High-quality coherent ghost imaging of a transmission target

Shihao Chang; Shihao Chang; Junjie Cai; Junjie Cai; Wenlin Gong; Wenlin Gong

doi:10.1364/OE.519158

1. Introduction

The first information acquisition based on the second-order correlation of light field could be traced back to the HBT experiment demonstrated in 1956, which was a brand-new method to obtain the angular diameter of the star by measuring the second-order auto-correlation function of light field [1]. Until 1995, the first proof-of-principle imaging demonstration based on the second-order cross-correlation function of the light field was experimentally verified by Shih’s group [2,3], which caused the research fever of ghost imaging (GI) in recent years. Different from conventional imaging, GI is a computational imaging technology that produces images by computing the correlation function between the intensity of modulation field and the target’s transmitted/reflected intensity recorded by a detector without spatial resolution [4–11]. Since the method of GI was invented, great development has been achieved especially in the area of remote sensing [12–15], three-dimensional imaging [16–19], X-ray imaging [20–22], and long-wavelength imaging with a single-pixel detector [23–26]. Up to now, the detection modes of GI can be divided into incoherent measurement and coherent measurement [20–29]. The typical optical system of incoherent measurement is GI with a bucket detector, where all the information transmitted/reflected from the target is collected onto the detector [27–29]. Therefore, this case is actually an incoherent imaging approach. When the test detector of GI system is a point-like detector, the case is corresponding to coherent measurement and is a coherent imaging method [4,20,28,29]. For example, Fourier-transform ghost diffraction [4,20], single-pixel ptychography [30], and pulse-compression ghost imaging lidar [31]. Compared with GI with a point-like detector, most of previous works focused on GI with a bucket detector because its imaging quality is much better, and lots of experiments demonstrated that the target’s intensity transmission function can be stably recovered only when all the information transmitted through a transmission target is collected by the bucket detector [20,28,29,32]. However, in practical applications, limited by the optical aperture of the receiving system, only a small part of the information transmitted through the target can be collected by the test detector, thus the condition of bucket detection is deviated and the imaging quality of GI will be sharply degraded [31,32]. Although GI quality of transmission targets can be improved by increasing the measurement number of image reconstruction [6], the target’s image can’t be perfectly reconstructed by GI in theory even if infinite random measurements are exploited [31,32], and it is too time-consuming to satisfy the requirement of the applications such as moving target imaging and medical imaging. Another approach is homodyne detection, but the system is complicated and phase-sensitive [33,34]. Therefore, it is natural to ask whether high-quality GI of a transmission target can be still achieved based on a simple GI system and a point-like detector. In this paper, we show that incoherent imaging and coherent imaging cases of GI can be realized by changing the detector’s transverse area, and the influence of the detector’s transverse area to GI quality has been clarified. Furthermore, we have designed a coherent GI (CGI) scheme of a transmission target and developed a corresponding CGI reconstruction algorithm, which can obtain an image with much better quality in comparison with GI with a bucket detector even if a point-like detector is adopted. The performance differences between traditional GI (TGI) reconstruction and CGI reconstruction are also discussed when the detection mode, the target’s property and the characteristics of modulation patterns are considered.

2. Theory and method

Figure 1 presents the proposed CGI scheme of a transmission target. The light emitted from a laser beam uniformly illuminates a digital micro-mirror device (DMD) and a series of random patterns are prebuilt by modulating the mirrors of the DMD. Then the patterns reflected by the DMD are imaged onto an object by a 4-$f$ optical imaging system with the focal length $f$. Next, the light transmitted through the object goes through an $f_1$-$f_1$ optical system and then is received by a single-pixel detector $D_t$. According to the theory of GI, the object’s image $O_{\rm {TGI}}$ can be reconstructed by computing the correlation function between the pattern’s distributions $I_r^i(x)$ modulated by the DMD and the intensities $I_t^i$ recorded by the detector $D_t$ [7–9]

(1)$$O_{\rm{TGI}}(x) = \frac{1}{{K }}\sum_{i =1}^{K}\left( {I_r^i(x)-\left\langle {I_r^i(x)} \right\rangle} \right)I_t^i.$$

where ${\left \langle {I_r^i(x)} \right \rangle } =\frac {1}{{K }}\sum _{s = 1}^{K}I_r^i(x)$ represents the ensemble average of $I_r^i(x)$ and $K$ is the total measurement number. In addition, the result recovered by Eq. (1) is usually called traditional GI (TGI) reconstruction.

