Ghost imaging through scattering medium by utilizing scattered light

Li-Xing Lin; Jie Cao; Jie Cao; Jie Cao; Dong Zhou; Huan Cui; Qun Hao; Qun Hao

doi:10.1364/OE.453403

1. Introduction

Ghost imaging (GI) is a nonlocal imaging technique that calculates the intensity fluctuation correlation of the object beam and reference beam. It was first performed by using entangled photon pairs in 1995 [1,2]. Alejandra Valencia et al. first demonstrated two-photon ghost imaging with a pseudothermal source [3] and F. Ferri et al. first considered the ghost diffraction and the resolution issues [4]. Lensless two-photon imaging with chaotic light was first demonstrated [5]. And Lorenzo Basano et al. presented an experiment in lensless ghost imaging (LGI) with thermal light, which confirmed the purely classical explanation of GI with thermal light [6]. There are two paths in the LGI system. One is the test path, in which the object exists. A bucket (single-pixel) detector is used to detect the total intensity of the transmitted or reflected light from the object. The other is the reference path, in which a spatial-resolving detector, such as a CCD camera, is used to detect the intensity distribution of illumination speckles. In applications over long distances, the influence of the scattering medium must be considered. Here, the scattering of light is usually classified into Rayleigh scattering and Mie scattering according to the size of the scattering particles and the wavelength [7]. Mie scattering is more common. The influence of the location of the scattering medium in GI has been widely discussed in Refs. [8–11]. Nicholas D. Hardy and Jeffrey H. Shapiro have explained the influence of turbulence on GI clearly [12]. It has been proved that compared to conventional imaging, GI is immune to the scattering medium between the object and the bucket detector. However, the image quality decreases when the scattering medium exists between the beam splitter and the object.

Generally, two kinds of solutions have been proposed: improve the illumination speckles [13,14] or post-process the bucket values [15,16]. Fu et al. [15] implemented reflective ghost imaging through specific kinds of scattering media based on binary GI [17] in which 1-bit quantization was used on the data of the bucket detector. Mark Bashkansky et al. [16] demonstrated the use of a high-pass filter on the detected bucket signals to suppress the effects of temporal variations of fog density and enable an effective reconstruction of the image. Nie et al. [14] proposed a computational ghost imaging (CGI) scheme using customized pink noise speckle pattern illumination, which was robust to several types of noise. Although part of the illumination light will be scattered when propagating in the scattering medium, after illuminating the object, the scattered light still contains information about the object. That is, after this light is detected by the bucket detector, the bucket values also contain the information. However, these methods ignored the object information contained in the scattered light. Therefore, in this paper, we prove that the scattered light can help GI to reconstruct images through a scattering medium. Considering the nonlocal nature of GI, we propose a method that places a scattering medium with certain characteristics in the reference path, which utilizes scattered light to extract the object information contained in scattered light. Theoretical analysis and experiment results show that compressive ghost imaging (CSGI) reconstruction algorithms [18,19] can be used to utilize scattered light for imaging through the scattering medium that exists in the test path when our method is used. And theoretical analysis shows that our method under the second-order correlation reconstruction algorithm [20,21] holds only if the number of measurements is large (close to infinity). Otherwise, our method will cause noise under this reconstruction algorithm. Simulation results show this situation. (See Supplement 1 for the simulation results). It is impractical to meet this condition. However, compressed sensing can be utilized to reconstruct objects with fewer measurements than dictated by Nyquist’s sample theorem [18], so we experimentally show the imaging results of CSGI. All experiments were carried out in a lensless ghost imaging system (LGI), which is simpler and easier to implement [6,12]. And whether our method is valid for entangled photons GI needs further discussion. The contrast-to-noise ratio (CNR) and visibility are used to quantify the image quality of the reconstruction results. Experimental results prove that our method is able to improve the imaging quality when there is a scattering medium between the beam splitter and the object.

2. Theoretical analysis

A schematic diagram of LGI is shown in Fig. 1. A CW laser goes through rotating ground glass (RGG), which produces pseudothermal light. Then, it is divided into two parts by a 50:50 beam splitter (BS). One is the reference speckle, the other is the test speckle. The reference speckle propagates freely through a reference path, and the intensity distribution of every speckle at a distance Z₁ from the BS is detected by a spatial-resolving detector, such as a CCD camera. The detection results are expressed as $I_r^i(x,y)$. The test speckle propagates through a test path and illuminates the binary object at a distance Z₂ from the BS, and then the light transmitted through the object is detected by the bucket detector which is used to collect the total intensity. The bucket values are expressed as Bⁱ. Z₁ = Z₂ needs to be satisfied in the LGI system.

