterns, as a special case of Bernoulli modulation, can usually obtain better reconstruction results compared with other random Bernoulli encoded patterns [29,30]. What’s more, because of the modulation property of DMD, the quality of SPC can be also enhanced by complementary detection [31]. However, different from traditional imaging, lots of measurements are required for the image reconstruction of SPI. Therefore, the intensity fluctuation of the background/disturbance light will lead to the degeneration of SPI in the sampling process, especially for the application of SPI Lidar where the light disturbance through the atmosphere is inevitable [27,32,33]. In order to reduce the influence of the disturbance light to SPI Lidar, there are two approaches up to now. One is to increase the transmitting energy of Lidar and to decay the intensity of the disturbance light by spectral filtering. However, the enhancement degree is usually confined in practical application. The other is based on differential measurement, but it is valid only for the case of disturbance light with periodic and slight intensity fluctuation [34]. Furthermore, the intensity fluctuation of the disturbance light is usually time-variation and random, which is impossible to be predicted in practice. Therefore, it is imperative to develop a method of SPC against the background/disturbance light where its intensity fluctuation is rapidly variable and even is much larger than the signal’s intensity. In this paper, we propose a single-pixel imaging camera based on complementary detection and optimized encoded modulation (called CSPI camera) that is disturbance-free when the disturbance light illuminating on the DMD plane of CSPI camera is spatially uniform and its intensity fluctuation is random in time domain. Combining with previous SPI research achievement, this work can dramatically quicken real application of SPI Lidar.
2. Model and image reconstruction
Figure 1 presents the proof-of-principle schematic of CSPI camera in light disturbance environment. The light emitted from a laser uniformly illuminates the target and the target is imaged onto a DMD by an optical imaging system with the focal length $f_1$. At the same time, the disturbance light directly enters into the imaging system and illuminates the same DMD. By controlling the mirrors of the DMD, both the disturbance light and the target’s image are modulated, and then the photons reflected by the DMD are collected onto two single-pixel detectors $D_{\textrm{up}}$ and $D_{\textrm{down}}$ by using another two conventional imaging system with the focal length $f_3$, respectively. According to the inherent property of DMD, the patterns at the plane of the detectors $D_{\textrm{up}}$ and $D_{\textrm{down}}$ are complementary. In addition, the intensity of the laser illuminating the target is assumed to be stable, and the disturbance light on the DMD plane is spatially uniform but its intensity is randomly changed for each measurement. The intensity $Y_{\textrm{up}}^i$ recorded by the detector $D_{\textrm{up}}$ can be expressed as [35]
Fig. 1. Proof-of-principle schematic of CSPI camera in light disturbance environment. The disturbance light directly enters into the imaging system and uniformly illuminates the DMD, but its intensity is time-variation.
Because the detection process is complementary, the intensity $Y_{\textrm{down}}^i$ recorded by the detector $D_{\textrm{down}}$ can be described as
According to the principle of SPI, the target’s image $O_{\textrm{SPI}}$ can be reconstructed by computing the correlation function between the pattern’s intensity distributions $A^i(x)$ modulated by the DMD and the intensities $Y_s^i$ recorded by the detector [1,4,10,27]
Based on the idea of complementary detection [31], the image of CSPI is the summation of the reconstruction results of $O^{\textrm{up}}_{\textrm{SPI}}(x)$ and $O^{\textrm{down}}_{\textrm{SPI}}(x)$. By some deviations, $O_{\textrm{CSPI}}(x)$ can be expressed as the correlation function between the pattern’s intensity distributions $A^i(x)$ and $Y_{\textrm{CSPI}}^i$, namely
From Eq. (5), we find that if the condition of $\int {\left ( {2A^i (x)-1} \right )} dx=0, {\textrm{\ \ }} \forall _i = 1 \cdots K$ is satisfied, then the disturbance light will have no influence to the quality of CSPI. Considering the binary modulation property of DMD, when the speckle patterns $A^i (x)$ obey Bernoulli distribution (supposed that the probability of the value “1” is $P$ for each speckle pattern, then the probability of the value “0” is 1-$P$) and $P$=0.5, the second item of Eq. (5) will be 0 and Eq. (5) can be simplified as
Eq. (6) means that CSPI is disturbance-free in the case of Bernoulli encoded patterns with $P$=0.5.