Fig. 1. Schematic of coherent ghost imaging for a transmission target. CA: circular aperture.

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For the schematic shown in Fig. 1, the intensity distribution $I_t^i(x_t)$ at the detection plane can be related to the light field $E_0^i(x_0)$ at the source plane and the optical transmission system by the Fresnel diffraction integral [35]

(2)$$I_t^i(x_t) = \left| \int {dx_0}E_0^i(x_0)h(x_t,x_0)\right|^2,{\rm{ \ }} i=1\cdots K,$$

where $h(x_t,x_0)$ denotes the impulse response function from the plane $x_0$ to the plane $x_t$.

When the transmission apertures of both the lens $f$ and the lens $f_1$ are large enough and under the paraxial approximation, the impulse response function $h(x_t,x_0)$ is

(3)$$h(x_t,x_0) \propto t(x_0)\exp \left\{ {-\frac{{2j\pi }}{{\lambda f_1 }}x_t x_0}\right\},$$

where $t(x)$ is the transmission function of the target. Then the intensity distribution $I_t^i(x_t)$ at the detection plane can be described as

(4)$$I_t^i(x_t) = \left| \int {dx_0}E_0^i(x_0)t(x_0)\exp \left\{ {-\frac{{2j\pi }}{{\lambda f_1 }}x_t x_0}\right\}\right|^2,{\rm{ \ }} i=1\cdots K,$$

usually, a bucket detector is used to record the whole intensity at the detection plane, namely

(5)$$\begin{aligned}I_t^i =&\int \limits_{ - \frac{D}{2}}^{\frac{D}{2}} dx_t\left| \int {dx_0}E_0^i(x_0)t(x_0)\exp \left\{ {-\frac{{2j\pi }}{{\lambda f_1 }}x_t x_0}\right\}\right|^2,{\rm{ \ }} i=1\cdots K,\\ =&\int {dx_0dx'_0}E_0^i(x_0){E_0^i}^*(x'_0)t(x_0)t^*(x'_0)\sin c \left( \frac{D}{\lambda f_1}(x'_0-x_0) \right), \end{aligned}$$

where $D$ is the diameter size of the detector $D_t$ and $\sin c(x)=\frac {\sin (\pi x)}{\pi x}$. For TGI reconstruction, when the size of $D$ is large enough such that all the information transmitted through the target are collected by the detector $D_t$, Eq. (5) can be simplified as

(6)$$\begin{aligned}I_t^i =& \int {dx_0}\left|E_0^i(x_0) \right|^2\left|t(x_0) \right|^2,\\ =&\int {dx_0}I_r^i(x_0)\left|t(x_0) \right|^2, {\rm{ \ }} i=1\cdots K, \end{aligned}$$

where $I_r^i(x)=\left |E_0^i(x) \right |^2$ is the intensity distribution of the $i$th pattern modulated by the DMD, and this detection mode described by Eq. (6) is corresponding to the case of incoherent GI [27–29].

