Differential Hadamard ghost imaging via single-round detection

Zhuo Yu; Zhuo Yu; Xiao-Qian Wang; Xiao-Qian Wang; Chao Gao; Zhe Li; Huan Zhao; ZhiHai Yao; ZhiHai Yao

doi:10.1364/OE.441501

1. Introduction

Ghost imaging (GI) is a remarkable imaging method where the image of an unknown object may be reconstructed by intensity correlation measurements. In traditional ghost imaging systems, the light signal is split in two correlated beams, one beam illuminates the object and it is referred to as the object beam. After passing through the object, the transmitted light is measured by a single-pixel detector. The other beam, usually referred to as the reference beam, does not pass through the object, and is measured by a charge coupled device (CCD). The image of the object can be retrieved by performing correlation measurements of the intensities at the two detectors. Ghost imaging was first proposed by Pittman et al. [1,2]. Later on, Shapiro et al. proposed a computational ghost imaging (CGI) scheme [3]. Compared to traditional GI, the experimental setup of CGI is significantly simplified. The reference beam is no longer present, and a single-pixel detector is used to detect the object beam [4]. A deterministic illumination pattern at the object plane is created by a spatial light modulator (SLM) or a Digital Micromirror Device (DMD), and the image may be reconstructed by recording only the total reflected light (or the transmitted light) intensity by a single-pixel detector. In recent years, several sophisticated algorithms have been developed to improve the computing efficiency and imaging quality of CGI, such as differential computational ghost imaging (DCGI) [5], compressed sensing computational ghost imaging (CSGI) [6], Singular value decomposition ghost imaging [7], deep learning ghost imaging (DLGI), and few more methods [8–17].At present, the applications of ghost imaging include but are not limited to three-dimensional imaging [18–22], Acoustic Imaging [23], multispectral imaging [24,25], infrared imaging [26,27], and terahertz imaging [28,29] .

In addition, since structured patterns are key factors affecting the CGI imaging efficiency, various types of structured illumination patterns have been suggested using sparse bases that are also orthogonal. These include Hadamard, Fourier, and wavelets, which are commonly used in single-pixel imaging [30–43].

CGI based on orthogonal illumination patterns, however, is affected by the so-called negative illumination problem. In order to illustrate the issue, let us consider CGI based on the Hadamard pattern (HCGI). Since the Hadamard matrix is a square matrix whose entries are either "+ 1" or "-1", the SLM (or DMD) is unable to project the pattern with "-1". Some researchers proposed to transform the "-1"’s to "0"’s to obtain a special Hadamard matrix, modulate it to scatterer for sampling, and the reconstruct the image. The corresponding reconstruction, however, turned out to be not very accurate. Welsh et al. suggested the use of a differential method to split the Hadamard matrix into positive and negative illumination patterns to solve the problem [44]. Currently, this method is widely used in the computational ghost imaging experiments based on Hadamard matrix such as the "Russian Dolls ordering", the "Cake-Cutting strategy", the "Fast Walsh transform", and few others [45–50]. The method has been experimental advantages but involves positive and negative patterns, i.e. it needs twice the number of samplings. In this paper, we propose a new scheme which only needs 1 round of measurement while maintaining good image quality.

The paper is arranged as follows. In Section 2, use HCGI as a case study, we illustrate the feasibility of our scheme. In Section 3, we show experimental results and discuss their significance. Finally, conclusions are given in Section 4.