Fig. 1. Schematic diagram of LGI. Z₁ = Z₂; Object is closed to bucket detector. S.C.: second-order correlation algorithm (see text for details).

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2.1 Theoretical analysis of the proposed method under second-order correlation reconstruction algorithm

The second-order correlation reconstruction algorithm is expressed as [20,21]

(1)$$\begin{array}{c} {T_{GI}}(x,y) = \frac{1}{N}\;\sum\limits_{i = 1}^N {\;\left( {I_r^i(x,y) - \left\langle {I_r^i(x,y)} \right\rangle } \right)\;\left( {{B^i} - \left\langle {{B^i}} \right\rangle } \right)} \\ = \left\langle {I_r^i(x,y){B^i}} \right\rangle - \left\langle {I_r^i(x,y)} \right\rangle \left\langle {{B^i}} \right\rangle , \end{array}$$

Here, <·> denotes an ensemble average over N measurements.

Take the theoretical derivation of a one-dimensional object as an example. The bucket values can be expressed as [20,21]

(2)$$B = \int {{I_{obj}}} (x )T(x )dx,$$

where I_obj (x) denotes the intensity distribution of the speckle illuminating the object plane and T(x) denotes the transmission function of the object.

In TGI, the reconstruction result of the object, O(x), is retrieved by calculating the second-order correlation

(3)$$O(x )\;\; = \;\;\left\langle {{I_{ref}}(x )B} \right\rangle - \left\langle {{I_{ref}}(x )} \right\rangle \left\langle B \right\rangle ,$$

where I_ref (x) represents the intensity distribution detected by CCD_R.

When a scattering medium exists between the BS and the object, the test speckle propagating to the object plane can be classified into transmission light that is not disturbed by the scattering medium and scattered light that is disturbed. The effects of scattering can be written as

(4)$${I_{obj}} \to \alpha {I_{obj}}(x )+ I_{obj}^s(x ),$$

where α (0 < α < 1) denotes the percentage of the transmission light, which weights the transmission ratio of the scattering medium. αI_obj(x) and $I_{obj}^s(x)$ denote the intensity distribution of the transmission light and the scattered light of the test speckle, respectively.

Then, the bucket value can be obtained by substituting Eq. (4) into Eq. (2):

(5)$$\begin{aligned} B &\to \int {[{\alpha {I_{obj}}(x) + I_{obj}^s(x)} ]} \;T(x )dx\\ &= \alpha B + {B^s}, \end{aligned}$$

where B^s denotes the bucket value from the scattered light, which is written as

(6)$${B^s} = \int {I_{obj}^s(x)T(x)dx} ,$$

Therefore, the reconstructed imaging result is written as

(7)$$\begin{aligned} {O^s}(x )\;\; &= \;\;\left\langle {{I_{ref}}(x )({\alpha B + {B^s}} )} \right\rangle - \left\langle {{I_{ref}}(x )} \right\rangle \left\langle {({\alpha B + {B^s}} )} \right\rangle \\ &= \alpha \cdot O(x )+ \left\langle {{I_{ref}}(x ){B^s}} \right\rangle - \left\langle {{I_{ref}}(x )} \right\rangle \left\langle {{B^s}} \right\rangle \\ &= \alpha \cdot O(x )+ \int {\left\langle {{I_{ref}}(x )I_{obj}^s(x )} \right\rangle } T(x )dx - \int {\left\langle {{I_{ref}}(x )} \right\rangle \left\langle {I_{obj}^s(x )} \right\rangle T(x )} \;dx, \end{aligned}$$

Mie scattering destroys the coherence of the incident light [22], i.e., the second-order correlation between the scattered light and the transmission light is lost, that is

(8)$$\left\langle {{I_{ref}}(x )I_{obj}^s(x )} \right\rangle = \left\langle {{I_{ref}}(x )} \right\rangle \left\langle {I_{obj}^s(x )} \right\rangle ,$$

Substituting Eq. (8) into Eq. (7), we obtain

(9)$${O^s}(x )\;\; = \;\;\alpha \cdot O(x ).$$

As the scattering becomes stronger, α decreases, and $I_{obj}^s(x)$ increases. In addition, $\left\langle {{I_{ref}}(x )} {I_{obj}^s(x )} \right\rangle$ is not strictly equivalent to $\left\langle {{I_{ref}}(x )}\right\rangle \left\langle{I_{obj}^s(x )} \right\rangle$ in real experiments, i.e., $\left\langle {{I_{ref}}(x ){B^s}} \right\rangle$ is not strictly equivalent to $\left\langle {I_{ref}}(x ) \right\rangle \left\langle{B^s} \right\rangle$. This distorts the reconstructed image in this case [9].