Generally speaking, Eq. (1) can be expressed as a series of matrix operations for TGI reconstruction. Provided that each of the pattern’s intensity distributions $I_r^i(x)$ is a $1\times M$ image and the target’s intensity function $\left |t(x) \right |^2$ is an $M\times 1$ image ($X$). What’s more, the intensities $I_t^i$ is corresponding to a $K\times 1$ column vector ($B$=$[I_t^1{\rm {\ \ }}I_t^2{\rm {\ \ }}\cdots {\rm {\ \ }}I_t^i{\rm {\ \ }}\cdots {\rm {\ \ }}I_t^K]^T$), then Eq. (6) can be expressed as

(7)$$B=\Phi X,$$

and Eq. (1) can be rewritten as

(8)$$\begin{aligned}O_{\rm{TGI}}(x)=&\frac{1}{{K }}\left(\Phi-I{\left\langle {\Phi} \right\rangle}\right)^T B\\ =&\frac{1}{{K }}\left(\Phi-I{\left\langle {\Phi} \right\rangle}\right)^T \Phi X, \end{aligned}$$

where $\Phi$ is a $K\times M$ matrix and $\Phi ^T$ denotes the transposition of the matrix $\Phi$, namely

(9)$$\Phi = \left[ \begin{array}{clll} I_r^1 (1) & I_r^1 (2) & \cdots & I_r^1 (M) \\ I_r^2 (1) & I_r^2 (2) & \cdots & I_r^2 (M) \\ \vdots & \vdots & \ddots & \vdots \\ I_r^i (1) & I_r^i (2) & \cdots & I_r^i (M) \\ \vdots & \vdots & \ddots & \vdots \\ I_r^K (1) & I_r^K (2) & \cdots & I_r^K (M)\\ \end{array} \right],$$

here ${\left \langle {\Phi } \right \rangle }=\frac {1}{{K }}\sum _{s = 1}^{K}\Phi ^i$ represents a $1\times M$ row vector and $I$ denotes a $K\times 1$ column vector whose elements are all 1.

When the size of $D$ is decreased and only a part of the information transmitted through the target is detected by the detector $D_t$ (namely the condition of incoherent GI doesn’t satisfy), then Eq. (5) can be rewritten as

(10)$$\begin{array}{r}I_t^i = \int_{{x_0} \ne x{'_0}} {dx_0dx'_0}E_0^i(x_0){E_0^i}^*(x'_0)t(x_0)t^*(x'_0)\sin c \left( \frac{D}{\lambda f_1}(x'_0-x_0) \right)\\ +\int {dx_0}\left|E_0^i(x_0) \right|^2\left|t(x_0) \right|^2,\; i=1\cdots K, \end{array}$$

in this case, Eq. (10) can be described as $B=\Phi X+N$ and $O_{\rm {TGI}}(x)=\frac {1}{{K }}\left (\Phi -I{\left \langle {\Phi } \right \rangle }\right )^T \Phi X +\frac {1}{{K }}\left (\Phi -I{\left \langle {\Phi } \right \rangle }\right )^T N$. Therefore, the first term in Eq. (10) is corresponding to $N$ and can be considered as an additive noise, which means that the quality of TGI will be reduced when the size of $D$ is decreased.

For Eq. (4), if the distribution of the light field $E_0^i(x)$ is non-negative, then the zero-frequency information of the signal $E_0^i(x)t(x)$ will be always positive for a transmission target. Therefore, when the detector $D_t$ is a point-like detector and it is positioned at $x_t=0$, the detection mode is corresponding to the case of CGI and Eq. (4) can be simplified as

(11)$$\sqrt {I_t^i(x_t=0)}= \int {dx_0}E_0^i(x_0)t(x_0),\; i=1\cdots K,$$

similar to Eq. (7), the matrix description of Eq. (11) can be expressed as

(12)$$B'=\sqrt{\Phi} \sqrt{X},$$

where $B'$=$[\sqrt {I_t^1(x_t=0)}\;\cdots \;\sqrt {I_t^i(x_t=0)}\;\cdots \;\sqrt {I_t^K(x_t=0)}]^T$. Similar to Eq. (8), the corresponding image reconstruction algorithm, which is called CGI reconstruction in this work, can be described as

