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Abstract
Computational ghost imaging (CGI) uses preset patterns and single-pixel detection, breaking through the traditional form of point-to-point imaging. In this paper, based on the Monte Carlo model, a reflective polarization based CGI (PCGI) system has been proposed and constructed under the foggy environments. And the imaging performances of the PCGI at different optical distances have been investigated and analyzed quantitatively. When the targets and the background have a small difference in reflectivity, the difference of polarization characteristics between the targets and the background can help the CGI to remove the interference of scattering light and improve the imaging contrast. Besides, in order to further improve imaging efficiency, a scanning-mode polarization based CGI (SPCGI) has also been proposed, in which the combination of polarization characteristics and the scanning-mode plays an important role to improve the CGI’s imaging efficiency and imaging quality.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Ghost imaging (GI) based on the second order correlation of light fields has attracted more and more attentions. The GI systems have two arms commonly, in which one arm captures reference light fields and the other arm receives light intensity using a bucket detector without spatial resolution. Subsequently, the reference light fields can be modulated by a spatial light modulator (SLM) and correlate with intensity received by a bucket detector to reconstruct the object, which is called the computational ghost imaging (CGI). In the beginning, people consider that the correlation property is only derived from entangled photons [1] until it is proved that classic light can also achieve the GI both in experiment [2] and theory [3,4]. Since then, the pseudo-thermal source GI [5–7], and the CGI are raised gradually, in which a pseudo-thermal source is created by lighting a laser on a rotating ground glass when the CGI relies on the SLM [8] or a digital micro-mirror device (DMD) [9] to generate a rapidly-shifting sequence of patterns.
The GI has unique advantages against scattering effects. Han et al. [10] have presented that the characteristics of the second-order correlation of light fields can decrease the effect of multiple scattering on the imaging quality and many researches have shown that the GI can eliminate or weaken the influence of a turbulence system than direct imaging [11–16]. Bina [17] has found the retrieval of absorbing objects immersed in turbid media better than direct imaging for the Rayleigh scattering. Xu et al. [18] have proposed that the GI behaves better than traditional imaging under the same experimental conditions in the Mie scattering media. However, the GI usually can’t perform well in dealing with complex scenes and long distances transmission, which are severe problems that need to be solved [19].
Polarization [20–24] as an intrinsic characteristic of electromagnetic wave has been proven to be a powerful tool to characterize targets [25–28], which can obtain more-dimension information of targets. Thus, polarization has been widely used to improve the ability of detecting and reconstructing the object from complex scenes. Recently, polarization based ghost imaging (PGI) has been proposed to distinguish the objects [29,30] such as the man-made object versus the natural background. Li et al. [31] have proposed a PGI system to suppress backscattering from volumetric media and a deep-learning method for fast reconstructions, in which it requires retraining networks with changing sampling rates, and costs too much time. Thus, it is crucial to improve the performance and the GI efficiency in practical applications.
In this paper, we have proposed a novel method of combining polarization information and scanning sampling together for giving a birth to Scanning-mode Polarization based CGI (SPCGI) to improve the performance and efficiency of the CGI and investigated its performances in foggy environments when the object and background have similar reflectivity. Firstly, based on the Mie scattering theory, we have established a Monte Carlo (MC) model to simulate the behavior of light in foggy environments, in which we have utilized polarization to enhance the imaging quality. However, the imaging time of polarization based CGI (PCGI) is so long that seriously affects the detection efficiency, so we have supplemented the scanning modes to the PCGI to shorten imaging time and improve imaging quality, which can be called the SPCGI. Finally, the performances of SPCGI on different sampling ratios and scanning scales have been investigated and discussed in details.