Equation (6) indicates that the bucket value B^s contains the information of the object. However, the information cannot be extracted with Eq. (7). We propose placing a scattering medium with certain characteristics in the reference path to utilize the scattered light to extract the information and reconstruct the image. We find a kind of scattering medium with scattering characteristics similar to those of the scattering medium existing between the BS and the object and place it in the reference path. And if the light intensity distribution of the light scattered by the two scattering media converges, it can be said that the scattering characteristics of the two scattering media are similar. In this case, the effects of scattering can be expressed by the same function as Eq. (4), i.e., the reference speckle detected by CCD_R can also be classified into transmission light that is not disturbed by the scattering medium and scattered light that is disturbed.

(10)$${I_{ref}} \to \beta {I_{ref}}(x )+ I_{ref}^s(x ),$$

where β (0< β <1) denotes the percentage of the transmission light of the reference speckles, which weights the transmission ratio of the placed scattering medium. $\beta {I_{ref}}(x )$ and $I_{ref}^s(x)$ denote the intensity distribution of the transmission light and the scattered light of the reference speckle, respectively. $I_{ref}^s(x)$ and $I_{obj}^s(x)$ maintain the second-order correlation when a similar scattering medium is put in the reference path.

Substituting Eq. (10) into Eq. (7), we obtain the reconstruction result

(11)$$\begin{aligned} {O^{ss}}(x )\;\; &= \;\;\left\langle {[{\beta {I_{ref}}(x )+ I_{ref}^s(x )} ]({\alpha B + {B^s}} )} \right\rangle - \left\langle {[{\beta {I_{ref}}(x )+ I_{ref}^s(x )} ]} \right\rangle \left\langle {({\alpha B + {B^s}} )} \right\rangle \\ &= \alpha \cdot \beta \cdot O(x )+ \alpha \left\{ {\left\langle {I_{ref}^s(x )B} \right\rangle - \left\langle {I_{ref}^s(x )} \right\rangle \left\langle B \right\rangle } \right\}\\ &+ \beta \left\{ {\left\langle {{I_{ref}}(x ){B^s}} \right\rangle - \left\langle {{I_{ref}}(x )} \right\rangle \left\langle {{B^s}} \right\rangle } \right\}\; + \left\{ {\left\langle {I_{ref}^s(x ){B^s}} \right\rangle - \left\langle {I_{ref}^s(x )} \right\rangle \left\langle {{B^s}} \right\rangle } \right\}. \end{aligned}$$

There are four terms in Eq. (11). When the scattering becomes stronger, α and β decrease, and $\langle I_{ref}^s(x) \rangle$ and $\langle {B^s} \rangle$ increase. Although $\left\langle {I_{ref}^s(x ){B}} \right\rangle$ is not strictly equivalent to $\left\langle {I_{ref}^s(x )} \right\rangle \left\langle B \right\rangle$ and $\left\langle {{I_{ref}}(x )} {{B^s}} \right\rangle$ is not strictly equivalent to $\left\langle {{I_{ref}}(x )} \right\rangle \left\langle {{B^s}} \right\rangle$ in the real experiments, after multiplying by the transmission ratio, the first three terms all become very small and can be neglected compared to the fourth term.

Next, we focus on the fourth term of Eq. (11). The scattered light of the reference speckles and the scattered light of the test speckles still maintain the second-order correlation, i.e., we can extract the object information from the fourth term. Therefore, the scattered light can be utilized to reconstruct the image of the object by Eq. (11) when we place a scattering medium with certain characteristics in the reference path. But Eq. (11) holds if the number of measurements is large (close to infinity). Otherwise, adding a scattering medium into the reference path (our method) will cause noise for TGI. (See Supplement 1 for the simulation results).