In Ref. [34], the optical background noise can be restrained by instant ghost imaging to some extent. To compare with the result of CSPI, we introduce this method into the reconstruction of SPC, which is called differential single-pixel imaging (DSPI) in this manuscript, the reconstruction process can be described as
In order to evaluate quantitatively the quality of images reconstructed by the methods described above, the reconstruction fidelity is estimated by calculating the peak signal-to-noise ratio (PSNR):
3. Experimental demonstration
To verify the concept, based on the proof-of-principle schematic CSPI camera in light disturbance environment shown in Fig. 1, we have proposed the experimental demonstration setup, which is displayed in Fig. 2. The experimental parameters are set as follows: the center wavelength of the LED is 532 nm, $z_{11}$=$z_{12}$=$z_{21}$=$z_{22}$=500 mm, $f_1$=$f_2$=250 mm, $z_{31}$=$z_{32}$=200 mm, and $f_3$=100 mm. The transverse size of the patterns at the DMD plane is set as 109.2 $\mu$m and the modulated area of the DMD is 64$\times$64 pixels (one pixel is equal to the pattern’s transverse size). In order to improve the sampling efficiency, the speckle patterns are arranged as Hadamard patterns (where the position of the value “-1” is set as 0) and the first pattern (namely the value of all the element is 1) is removed for the experimental demonstration of Fig. 3-Fig. 5. Therefore, the measurement number used for this special case is $K$=4095. What’s more, in order to analyze the influence of the light disturbance to SPI, the light emitted from a group of Xeaon lamps is collimated by the lens $L_5$ and then illuminates the DMD, which satisfies the condition that the disturbance light at the DMD plane is spatially uniform but its intensity is different for each measurement (see Fig. 3(b)). The PIN-photodiode in Fig. 2 is a single-pixel detector (Thorlabs PDA100A2, gain 50 dB). Figure 3(b) displays the time-variation property of the disturbance light. The imaging target, as illustrated in Fig. 3(a), is a group of letters (“SUDA”, 64$\times$64 pixels). The ideal detection signal of imaging the object “SUDA”, namely $Y_{\textrm{s}}^i=\int { A^i (x) T(x)} dx$, is shown in Fig. 3(a). According to Eq. (1) and Eq. (2), the detection SNR $\delta = \frac {{\left \langle { I_0\int {A^i (x) T(x)dx}} \right \rangle }}{{std(I_{n-up}^i )}}$ denotes the signal power to the standard deviation of the noise power ratio, and the irradiation SNR $\varepsilon = \frac {{I_0}}{{std(I_b^i)}}$ denotes the signal power to the standard deviation of the light disturbance power ratio. In the experiments, the detection SNR $\delta$ is set as 21.5 dB. Figure 3 has given the detection signal and the reconstruction results of SPI/DSPI/CSPI when the irradiation SNR $\varepsilon$ is 6.72 dB. If there is no detection noise and no light disturbance in the detection process, both the ideal detection signal and the target’s reconstruction image are shown in Fig. 3(a), which is also corresponding to the target’s original image. Figure 3(c) and Fig. 3(d) present the signal recorded by the detector $D_{\textrm{up}}$ and the signal obtained by DSPI method, respectively. It is clearly seen that the target’s signal (Fig. 3(a)) is absolutely overwhelmed by the disturbance light, and both traditional SPI and DSPI reconstruction methods can not rebuild the target’s image (Fig. 3(c) and Fig. 3(d)). The detection signal and reconstructed image based on the method of CSPI are displayed in Fig. 3(e). By computing the correlation coefficient $\beta$ between the signal $Y_{\textrm{s}}$ and the signals $Y_{\textrm{up}}$/$Y_{\textrm{DSPI}}$/$Y_{\textrm{CSPI}}$ [26], the value $\beta$ of DSPI is only 0.02 whereas it approaches to 1 for CSPI, which means that the effect of the disturbance light is wiped off by CSPI method and the target’s image can be stably reconstructed. Furthermore, the reconstruction results of SPI/DSPI/CSPI in different irradiation SNR $\varepsilon$ are shown in Fig. 4. It is obviously observed that SPI is better than DSPI in large $\varepsilon$ and the quality of CSPI is still perfect even if the irradiation SNR $\varepsilon$ is lower than -3.5 dB.
Fig. 2. Experimental demonstration setup of CSPI in light disturbance environment. BS: beam splitter; M: reflection mirror; PIN$_1$ and PIN$_2$: PIN-photodiode; L$_1$-L$_5$: lens.
Fig. 3. The influence of the disturbance light to the detection signals and different SPI methods, when the irradiation SNR $\varepsilon$ is 6.72 dB. (a) The target’s detection signal and reconstruction result without detection noise and light disturbance; (b) the time-variation intensity distribution of the disturbance light, the green area is corresponding to the amplified diagram labeled by the pink square area; (c) the signal recorded by the detector $D_{\textrm{up}}$ and corresponding reconstruction result; (d) the detection signal based on the method of DSPI and its reconstruction result; (e) the detection signal based on complementary detection and CSPI reconstruction result.