2. Methods
We have set up a PCGI system in foggy environments system, and the information transmission is simulated by the MC algorithm model, whose schematic is shown in Fig. 1. Firstly, the light with the wavelength of $\lambda = 550nm$ can be generated by the laser. Through a polarizer and modulated by the DMD, a series of spatially modulated linearly polarized beams $S = {({1,1,0,0} )^T}$ can be formed. The object will be illuminated by the modulated beams that are scattered and transmitted in a long-distance dynamic volumetric scattering medium. Secondly, the reflected light from the object will enter into the dynamic volumetric scattering medium again, and then it is splitted into two beams by a polarizing beam splitter (PBS), which will be collected by two bucket detectors respectively. The horizontal-component intensity and vertical-component intensity can be received and labeled as ${I_\parallel }$ and ${I_ \bot }$ accordingly. Thirdly, the intensities of both two bucket detectors (${I_\parallel }$ and ${I_ \bot }$) and the illumination patterns will be correlated by the second-order correlation algorithm to reconstruct the target on the computer.
The polarization state of light can be represented as Stokes vector [32]:
where I represents the total intensity, Q is the difference between horizontal polarization and vertical polarization, U is the difference between 45°and 135°linear polarizations, and V is the difference between left circular polarization and right circular polarizations. When the light interacts with scattering medium or objects, the relationship between the polarization states of the incident light and the output light can be written as:The target scenes are composed of three different materials with different reflectivities, where the background is consisted of wood and stone, whose reflectivities are 0.2 and 0.9 respectively. The target “HFUT” is made up of steel with the reflectivity of 1. The difference of reflectivities between stone and steel is smaller than that between wood and steel, and the Mueller matrix elements [34,35] of three materials are shown in Table 1.
Table 1. Mueller matrix elements for different materials
In our model, firstly, we emit the number of 109 photons to the foggy environments. After many scattering events, the scattering photons will arrive at the target. Then, the reflected photons will interact with the particles and then be analyzed by the PBS at different transmission paths. The concrete scattering process of the photons is shown in Fig. 2, where the light, called backscattered light, does not interact with the target and is reflected back due to scattering with particles, and the light reflected from the target is called transmission light.
The detectors record the intensity splitted by the PBS as ${I_\parallel }$ and ${I_ \bot }$. The target scene can be reconstructed by the second order correlation:
For Eq. (5), when $S = {I_t}$, it is the conventional CGI, and when $S = {I_p}$, it is the PCGI. The Hadamard matrixes have special advantages for the CGI, corresponding to an orthogonal matrix composed only by ${\pm} 1$. Fu [36] has proposed that the -1 in the Hadamard matrix can be replaced with 0 directly, which modulate the DMD only once and can remove the correlation noise as same as the Hadamard matrixes. Walsh-Hadamard matrix [37] is formed by a set of Walsh functions, which performs better than Hadamard matrix. We have replaced -1 of Walsh Hadamard matrix with 0. Thus, the patterns in our simulations are composed only by 1 and 0 in the matrix.
3. Results and discussion
3.1. Traditional polarization ghost imaging
Foggy environment is one of typical complex atmosphere environments, which can be divided into radiation fog and advection fog according to different regions [38]. In this paper, we focus on radiation fog, also called inland fog, which is formed by the condensation of surface gas vapor due to ground radiation cooling, and its particle sizes are usually less than 10 µm [38]. For real foggy conditions, the particle sizes obey some distributions. There have been already some researches focusing on measuring the foggy distribution model that obey lognormal distribution with the mean particle size of 2µm [39,40]. Here, we also use this model, which can be expressed as:
Our works simply focus on homogeneous media systems. According to Table 2, the refractive index of fog@550nm is 1.335 + 1.0×10−9i. The refractive index of air is 1. Based on the Eq. (8), the foggy environment can be generated by six types of foggy particles with sizes of 0.409, 1.491, 2.574, 3.657, 4.740, 5.823µm and the number densities of 2×1.0−12, 9×1.0−11, 5.6×1.0−11, 2.2×1.0−11, 8×1.0−12, 4×1.0−12 N/µm3 respectively, and those particles can be considered as roughly equivalent spherical particles. The scattering coefficient and absorbing coefficient of the medium can be calculated by the Mie scattering theory. According to Beer-Lambert theory, when the light transmits for a distance, the relationship between input light intensity and the output light intensity can be described as:
here $T$ is the transmittance, I and ${I_0}$ are the intensities of scattered light and incident light, respectively, ${\mu _e}$ is the extinction coefficient of the medium equaling to the sum of scattering coefficient and absorption coefficient, and d is the transmission length.Table 2. Complex refractive index of water mist particles at different wavelengths [41]
In order to quantitatively analyze the imaging performances through scattering medium, there is a laser with the intensity ${I_0}$, and then we can receive the center intensity of detector as $I$. The light transmittance (T) in the environment can be calculated from Eq. (10) and we define the length whose transmittance is ${e^{ - 1}}$ as L. Because the scattering environment we have built is highly strong, the L is roughly equal to 600m. As shown in Fig. 3, we compare the results between the CGI and direct imaging (DI) that means detecting light intensity using CCD directly, in which the first row shows the DI results with increasing optical depths, and the second row is those of CGI. Both the DI and CGI go blurred because of the strong background light which will reduce the imaging visibilities and contrasts sharply. From the imaging results, brightness is related to reflectivity of materials. The image of steel is always brighter than marble and wood, because reflectivity of steel is larger than those of marble and wood. As for the DI, the target information carried by light is destroyed by the scattering medium, which result in the imaging worse and worse with the enhancement of scattering effects. Besides, we can observe that the quality of CGI also deteriorates dramatically when the optical length arrives at 3L, because the second order function between scattering light and transmission light is not strictly equal to 0 when the scattering disturbance is strong enough [18].
And then, we take the ${I_\parallel }$, ${I_ \bot }$ and ${I_p}$ for the CGI, respectively. The results are shown in Fig. 4. Compared to the CGI, the horizontal-component (${I_\parallel }$) CGI (HCGI) has better visibility and contrast. The CGI cannot observe the target when the optical length is larger than 3L, but the HCGI can still obtain the clear imaging. When we use the bucket detector after the PBS to receive the horizontal polarization, about half of backscatter light can be filtered out and the reflected light from stone and wood can be removed because of the polarization properties of stone and wood materials. However, due to the polarization-maintaining property of steel, a great number of reflected lights from steel can be received. Thus, we can observe the letter clearly and big contrast between target and background. When the vertical-component (${I_ \bot }$) polarized light is received, about half reflect light from the stone and wood is filter out, and almost all of the ballistic light reflected from target light is filtered out, resulting in target being black. The difference between horizontal and vertical components, that is the Q-component (${I_p}$), can filter out the backscatter light and reflected light from stone and wood. Besides, it can retain the ballistic light reflected from steel which have better correlation. In this case, the polarization component is used for correlation calculation to obtain a high-contrast image, which can get good performance until the optical length up to 5L, because the photons will interact with the scattering medium and experience multiple scattering events with increasing optical length, and the transmitting directions and attenuation of energy will be changed for the scattering photons, resulting in reducing even losing their second-order correlation with the transmitted photons
Fig. 4. Comparison of CGI performances of different polarization components under different optical lengths : Intensity (CGI); Horizontal-component (${I_\parallel }$) (HCGI); Vertical-component (${I_ \bot }$) (VCGI); Q-component (${I_p}$) (PCGI).