2.2 Theoretical analysis of the proposed method under CSGI reconstruction algorithm

In CSGI, the bucket value matrix after K measurement times is formulated as

(12)$$\textrm{B} = {\textrm{I}_{\textrm{mea}}}\textrm{X} = {\textrm{I}_{\textrm{mea}}}\Psi \textrm{S,}$$

where I_mea is a K×N measurement matrix (K<N) which consists of all detect data of CCD_R. X is an N×1 column vector which stands for an unknown N-pixel object image. B is a K×1 column vector that consists of bucket values. $\Psi $ is an N×N matrix which is the transform operator to the sparse basis (natural images tend to be compressible in the discrete cosine transform (DCT) [18], so we use 2D-DCT in the experiment). S is the N×1 vector composed of the expansion coefficients in the above basis.

The relationship between sparse transformation and thermal intensity distribution satisfies the Restricted Isometry Property (RIP) [23]. In LGI, a pseudothermal source is used, so CSGI can be used in our experiments. Resolving X form Eq. (12) is a convex optimization problem when K<N and the problem can be converted to L₀-norm minimization (we use Orthogonal Matching Pursuit, OMP [24] in our experiment)

(13)$$\begin{array}{l} {\begin{array}{ccc} {\mathop {\min }\limits_\textrm{S} {{\left\| \textrm{S} \right\|}_0}}&{\textrm{subject}\,\textrm{to}}&{\left\| {{\textrm{B}_{\det }} - {\textrm{I}_{\textrm{mea}}}{\Psi }\textrm{S}} \right\|_2^2 \le t,} \end{array}}\\ { \to {\Psi }\textrm{S} = \textrm{X}} \end{array}$$

where t is a small number and ${||\textrm{S} ||_0}$ represents the L₀-norm of S. B_det = I_samX, I_sam is a K×N sampling matrix that denotes the speckle beamed onto the object. B_det is also a K×1 column vector that consists of bucket values.

In no scattering condition, I_sam = I_mea, that is B_det = B and we can obtain S from Eq. (13).

When there is a scattering medium exists before the object

(14)$${\textrm{I}_{\textrm{sam}}} = {\textrm{M}^\alpha }.\ast {\textrm{I}_{\textrm{mea}}} + {\textrm{I}_{\textrm{mea}}}.\ast {\textrm{F}_{\textrm{sca}}}$$

and we obtain a superimposed image of the object and the scattering medium:

(15)$$ \begin{aligned} &\min _{\mathrm{S}}\|\mathrm{S}\|_{0} \quad \text { subject to }\left\|\mathrm{B}_{\mathrm{det}}-\mathrm{I}_{\text {mea }} \Psi \mathrm{S}\right\|_{2}^{2} \leq t \\ &\rightarrow \Psi \mathrm{S}=\mathrm{M}^{\alpha} \cdot *+\mathrm{F}_{\mathrm{sca}} * \mathrm{X} \end{aligned} $$

where “.*” represents the operation: (G.*J)_ij= g_ijb_ij (element-wise product). M^a is a K×N matrix whose elements are a (0 < a < 1) and it weights the transmission ratio of the scattering medium. I_mea.*F_sca denotes the scattered light matrix. (F_sca is a K×N scattering matrix that denotes the scattering effect of the scattering medium.)

With the increase of scattering (the element's value of F_sca becomes larger and the element's value of M^a becomes smaller), the error between $\Psi \textrm{S}$ (the result of Eq. (15)) and X becomes larger and larger. But our method is helpful to reduce the reconstruction error:

If we find a kind of scattering medium with scattering characteristics similar to those of the scattering medium existing between the BS and the object and place it in the reference path. We have

(16)$${\textrm{I}_{\textrm{mea}}} = {\textrm{M}^\beta }.\ast {\textrm{I}_{\textrm{mea}}} + {\textrm{I}_{\textrm{mea}}}.\ast {\textrm{H}_{\textrm{sca}}}$$

and we obtain

(17)$$\begin{array}{l} {\begin{array}{ccc} {\mathop {\min }\limits_\textrm{S} {{||\textrm{S} ||}_0}}&{\textrm{subject}\,\textrm{to}}&{||{{\textrm{B}_{\det }} - {\textrm{I}_{\textrm{mea}}}\Psi \textrm{S}} ||_2^2 \le t} \end{array}}\\ { \to \Psi \textrm{S} = [{({{\textrm{M}^\alpha } + {\textrm{F}_{\textrm{sca}}}} )\textrm{./}({{\textrm{M}^\beta } + {\textrm{H}_{\textrm{sca}}}} )} ]\mathrm{.^\ast X}} \end{array}$$

where “./” represents the operation: (G./J)_ij= g_ij/b_ij. M^β is a K×N matrix whose elements are β (0< β <1) and it weights the transmission ratio of the scattering medium. I_mea.*H_sca denotes the scattered light matrix detected by CCD_R. (H_sca is a K×N scattering matrix that denotes the scattering effect of the scattering medium put in the reference path.)