2. Theoretical model analysis

The schematic diagram of the conventional CGI is shown in Fig. 1. A series of patterns are generated by a computer and projected onto the object by a projector. The quantity $R_i(x,y)$ is a $N_x \times N_y$ matrix which denotes the $i$-th illumination pattern. The patterns $R_i(x, y)$ interacts with a target object, and then it is measured by a bucket detector. The i-th detector signal detected is denoted by $B_i$. After $N$ measurements, the image could be reconstructed by calculating the second-order correlation function as:

(1)$$G^{(2)}(x, y)=\frac{1}{N} \sum_{i=1}^{N} R_{i}(x, y) B_{i}$$

Since the Hadamard matrix is the simplest orthogonal matrix, we introduce our single-measurement technique using this example. The Hadamard basis is given by

(2)$$\hat H_{2^{n}}=\hat H_{2} \otimes \hat H_{2^{n-1}}=\left[\begin{array}{cc} \hat H_{2^{n-1}} & \hat H_{2^{n-1}} \\ \hat H_{2^{n-1}} & -\hat H_{2^{n-1}} \end{array}\right].$$

where $n \geq$2, $\otimes$ denotes the Kronecker product, $n$ is a positive integer and and the Hadamard matrix $\hat H_{2}$ is the square matrix given by

(3)$$\hat H_{2}=\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]\,.$$

The Hadamard basis may be written a column of Hadamard matrices

(4)$$\mathrm{\hat H}=\left[\vec{h}_{1}, \vec{h}_{2}, \ldots, \vec{h}_{N}\right]^T.$$

and then reshaped into $N_x \times N_y$ 2$D$ patterns. To facilitate theoretical analysis, both the patterns and the transmission function of the image are transformed into 1$D$ form, where the 2$D$ object $O(x,y)$ is represented by a $1\times N$ ($N=N_xN_y$) column vector $\textbf {O}$

(5)$$O=\left[O_{1}, O_{2}, O_{3}, \ldots, O_{N}\right]^{T}.$$

The second-order correlation function of Hadamard ghost imaging may be then written as

(6)$$G^{(2)}=\frac{1}{N} \sum_{i=1}^{N} \vec{h}_{i}^T B_{i}.$$

Throughout the paper, we are going to refer to this quantity as the bucket signal, which itself may be rewritten as

(7)$$B_{i}=\vec{h}_{i}^{T} O$$

where T denotes the transpose operator. The second-order correlation function may be rewritten accordingly as

(8)$$G^{(2)}=\frac{1}{N} \hat H^{\prime} \hat H^{\prime T} \mathrm{O}$$

where $H^\prime$ is the $N\times n$ submatrix of the Hadamard matrix. If a complementary matrix $\hat H$ exists, then we have $\hat H\, \hat H^T= N I$, where $\hat I$ is the identity matrix of dimension $N$, and the second-order correlation function is given by $G^{\left (2\right )}=\hat I O= O$. In this case, the original image may be fully reconstructed.

Fig. 1. Schematic diagram of computational ghost imaging.

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Assuming that the binary patterns illuminated by the projector is made of "1" and "0", the Hadamard matrix may be divided into two complementary matrices $\hat H_+=\frac {1+\hat H}{2}$ and $\hat H_-=\frac {1-\hat H}{2}$ , where "1" stands for a matrix with all the entries equal to 1. Then the value range of the differential patterns is given by

(9)$$\hat H=\hat H_{+}-\hat H_{-}.$$

For the sake of simplicity, in the following we assume a square picture ($N_x=N_y$). If the image is reconstructed using the $\hat H_+$ matrix, we refer to this scheme as the "Positive Hadamard ghost imaging (PHGI)". The reconstructed image of HCGI can be written as

(10)$$\begin{aligned} & G^{(2)}=\frac{1}{N} \hat H^{T}_{N+} \hat H_{N+} O \\ & =\frac{1}{4}\left(\sum_{n=1}^{N} O_{n}+O_{1}\right)+\frac{1}{4}\left[\begin{array}{c} O_{1}+\sum_{n=1}^{N} O_{n} \\ O_{2} \\ O_{3} \\ \vdots \\ O_{N} \end{array}\right]_{N \times 1} \end{aligned}$$