3.2. Scanning mode
The imaging time of CGI is determined by the data acquisition time and the imaging reconstruction time. Generally, the data acquisition time is related to the number of samples and the performance of optical modulation devices [42]. Using statistically average correlation algorithm to reconstruct object image requires much more samples than the target's number of pixels, which can reconstruct a high-quality image but cost more times. Scanning imaging technologies have been commonly used in underwater optical systems, using a highly collimate laser beam to scan the target point-by-point and gradually covering the entire scene to obtain the two-dimensional reflectivity distribution image. The pixels of the image in laser scanning sampling are obtained in chronological order, which can avoid interference of reflected light between different pixels. Based on this, the sliding scan sampling method is proposed to reconstruct the target using the compressed sensing algorithm. The scheme not only greatly reduces the time of data acquisition but also improves the imaging quality. The schematic diagram for the scanning model is shown in Fig. 5
The difference between the scanning method and traditional ghost imaging is that the scanning method recovers the target piece by piece until the scanning window scans the entire scene. In the scanning mode, the laser produces visible light with the wavelength of 550nm, modulated to linear polarized light by a polarizer, and the DMD loads a series of patterns to modulate the linear polarized light. Each pattern in the scanning method consists of a Hadamard matrix in the scanning window and 0 in the pixels beyond the scanning window which transmits through scattering medium to irradiate the target. And then the reflected light from the target transmits back and will arrive to the PBS that divides the refractive light into horizontal polarized light and vertical polarized light. The relationship [9] between output and input can be expressed as follow:
The sliding window continues to scan until the sampling block covers the whole scene to reconstruct the whole scene image by the OMP. The algorithm flowchart for this scheme is shown in the Fig. 6
3.3. SPCGI (scanning-mode polarization ghost imaging)
The size of the scene is 64cm×64cm. Under the size of sliding window as 4cm×4cm, the imaging performances of the SPCGI, SHCGI, SVCGI and SCGI at different optical distances have been obtained and shown in Fig. 7(a). When photons experience a few scattering events in the media, the target surface will be covered by “mist”. For the SCGI, as the optical distance increases, the energy of photons reaching to the detector decrease and the number of ballistic photons decreases significantly, making the reconstructed target image invisible. When the optical length reaches 5L, a large amount of stray light makes the whole scene completely indistinguishable. As for the SPCGI, when the optical length is 1L, the target is clearly visible and the image is smooth and noise-free. As the optical distance increases, the target and background remain high contrast, but the image’s signal-to-noise ratio decreases due to the significant reduction in the number of ballistic photons. When the optical length reach to 5L, although the boundary between stone and wood are not obvious throughout the scenes, but the different objects information in the scenes are still distinguishable. The performances of the SHCGI and SVCGI lie between those of the SCGI and SPCGI.
Fig. 7. Comparisons of the SCGI, SHCGI, SVCGI and SPCGI at different optical distances with the different sizes of scanning windows: (a) 4cm×4cm (b) 16cm×16cm.
Under the size of slide window as 16cm×16cm, the imaging performances of the SPGI and SGI at different optical distances are shown in Fig. 7(b). Comparing the results between Fig. 7(a) and Fig. 7(b), we find that the imaging is clearer in the small window. With increasing the size of scanning window, the reconstructed image is smoother, but the contrast of the whole image decreases. The increased size of the scanning window means that the detector captures more photons from other pixels in the scanning window, reducing imaging quality of the piece for the target. However, with the increase of scanning window size, the number of target piece decreases, making the reconstructed image look smoother.
In order to quantitatively describe the effect of sliding window on imaging quality, Peak Signal to Noise Ratio (PSNR) and Structural Similarity Index Measure (SSIM) have been selected as evaluation indicators. The higher value of PSNR or SSIM, the higher imaging quality. The formula of which can be written as follows:
Fig. 8. The PSNR and SSIM of the SCGI, SHCGI, SVCGI, SPCGI under different optical distances, (a) PSNR of 4cm×4cm, (b) SSIM of 4cm×4cm, (c) PSNR of 16cm×16cm, (c) SSIM of 16cm×16cm.