Comparing the results of Eq. (17) and Eq. (15), we note that if the characteristics of two scattering mediums are similar, i.e., (M^a + F_sca) is closed to (M^β + H_sca), our method helps to reduce the reconstruction error of CSGI when there is a scattering medium exists between BS and object.

To summarize our method, for the TGI reconstruction algorithm, the scattered light of the reference speckles and the scattered light of the test speckles still maintain the second-order correlation [the fourth term of Eq. (11)]. Equation (10) is a prerequisite for the method, i.e., the scattering characteristics of the scattering medium placed in the reference path must meet conditions similar to those of the scattering medium in the object path. For the CSGI reconstruction algorithm, (M^a + F_sca) is closed to (M^β + H_sca) when the characteristics of the two scattering mediums are similar. In our experiments, this prerequisite was satisfied. In addition, when the scattering is so strong that the scattered light of the speckles reaching the CCD_R and the object plane is too weak. For the TGI reconstruction algorithm, the second-order correlation of these two parts of scattered light [the fourth term of Eq. (11)] is smaller as well. In this case, the reconstruction results are difficult to highlight from the background. For the CSGI reconstruction algorithm, the element's value of I_mea is so small that the RIP is not satisfied. That is our method fails in this case. We show this situation in our experiments.

3. Experiments and results

The experimental setup is sketched in Fig. 2. In the test path, the test speckle goes through the scattering medium and then illuminates the object. A suspension is used to simulate the effect of the scattering medium. The suspension is created by mixing a certain concentration of latex spherical calibration particles (GBW(E)120002a with particle diameter α = 3.5 µm; additional details of the particles can be found on the website [25]) into the NaCl solution with a density of ρ = 1.19 g/cm³. This satisfies the condition of Mie scattering (2πα/λ = 41.31 < 50) [7] and is held in a transparent acrylic cube with a side length of d = 35 mm in our experiments. The transparent cube in the test path is placed at distance Z₃ from the BS. The scattering intensity of the scattering medium is controlled by adding different volumes of the calibration particles to the NaCl solution. Here, the scattering media with concentrations of 0% and 5.3% correspond to no scattering and strong scattering, respectively. Concentrations of 2.8%, 3.7%, and 4.2% present moderate scattering. A black binary transmissive double slit is used as the object. The length of the slit is 2.0 mm, the width is 0.5 mm, and the centers of the two slits are 1.0 mm apart. A CW laser of wavelength λ = 532 nm and beam diameter D₀ = 3 mm is beamed onto RGG to produce pseudothermal light. The laser power is 8 mW, and the correlation time τ of the pseudothermal source is controlled to 0.1 s. The distances satisfy Z₁ = Z₂ = 200 mm and Z₃ = 100 mm. A CCD camera (named CCD_T in Fig. 2) is used to serve as the bucket detector, which collects the total integrated intensity of the light transmitted through the object. The parameters of CCD_T and CCD_R are set to the same in our experiments: the exposure time is 500 µs, which is much shorter than τ; the gain is set to 7 dB; the frames are grabbed at a rate of 6 Hz; the pixel size is 5.29 µm, and the photosensitive area is set to 960×1280 pixels. The size of the raw images detected by CCD_T and CCD_R is 960×1280. To reduce the memory usage of the computer while running the algorithm, we resize these raw images to 128×128 before performing image reconstruction. To quantitatively analyze the image quality, the contrast-to-noise ratio (CNR) [26] and visibility (V) of the reconstruction results are calculated. The CNR is defined as Eq. (18), and the visibility (V) is defined as Eq. (19).