(13)$$\begin{aligned}O_{\rm{CGI}}(x)=&\frac{1}{{K }}\left(\sqrt{\Phi}-I{\left\langle {\sqrt{\Phi}} \right\rangle}\right)^T B'\\ =&\frac{1}{{K }}\left(\sqrt{\Phi}-I{\left\langle {\sqrt{\Phi}} \right\rangle}\right)^T \sqrt{\Phi} \sqrt{X}, \end{aligned}$$

and on the basis of Eq. (1), CGI reconstruction can be also expressed as

(14)$$O_{\rm{CGI}}(x) = \frac{1}{{K }}\sum_{i =1}^{K}\left( {\sqrt{I_r^i(x)}-\left\langle \sqrt{{I_r^i(x)}} \right\rangle} \right)\sqrt{I_t^i(x_t=0)},$$

in comparison with TGI reconstruction described by Eq. (1) and Eq. (8), both the pattern’s intensity distribution and the intensity recorded by the detector are taken a root for CGI reconstruction. Therefore, it is easy to realize real-time image reconstruction, similar to TGI reconstruction. What’s more, from Eq. (8) and Eq. (13), it can be find that when both $\left (\Phi -I{\left \langle {\Phi } \right \rangle }\right )^T \Phi$ and $\left (\sqrt {\Phi }-I{\left \langle {\sqrt {\Phi }} \right \rangle }\right )^T \sqrt {\Phi }$ are diagonal matrixes, both TGI and CGI can obtain high-quality images. For example, when the patterns modulated by the DMD are Bernoulli distribution, the image’s quality reconstructed by both TGI with a bucket detector and CGI with a point-like detector will be the same without detection noise. However, if there is detection noise in the sampling process, namely $B=\Phi X+N_s$ ($N_s$ denotes the detection noise), then the detection signal-to-noise ratio (DSNR) of TGI is $10 \log _{10}\left (\frac {{\left \langle {\Phi X} \right \rangle }}{std(N_s)}\right )$ ($std(x)$ denotes the standard deviation of the vector $x$). Correspondingly, $B'=\sqrt {B}\approx \sqrt {\Phi X}\left (1+\frac {1}{2}\frac {N_s}{\Phi X}\right )$ and the DSNR of CGI is $10 \log _{10}\left (\frac {{2\left \langle {\Phi X} \right \rangle }}{std(N_s)}\right )$, which is 3 dB higher than that of TGI. Therefore, the result reconstructed by CGI will be better than TGI in practical application because the noise is inevitable. In addition, the result of TGI is corresponding to the target’s intensity transmission function $\left |t(x) \right |^2$ whereas the target’s transmission function $t(x)$ can be directly reconstructed by CGI.

In order to evaluate quantitatively the quality of images reconstructed by TGI and CGI methods, the reconstruction fidelity is estimated by calculating the peak signal-to-noise ratio (PSNR):

(15)$${\rm{PSNR} } = 10 \times \log _{10} \left[ {\frac{{(2^p - 1)^2 }}{{{\rm{MSE} }}}} \right],$$

where the bigger the value PSNR is, the better the quality of the recovered image is. For a 0$\sim$255 gray-scale image, $p$=8 and MSE represents the mean square error of the reconstruction images $O_{\rm {re}}$ with respect to the original object $O$, namely

(16)$${\rm{MSE} }=\frac{1}{{N_{pix}}}\sum_{i = 1}^{N_{pix}}{\left[ {O_{{\rm{re }}} (x_i) - O (x_i)} \right]} ^2.$$

where $N_{pix}$ is the total pixel number of the image.