As demonstrated in Fig. 8, The PSNR and SSIM of the images show a downward trend with increasing optical distance whatever the size of the scanning window is, and when the optical distance is greater than 3L, the SSIM drops sharply. Compared with the SCGI, the PSNR and SSIM of the SPCGI, are always higher, so the SPCGI results are always better than those of intensity CGI because polarization can filter out background light. Comparing Fig. 8(a) with (c), we find that the SCGI quality deteriorates when the scanning window becomes larger. It is because when the window goes large, the total intensity value received in the bucket detector will contain more stray light from other pixels, which are taken to each pixel when the image is reconstructing, resulting in the decreased PSNR. Moreover, the SPCGI’s PSNR in Fig. 8 (c) is greater than that in Fig. 8 (a). Although the SPCGI filters out a large amount of background noise in both Fig. 8 (a) and Fig. 8 (c), but the larger the scanning window, the fewer splitting blocks the whole image, and the difference between the image blocks has small impacts on the overall image. The SSIM values in Fig. 8 (b) and Fig. 8 (d) also indicate that the SPCGI is more similar to the original image than that of 4cm×4cm.
In order to further reduce the acquisition time of imaging data and investigate the relationship between the sampling rates and the imaging quality, the sampling rates have been set as 25%, 50% and 100% under different scanning windows, and the imaging results can be seen in Fig. 9 and Fig. 10. When the scanning window is set as 4cm×4cm and the sampling rate is set 50% or less, there are a lot of mosaic effects in the imaging results for both SCGI and SPCGI. The Hadamard matrix is consisted of mosaic-like block [44], and the Walsh-Hadamard pattern obtains low-frequency information of the scene in the case of under-sampling [45], which makes the contours of the target clearly visible while the details around the edge are distorted. As for 16cm×16cm scanning window shown in Fig. 10, due to the incomplete information of the under-sampled reconstructed image blocks, the difference between adjacent blocks becomes larger, which makes the block effect more obvious, but the block effect gradually disappears with increasing the sampling rate.
Fig. 9. Comparison of SCGI and SPCGI reconstruction results at different sampling rates under different optical distances with the size of sliding window as 4cm×4cm.
Fig. 10. Comparison of SCGI and SPCGI reconstruction results at different sampling rates under different optical distances with the size of sliding window as 16cm×16cm.
In the case of multiple scattering events, it is found that the imaging quality of SCGI decreases as the sample rate increases from Fig. 9 and Fig. 10. In other words, the increased sampling rate does not make the signal-to-noise ratio of the image increase, but make the noise accumulate. However, the SPGI remove stray light, with increasing sampling rate, the SPCGI quality continuously improves. Additionally, it also shows that the number of ballistic photons arriving at the detector is small and most photons are scattered after many collisions, and scattering light is strong enough for the SCGI with increasing optical length, so that the target information is submerged in the background, resulting in the target invisible.
In order to compare the imaging efficiency of traditional PCGI and SPCGI, we select the simulation platform Matlab (R2018a) on Intel Core i7-5820K CPU@3.3GHz, RAM 64 GB to work and record imaging time, in which the imaging process are completely consistent and the difference is the imaging reconstruction algorithm. The SPCGI methods are based on scanning mode, while the PCGI methods have no scanning mode. To make the data more intuitive, the imaging time with the sampling rate as 100% at 3L and the scanning window size as 4cm×4cm has been taken as the standard. The ratios of imaging time to the standard time have been recorded in the Table 3, from which we can see the SPCGI efficiencies is better than those of the PCGI, no matter how to change the sampling ratio.
Table 3. Comparison of PCGI and SPCGI imaging efficiencies
4. Conclusions
In summary, a CGI simulation system under the dynamic volumetric scattering system (real foggy environment) has been set up. In view of the problem that the CGI is seriously degraded in the case of the difference of reflectivity between target and background is tiny, the PCGI has been proposed to enhance imaging contrast and imaging quality. In addition, an SPCGI scheme has also been proposed for improving the imaging efficiency. The results show that the imaging performance of SPCGI can improve the CGI’s function and efficiency dramatically. There are two reasons, on the one hand, the advantage of the OMP algorithm greatly reduces the sampling ratio and shortens the sampling time, and on the other hand, with the help of block scanning, each sub-image of the image is constructed independently, resulting in the reduced interference in the imaging reconstructions’ process. We believe that the proposed SPCGI will pave the way to efficient and high-performance CGI.
Funding
National Natural Science Foundation of China (61775050).
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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