(18)$$CNR = \frac{{\left\langle {O({{x_{\textrm{in}}}} )} \right\rangle - \left\langle {O({{x_{\textrm{out}}}} )} \right\rangle }}{{\sqrt {\frac{1}{2}[{{\mathrm{\Delta }^2}O({{x_{\textrm{in}}}} )+ {\mathrm{\Delta }^2}O({{x_{\textrm{out}}}} )} ]} }},$$

(19)$$V = \frac{{\left\langle {O({{x_{\textrm{in}}}} )} \right\rangle - \left\langle {O({{x_{\textrm{out}}}} )} \right\rangle }}{{\left\langle {O({{x_{\textrm{in}}}} )} \right\rangle + \left\langle {O({{x_{\textrm{out}}}} )} \right\rangle }}.$$

where O(x_in) and O(x_out) represent the pixel positions inside and outside the transmitting regions of the binary object, respectively. Δ²O($\cdot$) denotes the statistical variance. Considering the object is a double slit, both these quantities are evaluated by comparing the signal in the two slits with the “dark” between the two slits, and the background in areas very far away from the slits is not particularly relevant and is ignored.

Fig. 2. Experimental configuration utilizing scattered light for image reconstruction. A container filled with a specific scattering medium is placed at the symmetrical position of BS in the reference path. The object is close to the detection plane. CCD_T serves as a bucket detector. SMT: scattering medium existing in test path. SMR: scattering medium placed in reference path.

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It has been proven that the scattering medium between the object and the bucket detector does not affect GI [8–11]. Therefore, we keep the object close to the detection plane of the bucket detector, i.e., we only consider how to improve the image quality when the scattering medium exists between the BS and the object.

To verify our method, in the reference path, we need to place another scattering medium with scattering characteristics similar to those in the test path. Another empty transparent acrylic cube is placed at the symmetrical position of the BS in the reference path, this cube can be considered the same as the cube located in the test path within the error range. The scattering medium placed in the reference arm is prepared in the same way as the scattering medium in the reference arm. The scattering intensity of the scattering medium is changed by changing the volume ratio of latex spherical calibration particles to NaCl solution. Experiments were performed to validate our method, and all CSGI results were reconstructed from N = 720 illumination patterns, as shown in Figs. 3(1)-(8). The values of the CNR and the visibility of every image are marked at the bottom. The characteristics of the scattering medium are changed by varying its concentration. In this way, the prerequisite of Eq. (17) was satisfied when performing our method. The results are shown in Figs. 3(3), 3(5), 3(6), and 3(8).

Fig. 3. Experimental results of CSGI. Scattering conditions in two paths are marked at beginning of each row as (SMT, SMR). (1): No scattering medium in test path. (2), (7): Scattering medium exists in test path with a concentration of 2.8% and 4.2%, respectively. (4): Scattering mediums that exist in two paths are not similar. (3), (5), (6) and (8): Reconstruction results of our method when concentration is 2.8%, 3.7%, 5.3% and 4.2%. Values of CNR and visibility of each image are marked at the bottom as (CNR, visibility). “Milk”: scattering medium prepared using skimmed milk.

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Figure 3(1) shows the reconstruction results under no scattering condition, and Fig. 3(2) shows the reconstruction results scattering medium existing with a concentration of 2.8% in the test path. By comparing Figs. 3(2) and 3(1), the image quality worsens when there exists a scattering medium in the test path. Then, we filled the cube located in the reference path with a scattering medium of concentration 2.8% to make the scattering characteristics of the scattering medium in both paths essentially the same (our method). The reconstruction results are shown in Fig. 3(3). The image quality is worse than the results under the no scattering condition but does be improved compared to Fig. 3(2), i.e., our method improves the image quality when a scattering medium exists in the test path. We note that Fig. 3(2) is not completely confused, that is because the scattering is not so strong. In this case, the elements’ value of F_sca is smaller than that of M^a, so the reconstruction object image can still be slightly highlighted from the background.

As mentioned in the theoretical analysis section, the RIP is not satisfied when the scattering is so strong that the scattered light of the speckles reaching the CCD_R and the object plane is too weak, and then our method fails. To show this situation, we increased the concentration of the scattering medium in the test path. The image quality worsened again when the concentration was 3.7%, as shown in Fig. 3(4). However, the image quality improved when the concentration of the scattering medium in the reference path was also increased to 3.7%, as shown in Fig. 3(5). That is, our method works in this case. We continued to increase the concentration of the scattering medium in both paths to 5.3% (strong scattering). In this case, our method failed to reconstruct the image, as shown in Fig. 3(6), as the scattering was so strong that the speckle field reaching the CCD_R and the object plane was too weak. The raw images of the reference speckles detected by CCD_R under different scattering conditions are shown in Fig. 4. Obviously, the intensity decreases as the scattering increases.