Looking at Eq. (10), one sees that $G^{(2)}$ is not (strictly and positively) correlated with the object $O$. The reconstructed image may be thus significantly distorted. This is reflected by the fact that the intensity value of the first pixel is significantly increased. This usually results in making visible only the upper left corner of the reconstructed image. We correct the value of the first-pixel point for the second term in the equation. Since the first Hadamard scatter is an all-ones matrix, and the error is $\Delta =\sum _{n=1}^{N} O_{n}$, the resulting bucket signal can be written as

(11)$$B_{1}=\vec{h}_{1} O=\sum_{n=1}^{N} O_{n}.$$

The exact value is obtained by subtracting $B_1$ from the value of the first-pixel point. However, since the first term in Eq. (10) is a background term, this results in a lower contrast in the recovered image. In order to correct this behaviour, the following scheme is employed. In HCGI experiments, we can only use the "1" and "0" special Hadamard patterns for sampling, but in the image reconstruction process, the illumination patterns are controllable, and thus we use the normal Hadamard scatterers for image reconstruction. We refer to this scheme as the "Modified positive Hadamard ghost imaging (MPHGI)". The reconstructed image of HCGI can be written as

(12)$$G^{(2)}=\frac{1}{N} \hat H \hat H_{N+}^{T} O=\frac{2}{N}\left[\begin{array}{c} O_{1}+\sum_{n=1}^{N} O_{n} \\ O_{2} \\ O_{3} \\ \cdots \\ O_{N}. \end{array}\right]_{N \times 1}$$

Compared to PHGI, this scheme removes the background term, but the intensity of the first-pixel point is still much larger than the values of the other pixel points. After correcting the value of the first-pixel point according to the method mentioned above, the correlation function may be re-defined as

(13)$$G^{\prime(2)}=\frac{2}{N} G^{(2)}-B_{1} \delta(n-1)$$

In the MPHGI scheme, one does not detect the bucket signal in the normal Hadamard mode, and therefore picture information cannot be recovered accurately. Inspired by Eq. (9), we argue that the correct signal can be obtained by projecting two opposite illumination patterns. The resulting scheme is termed "Differential Hadamard ghost imaging (DHGI)". The splitting of Hadamard illumination pattern into positive and negative patterns is illustrated in Fig. 2. The two groups, $\hat {H}_+$ and $\hat {H}_-$ patterns, are measured separately to obtain the detection results of Hadamard pattern in normal mode, and the bucket signal may be written as

(14)$$\begin{array}{l} B_{i}=\left(\vec{h}_{i+}-\vec{h}_{i-}\right) O=\vec{h}_{i} O=B_{i+}-B_{i-}. \end{array}$$

In this case, the second-order correlation function is given by

(15)$$G^{(2)}=\frac{1}{N} \sum_{i=1}^{N} \vec{h}_{i}\left(B_{i+}-B_{i-}\right)=O.$$

This scheme obtains the bucket signal under normal Hadamard pattern, however at the price of doubling the number of measurements.

Fig. 2. Splitting the Hadamard illumination pattern into positive and negative patterns (we show a $8\times 8$ Hadamard matrix as an example).

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In order to overcome this drawback, we suggest to employ a single measurement to obtain the bucket signal. Let us consider $\hat H_{N+}^T$ in Eq. (13) as the observation matrix, and introduce $\hat H_+=\frac {1+\hat H}{2}$. Here, since we add one to the elements of $\hat H$, $\hat H_+$ can be written as an all-ones matrix of the same size as $H$. The positive Hadamard observation matrix can be thus written as

(16)$$\vec{h}_{i+}=\frac{\vec{h}_{1}+\vec{h}_{i}}{2},$$

and the corresponding measurement intensity as

(17)$$B_{i+}=\vec{h}_{i+}^{T} O=\frac{\vec{h}_{1}^{T}+\vec{h}_{i}^{T}}{2} O=\frac{1}{2}\left(B_{1}+B_{i}\right).$$

In this case, we reconstruct the image using the metric intensity $B_{i+}$ and Hadamard matrix $H$. The second-order correlation function can be written as

(18)$$G^{(2)}=\frac{1}{N} \sum_{i=1}^{N} \vec{h}_{i}\left(2 B_{i+}-B_{1}\right)=O,$$

and the original image can be completely reconstructed. We term this scheme "Differential positive Hadamard ghost imaging (DPHGI)".