3. Simulation and experimental demonstration

To demonstrate the property of CGI and the theoretical analysis above, the parameters of both numerical simulation and experimental demonstration based on the setup of Fig. 1 are set as follows: the wavelength of the laser is $\lambda$=532 nm, the transverse size of the patterns at the DMD plane is set as $\Delta x$=41.0 $\mu$m and the modulated area of the DMD (DLP7000, Texas Instruments) is 64 $\times$ 64 pixels (one pixel is equal to the pattern’s transverse size). The speckles modulated by the DMD are Hadamard patterns (where the position of the value “-1" is set as 0). In addition, $f$=100 mm and $f_1$=175 mm, the DSNR is set as 20 dB in the process of numerical simulation. The imaging target, as displayed in Fig. 2(a) (namely $\left |t(x)\right |^2$), is a binary transmission three-slit (64 $\times$ 64 pixels) with slit width $a$=125 $\mu$m, slit height $h$=625 $\mu$m, and center-to-center separation $d$=250 $\mu$m (chosen from the USAF resolution test target). When one of the Hadamard patterns is loaded by the DMD, the corresponding intensity distribution of the light field at the detection plane is shown in Fig. 2(b) (namely $\left |{\rm {FFT}}\{E^i(x)t(x)\}\right |^2$). By changing the diameter size $D$ of the detector $D_t$, Fig. 2 have given the simulated and experimental results of CGI and TGI in different $D$ (in experiments, the reconstruction results obtained by a similar complementary single-pixel imaging method is as the reference object of PSNR computation, namely two complementary patterns successively illuminate the same target and the signals used for reconstruction are corresponding to the subtraction of two detection signals [36]). Besides, the curves of PSNR-$D$ for both CGI and TGI based on Fig. 2(c)-(h) are illustrated in Fig. 3. It is clearly seen that as the detector’s diameter size $D$ is increased, as displayed in Fig. 2 and Fig. 3, the quality of both TGI and CGI is firstly decreased and then improved. When the size $D$ is smaller than $\lambda f_1/2m\Delta x$ (namely only the zero-frequency information of the target is detected by the detector $D_t$, and $m$ is the pattern’s pixel number in $x$ direction), the target’s image can be stably reconstructed by CGI, which accords with the theoretical analysis described by Eq. (13). Moreover, as predicted by the theory, CGI with a point-like detector (namely $D=\lambda f_1/2m\Delta x=17.7 \mu$m, Fig. 2(c)) is better than TGI with a bucket detector (namely $D=\lambda f_1/\Delta x=2265.6 \mu$m, Fig. 2(h)) when detection noise exists in the sampling process. It is worth mentioning that in comparison with the simulation result, the intensity distribution of three-slit for experimental reconstruction results is not uniform, which originates from the pixel alignment deviation between the DMD plane and the target plane, and the heterogeneity of illuminating light in the experiment. What’s more, in practical application, imaging speed is very important to the scenes like moving target and medical imaging. Because GI is a computational imaging method with multiple random measurements, we should reduce the measurement number as few as possible and the image reconstruction process must be also fast, like the linear reconstruction algorithm of TGI and CGI described by Eq. (1) and Eq. (14). Most previous works have demonstrated that orthogonal measurements like Hardmard patterns and Fourier patterns are helpful to reduce the measurement number of GI reconstruction [37], the imaging speed in present experiments is 5 fps when the modulation speed of the DMD is 20480 fps and Hadamard patterns are adopted. Of course, the imaging speed can be further improved by methods with higher modulation speed and GI via deep learning where the measurement number used for image reconstruction can be further reduced [38,39].

Fig. 2. Simulated and experimental results of CGI and TGI for imaging a transmission binary three-slit in different $D$. (a) The imaging target; b) the intensity distribution of the light field at the detection plane when one speckle illuminates the target; (c) $D$=17.7 $\mu$m; (d) $D$=123.9 $\mu$m; (e) $D$=300.9 $\mu$m; (f) $D$=902.7 $\mu$m; (g) $D$=2000.1 $\mu$m; (h) $D$=2265.6 $\mu$m.

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Fig. 3. The relation of PSNR-$D$ for both CGI and TGI, based on the experimental and simulation results in Fig. 2.