Fig. 4. (1)-(4) Raw images of reference speckles detected by CCD_R under concentrations of 0%, 2.8%, 3.7%, and 5.3%.

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In addition, we show that our method works when a scattering medium prepared by different means than that in the test path is placed in the reference path. A certain amount of skimmed milk (GB25190) was added to the NaCl solution (with a density of ρ = 1.19 g/cm³) to serve as a scattering medium in the reference path. The reconstruction results are shown in Figs. 3(7) and (8). First, we keep the concentration of the scattering medium existing in the test path at 4.2% and replace the scattering medium in the reference path with a concentration of 0% (NaCl solution). The reconstruction results are shown in Fig. 3(7). Then, we randomly add approximately 1 mL of the prepared skimmed milk to the NaCl solution, and the reconstruction results are shown in Fig. 3(8). The image quality is improved after adding the prepared skimmed milk to the NaCl solution located in the reference path (comparing Figs. 3(7) and (8)). We note that the image quality of Fig. 3(8) is better than that of Fig. 3(5). Perhaps the noise caused by the laser light source in the last group of experiments (Fig. 3(c)) is less than that in the first two groups of experiments (Figs. 3(a) and (b)). And further experiments need to be done to explain this phenomenon. We measured the intensity distribution of a laser beam going through both scattering mediums, as shown in Figs. 5(a) and 5(b). The distribution of the two curves is similar, i.e., Eq. (17) is satisfied.

Fig. 5. (a) Laser beam goes through suspension existing in test path. (b) Laser beam goes through skimmed milk liquid placed in reference path.

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The above experiments prove that the object information embedded in the scattered light can be utilized for CSGI through a scattering medium by placing a scattering medium with certain characteristics in the reference path.

4. Discussion

By comparing the reconstruction results under the no scattering condition (Fig. 3(1)) with the other reconstruction results, we find that the reconstruction results are slightly shifted. We infer that this is caused by the refraction of the illumination speckle as it passes through the scattering medium and propagates into the air. We believe that our method is also suitable for other GI reconstruction algorithms such as pseudo-inverse ghost imaging (PGI) [27]. Other methods, such as deep learning, are expected to extract object information contained in the scattered light to help GI to reconstruct images through a scattering medium. It is possible to increase the laser power to compensate for the attenuation caused by scattering media [9].

We demonstrated that the object information embedded in scattered light is important for improving the robustness of CSGI through scattering media. And our method is possible in practical situations: Another CCD camera should be used to capture the scattering medium environment as prior information. And the characteristic of the existing medium in the object arm can be measured by analyzing the series of image data. After analyzing the effect of the scattering medium on the optical field, this effect can be modulated algorithmically onto the reference speckles, which is equivalent to adding a similar scattering medium into the reference arm.

5. Conclusion

In conclusion, based on the fact that the scattered light also contains information about the object after illuminating it, and combining with the advantages of CSGI, we proposed a method to realize CSGI through a scattering medium by utilizing scattered light. Placing a scattering medium with certain characteristics in the reference path can extract the object information contained in the scattered light. Our method inspires that it is feasible and helpful to realize GI through a scattering medium by extracting object information contained in the scattered light. We believe our finding helps to pave the way for practical applications of GI under the Mie scattering environment.

Funding

Beijing Municipal Natural Science Foundation (4222017); Funding of Foundation Enhancement Program (2019-JCJQ-JJ-273); National Natural Science Foundation of China (61871031, 61875012, 61905014).

Acknowledgments

We are grateful to Chuanxun Chen and Zhikuo Li for helping make the transparent cubes.

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.

Supplemental document

See Supplement 1 for supporting content.

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Ghost imaging through scattering medium by utilizing scattered light

Abstract

Corrections

1. Introduction

2. Theoretical analysis

2.1 Theoretical analysis of the proposed method under second-order correlation reconstruction algorithm

2.2 Theoretical analysis of the proposed method under CSGI reconstruction algorithm

3. Experiments and results

4. Discussion

5. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (5)

Equations (19)

Optics Express