We also find that if $\vec {h}_{i-}^{T}$ is used for sampling in the DPHGI scheme, the detector signal obtained can be expressed as:

(19)$$B_{i-}=\frac{1}{2}(\hat{I}^{'}O-B_i)$$

where $\hat {I}^{'}$ denotes the all-ones matrix and the value $\hat {I}^{'}O$ denotes the total transmittance of the target object. Since $\vec {h}_{i-}^{T}$ is the inverse of $\vec {h}_{i+}^{T}$ and does not have a measurement base with all elements of 1, the information of the target object is lost by using $\vec {h}_{i-}^{T}$ only for measurement. The results of the above two schemes, MPHGI and DPHGI, illustrate that the full information of the target object can be obtained by using positive pattern measurements, and doubling measurements using negative pattern is redundant.

3. Experiments and discussions

To verify the feasibility of the scheme in practical computational ghost imaging, several experiments have been performed. The experimental setup is shown in Fig. 1. The Model of the projector is XGIMI-XE11F and we use a metal piece with a hollow pentagram in the center as the target object, and the Model of the photodiode in the optical detection circuit is LSSPD-2.5-3P-08.26. We reconstruct the images using all of the four methods introduced in Section 2, and compare their results sampling the target objects in the same conditions. The reconstruction results are shown in Fig. 3. We refer to the four schemes in Section 2 as PHGI, MPHGI, DHGI, and DPHGI, respectively. Among the four options, only PHGI fails to reconstruct the image. The experimental results are consistent with the theoretical derivation. In particular, we see that the value of the first-pixel point is much larger than the other pixel points, so only the first point in the reconstructed image is bright, and the others are dark. The other three reconstruction schemes provide better performance, but the number of measurements of MPHGI and DPHGI is reduced by 50$\%$ compared with DHGI.

Fig. 3. The experimental results of the image reconstruction of the object "star" by using PHGI, MPHGI, DHGI, and DPHGI, respectively.

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Let us now further analyze and compare MPHGI and DPHGI. DPHGI requires to calculate the bucket signal under normal Hadamard scatter irradiation after a single sampling, and then reconstruct the image. MPHGI allows us to reconstruct the image directly from the result of a single sampling, but then we have to correct the first-pixel value to obtain the correct image. It may be argued that DPHGI can further improve the efficiency of ghost imaging, and we demonstrated this possibility by experiments.

In the literature on compressed sensing imaging, stable and fast imaging schemes, such as Walsh-Hadamard transform and fast Fourier transform, have been proposed. Among them, Walsh-Hadamard transform is based on Hadamard matrices as observation matrices, and the number of sampling and image reconstruction is reduced by changing the order of the Hadamard matrix row order [32].

The experimental results are shown in Fig. 4. The resolution of the reconstructed image is $64$ $\times$ $64$ pixels. As the figure shows, the higher is the sampling rate, the better is the quality of the recovered images, but at the same sampling rate the number of DPHGI measurements is reduced by half compared to DHGI, and the image quality is not much different. If we look closely, the quality of the images recovered by DHGI is slightly better because DPHGI introduces a small amount of noise, while DHGI performs a differential operation on the measurements, which results in a more effective suppression of the noise. In order to compare the quality of the reconstructed image quantitatively, the Peak Signal-to-Noise Ratio (PSNR) is used as a criterion, which is defined as [51]