Fig. 4. Experimental results of different SPI reconstruction methods in different $\varepsilon$. (a) $\varepsilon$=-3.52 dB; (b) $\varepsilon$=6.72 dB; (c) $\varepsilon$=14.52 dB; (d) $\varepsilon$=24.75 dB; (e) the curve of PSNR-$\varepsilon$.
Fig. 5. Experimental demonstration of imaging a real scenario (“cock”) in different irradiation SNR $\varepsilon$ by CSPI reconstruction, using the same experimental parameters as Fig. 2. (a) the target; (b) $\varepsilon$=1.25 dB; (c) $\varepsilon$=6.72 dB; (d) $\varepsilon$=14.52 dB; (e) $\varepsilon$=24.75 dB.
In order to validate the applicability of CSPI for complex scenes, Fig. 5 gives the experimental demonstration of imaging a real scenario (“model of a cock”). Using the same experimental parameters as Fig. 2, the reconstruction results of CSPI are shown in Fig. 5(b)-Fig. 5(e) when the irradiation SNR $\varepsilon$ is 1.25 dB, 6.72 dB, 14.52 dB and 24.75 dB, respectively, which is similar to the results described in Fig. 4 and the scenario can be always rebuilt stably by CSPI.
The patterns $A^i (x)$ used in Fig. 3-Fig. 5 are corresponding to Hadamard codes, which is a special case of Bernoulli distribution. To analyze the influence of the value $P$ for Bernoulii modulation to CSPI, Fig. 6 has given the experimental results of CSPI in different $P$ when the patterns $A^i (x)$ are chosen as a random binary distribution. The measurement number used for image reconstruction is $K$=20000. We find that the quality of CSPI is the best in the case of $P$=0.5 and it will be degraded as the increase of the deviation from the value $P$=0.5, which accords with the theoretical analysis described by Eq. (5). In addition, compared with the results of CSPI described in Fig. 3 and Fig. 5, the reconstruction results in Fig. 6(d) is worse even if the measurement number $K$ is much larger than the case of Hadamard patterns, which originates from high efficiency in information acquisition for the Bernoulli speckle patterns with Hadamard codes.
Fig. 6. Experimental results of CSPI when the patterns $A^i (x)$ is a random Bernoulli distribution and the probability $P$ of the value “1” is different. (a) $P$=0.3; (b) $P$=0.4; (c) $P$=0.45; (d) $P$=0.5.
As described by Eq. (5), when the condition of $\int {\left ( {2A^i (x)-1} \right )} dx=0, \,\, \forall _i = 1 \cdots K$ is satisfied, CSPI will be disturbance-free. To further demonstrate that this method is also valid for other illumination patterns, Fig. 7 have displayed CSPI reconstruction results for the illumination patterns with uniform distribution or Gaussian distribution when the condition described above is satisfied. The measurement number used for image reconstruction is also $K$=20000. The irradiation SNR $\varepsilon$ is -5 dB and the detection SNR $\delta$ is 20 dB. It is clearly seen that CSPI is also disturbance-free for grayscale illumination patterns. Therefore, we demonstrate that CSPI is disturbance-free and may be applied to SPI Lidar in the environment of strong background/disturbance light. What’s more, the disturbance light in the present work is on the DMD plane, which mainly focuses on the case of stray light in receiving imaging system. When the disturbance light is on the target plane, it will be equal to the case that the intensity fluctuation of illumination source is time-variant for SPI system, CSPI method is also effective when the detection signal of CSPI is corrected as $Y_{\textrm{CSPI}}^i= \frac {Y_{\textrm{up}}^i-Y_{\textrm{down}}^i}{{Y_{\textrm{up}}^i+Y_{\textrm{down}}^i }}$. The detail description of the approach will be reported in the next work.
Fig. 7. Simulation results of CSPI when the illumination patterns $A^i (x)$ satisfy uniform distribution or Gaussian distribution. (a) the result of $\rm {SPI}_{\textrm{up}}$ for uniform illumination patterns; (b) the result of DSPI for uniform illumination patterns; (c) CSPI reconstruction result for uniform illumination patterns; (d) CSPI reconstruction result for Gaussian illumination patterns.
4. Conclusion
In conclusion, we have proposed a technique called CSPI that can remove the influence of the disturbance light to sing-pixel imaging. We also show that the technique is always valid even if the intensity of the disturbance light is much stronger than the signal’s intensity and its intensity fluctuation is random and rapidly variable. This technique can quicken real application of SPI Lidar and is useful to the imaging in the wavebands without cameras.
Funding
Startup Funding of Soochow University (NH15901123); Chinese Aeronautical Establishment (2020Z073012001); Natural Science Research of Jiangsu Higher Education Institutions of China (21KJA140001).
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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