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To further clarify the performance differences between TGI with a bucket detector and CGI with a point-like detector, Fig. 4 has displayed the experimental demonstration results of imaging the same binary three-slit, by changing the position of the target. For Fig. 4, the detector’s diameter size $D$ is fixed as 17.7 $\mu$m and other parameters are the same as Fig. 2. It is obviously observed that high-quality images of the target can be stably reconstructed by CGI whereas the reconstruction results of TGI are worse than CGI, and the pseudomorphism of TGI reconstruction will change with the position of the target. Furthermore, the effect of the property of the speckle’s pattern on CGI/TGI is shown in Fig. 5, by imaging a gray-scale three-slit and simulation demonstration. When the speckles (namely $\Phi$) modulated by the DMD are Hadamard patterns or cosine-basis patterns, the results of TGI and CGI are illustrated in Fig. 5(a) and Fig. 5(b), respectively. Inversely, when the speckles (namely $\sqrt {\Phi }$) modulated by the DMD are cosine-basis patterns, the corresponding reconstruction results are shown in Fig. 5(c). From Fig. 5, we can find that on the one hand, if $\left (\Phi -I{\left \langle {\Phi } \right \rangle }\right )^T \Phi$ is a diagonal matrix, TGI with a bucket detector can obtain high-quality images and it is corresponding to the target’s intensity transmission function $\left |t(x) \right |^2$. For CGI with a point-like detector, the target’s transmission function $t(x)$ can be perfectly reconstructed when $\left (\sqrt {\Phi }-I{\left \langle {\sqrt {\Phi }} \right \rangle }\right )^T \sqrt {\Phi }$ is a diagonal matrix. On the other hand, when the patterns modulated by the DMD are Bernoulli distribution (Fig. 5(a)), high-quality images can be reconstructed by both TGI with a bucket detector and CGI with a point-like detector, and the result of CGI is much better than that of TGI because of the influence of the detection noise. These simulated results completely match with the theoretical analysis described by Eq. (8) and Eq. (13).

Fig. 4. Experimental results of CGI and TGI for imaging the same binary three-slit described in Fig. 2 when the speckles modulated by the DMD are Hadamard patterns and the target is located in different positions of the illuminating speckle pattern.

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Fig. 5. Simulated results of performance differences between CGI with a point-like detector and TGI with a bucket detector for imaging a gray-scale three-slit when the DSNR is 20 dB. (a) $\Phi$ is Hadamard patterns; (b) $\Phi$ is cosine-basis patterns; (c) $\sqrt {\Phi }$ is cosine-basis patterns.

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Finally, to validate the applicability of CGI for complex scenes, Fig. 6 gives the simulation demonstration of imaging three different targets (namely the USAF resolution testing target, a gray-scale object “model of a man", and a real scenario “tower") when a point-like detector is adopted. The size of the target is chosen as 128 $\times$ 128 pixels and the modulated area of the DMD is also 128 $\times$ 128 pixels. In addition, the speckles modulated by the DMD are also Hadamard patterns, the DSNR is set as 30 dB and other parameters are the same as Fig. 2. Similar to the results displayed in Fig. 6(c), as shown in Fig. 6(a)-(c), CGI is always better than TGI for the testing targets especially for the USAF resolution testing target. Therefore, We demonstrate that CGI method is valid to dramatically improve the imaging quality of transmission targets.

Fig. 6. Simulation results of CGI and TGI for different objects. (a) the USAF resolution testing target; (b) a gray-scale object “model of a man"; (c) a real scenario “tower".

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4. Conclusion

In summary, we have clarified the incoherent imaging and coherent imaging cases of GI, which can be achieved by changing the detector’s transverse area. As the detector’s transverse area is decreased, the reconstruction quality of TGI will be sharply degraded especially for a transmission target. We have proposed a CGI method of stably imaging a transmission target even if a point-like detector is used. When the transmission target is illuminated by a series of random patterns with amplitude modulation and only its zero-frequency information is detected, high-quality imaging can be still obtained when CGI reconstruction algorithm is adopted. In comparison with TGI reconstruction algorithm, the target’s transmission function can be directly achieved by CGI and the imaging quality is much better when there is detection noise in the sampling process. This work is helpful to the system design of ghost imaging via coherent detection.

Funding

Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province (ZZ2307); 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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High-quality coherent ghost imaging of a transmission target

Abstract

1. Introduction

2. Theory and method

3. Simulation and experimental demonstration

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (16)

Optics Express