(20)$$PSNR=20\log _{10}\left[ \frac{\max Val-1}{MSE} \right],$$

where $MSE=\frac {1}{N}\sum _{x,y}{\left ( O_0(x,y)-O(x,y) \right ) ^2}$, $O_0(x,y)$ and $O(x,y)$ represent the original and reconstructed images, respectively, $N$ denotes the total number of pixel points for each of $O_0(x,y)$ and $O(x,y)$, and $max Val$ is the maximum gray value of $O(x,y)$.We calculated the PSNR using Eq. (21), as shown in Fig. 5, and the results show that the difference in imaging quality between the two schemes is not significant, with DHGI being slightly more noise resistant than DPHGI. However, DPHGI still has an advantage in terms of the number of measurements. From the second-order correlation functions of Eq. (16) and Eq. (19) and the relationship between the positive and negative pattern of the two schemes, it can be seen that the results of the two schemes are in principle the same, but the differences in the experimental operation produce differences in the noise suppression. DHGI effectively suppresses noise, while since DPHGI multiplies each measurement by 2 and then subtracts the first measurement (as in Eq. (18)), it doubles the error as well. Besides, the subtracted measurement is a constant, so the robustness of this scheme is poor. On the other hand, the noise has a very small order of magnitude and has little effect on the overall quality of the reconstructed images. Overall, this proves the feasibility of using HCGI with a single-round measurement scheme.

Fig. 4. Experimental results at different sampling rates and the corresponding number of measurements. Row (a) is the image reconstructed using the DHGI, row (b) is the image reconstructed using the DPHGI.

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Fig. 5. Comparison of PSNR of reconstructed images at different sampling rates for DPHGI and DHGI.

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From the experimental point of view, the problem of negative illumination is solved by adding a constant to the whole matrix (see Fig. 6 ), which removes all the negative elements in the matrix. For such an operation, one does not need to modify the observation matrix. However, since a known auxiliary element is added to each pattern, a correction is needed for each measured signal (in DPHGI, each detector value is $B_1$ larger than the detector value in normal mode). Remarkably, this method can be extended to other orthogonal matrices, as it is summarized in the following equations

(21)$$\begin{array}{l} \hat A_{+}=\hat A+\left|A_{\min }\right| \hat I^{\prime} \\ B_{+}=\left(\hat A+\left|A_{\min }\right| \hat I^{\prime}\right) O=B+\left|A_{\min }\right| \hat I^{\prime} O \\ B=B_{+}-\left|A_{\min }\right| \hat I^{\prime} O. \end{array}$$

where $\hat A$ is an orthogonal matrix and $\hat I^{\prime }$ denotes a matrix of ones.

Fig. 6. Schematic diagram of the single-round measurement scheme (The effect of projecting 1,0 elements with DMD is the same as projecting 2,0 elements, so there is no division by 2 here).

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The bucket signal obtained by the all-ones matrix projection contains the transmittance information of the whole target object, and plays an important role in the scheme as the basic unit for correcting the bucket signal. We have simulated this procedure with several common orthogonal pattern spots, and good reconstruction results have been obtained. We thus foresee application of our method to other orthogonal pattern spot-based computational ghost imaging.

4. Conclusions

In conclusion, we have proposed a single-round measurement scheme to solve the "negative illumination" issue in ghost imaging experiments based orthogonalized patterns. By analyzing the form of the second-order correlation function in HCGI experiments, we have put forward the DPHGI scheme, which has been experimentally tested and assessed. Results have shown that DPHGI is able to recover the image information with half the number of samples of DHGI. Upon combining this scheme with the Walsh Hadamard transform, one may recover high quality images using significantly less measurements.

In addition, by further analyzing the DPHGI scheme, we have proposed a method to solve the negative illumination problem by translating the pattern structure and eliminating the negative values, which is theoretically applicable to any computational ghost imaging experiment based on orthogonality patterns. We believe that the proposed scheme may be practically relevant for computational ghost imaging. In future research, we are going to apply the single-round measurement scheme to more CGI experiments.

Funding

Natural Science Foundation of Jilin Province (YDZJ202101ZYTS030).

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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Differential Hadamard ghost imaging via single-round detection

Abstract

1. Introduction

2. Theoretical model analysis

3. Experiments and discussions

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (21)

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