## Abstract

A spatial multiplexing reconstruction method has been proposed to improve the sampling efficiency and image quality of Fourier-transform ghost imaging. In this method, the sensing equation of Fourier-transform ghost imaging is established based on recombination and reutilization of the correlated intensity distributions of light fields. It is theoretically proved that the scale of the sensing matrix in the sensing equation can be greatly reduced, and spatial multiplexing combined with this matrix reduction provides the feasibility of ghost imaging with just a few measurements. Experimental results show better visibility and signal-to-noise ratio in the Fourier spectrums reconstructed via spatial multiplexing compared with previous methods. The transmittance of an object is also recovered in spatial domain with better image quality based on its spectrum of spatial multiplexing reconstruction. This method is especially important to x-ray ghost imaging applications due to its potential for reducing radiation damage and achieving high quality images in x-ray microscopy.

© 2018 Optical Society of America

## 1. Introduction

Ghost Imaging (GI) has raised more and more interest in the past decades due to its special characteristics and wide applications. The first experiment about GI was demonstrated with an entangled source generated by spontaneous parametric down-conversion in 1994 [1]. Bennink and his colleagues performed a coincidence imaging experiment with a classical source in 2002 [2]. And theoretical GI schemes exploiting the correlation of two classical beams obtained by splitting incoherent thermal radiation were proposed in 2004 [3, 4]. Soon the experimental demonstration of GI with pseudo-thermal sources and true thermal lights were reported [5, 6]. After that, lots of relevant results including computational, lensless, differential, high-order and turbulence-free GI have been achieved [7–12] and ghost imaging has found its applications in remote sensing, super-resolution, security inspection and x-ray diffraction etc [13–19].

Although ghost imaging have prospered with visible light, it was not realized in the x-ray regime until recently. X-ray Fourier-transform GI of a complex amplitude sample was experimentally demonstrated with a pseudo-thermal source in 2016 [20]. And the real-space x-ray GI experiment was also performed [21]. Nevertheless, in order to achieve high quality images of samples, a great many of measurements are required in x-ray ghost imaging to calculate the average of ensemble, which may result in low efficiency and radiation damage. Fortunately, ghost imaging via sparsity constraints(GISC) was proposed by combining GI methods with the theory of compressing sensing [14, 16, 18], which provides the approach of high quality ghost imaging with less measurements and has already been applied in three dimensional lidar and multispectral camera [22, 23]. In GISC, data acquisition can be treated as an information encoding process. The sample’s information is encoded by the fluctuation of light fields and can be reconstructed from the sensing equation with prior knowledge. Similar method was proved to be effective in Fourier-transform ghost imaging [20, 24]. However, the scale of the sensing matrix is proportional to the square of the image size theoretically, which makes the reconstruction very time consuming and less applicable to high resolution imaging. More importantly, in Fourier-transform ghost imaging, much more correlated information can be obtained simultaneously by simply using a multipoint detector behind the sample, but existing reconstruction method does not take full advantage of it.

In this paper, a spatial multiplexing reconstruction method for Fourier-transform GISC has been proposed to improve the sampling efficiency and image quality of Fourier-transform ghost imaging. The sensing equation of Fourier-transform GISC is established based on recombination and reutilization of the intensity data, and the scale of the sensing matrix is just proportional to the image size. Relevant experiments were performed, and the results are in agreement with theoretical predictions.

## 2. Theory and method

Figure 1 shows the scheme of Fourier-transform GISC. The laser illuminates a rotation ground glass to produce pseudo-thermal light. A beam splitter divides the pseudo-thermal light into two paths: in the reference arm, the pseudo-thermal light propagates freely to CCD-1; in the test arm, the light transmits the object and detected by CCD-2. Different from traditional single point detector scheme, the detector in the test arm is a multipoint detector. Thus, intensity correlation at different spatial points can be obtained simultaneously, and the intensity signals can be multiplexed to form a more effective sensing equation.

The light field on the detector plane in the reference arm can be described as:

*E(x*

_{0},

*y*

_{0}

*)*is the optical field in the source plane,

*h*(

*x*

_{0}, y_{0}; x_{1},

*y*

_{1}) is the free-space transfer function from source to detector in the reference arm. Suppose the area of the detector is large enough, we can treat the source as a phase-conjugation mirror [6, 24, 25]. Thus, the light field in the front plane of the object can be written as:

*h(x*

_{1},

*y*

_{1}

*; x, y)*is the free-space transfer function from source to object in the test arm.

Then, we have the light field on the detector plane of the test arm

*t(x, y)*is the transmittance of the object,

*h(x, y; x*

_{2},

*y*

_{2}

*)*is the free-space transfer function from object to detector in the test arm.

The free-space transfer function from plane *(x, y)* to plane *(ζ*, *η)* is

*λ*is wavelength,

*k*is the wave vector and

*z*is the propagation distance. Substituting Eqs. (1) and (2) into Eq. (3) and using the transfer function in Eq. (4), when satisfying ${d}_{1}={d}_{21}+{d}_{22}$, we have

*T*is the Fourier transformation of

*t(x*,

*y)*.

Ignoring several constants, the intensity distribution at the detector of the test arm [26] is

^{9}. To solve such an equation, is it is quite demanding for computation hardware.

In fact, Eq. (6) can be simplified. The light from a thermal source will become partially coherent after free-space transportation, and the intensity distribution of this kind of partially coherent light can be described as the Gaussian laser beam distribution [27]. Then the pseudo-thermal light has the degree of spatial coherence

*σ*is the linear dimension of the source,

_{Q}*σ*is the width of spatial coherence,

_{L}*A*is a constant. Considering the property of the first order intensity correlation function [27], we have

_{L}*q*indicates the

*q*-th measurement in ghost imaging. After

*m*measurements, for the fixed point $P({x}_{2i},{y}_{2j})$ on the detector of the test arm, the sensing equation is:

**Ι**

_{1}

*(*

^{q}*q*= 1, 2 …

*m*) is a row vector formed by the intensities recorded in the reference arm in each measurement,

**X**is a column vector corresponding to the squared modulus of the Fourier transformation of the object’s transmittance.

At present, we just use the intensity information of one point $P({x}_{2i},{y}_{2j})$ on CCD-2, however, the detector of the test arm is a multipoint detector. Assuming the detecting area is large enough, we can choose a window centered at point $P({x}_{2},{y}_{2})$ as the integral domain, the size of which is determined by the image resolution requirements. Then we have

**X**will be identical. That is to say, we can concatenate the sensing equation of each different point $P({x}_{2},{y}_{2})$ as following

*n*points here). If $X\in {R}^{N}$, the scale of the sensing matrix is $A\in {R}^{mn\times N}$. Compared with the theoretical scale $A\in {R}^{m\text{'}\times {N}^{2}}$ in [24], our method is more practical in computation, which is benefited from the simple form of Eq. (11). Furthermore, in terms of mathematical solution, the required measurements can be cut down. Especially, if

*n*is large enough,

*m*can be very small in practice. Even if

*N*

^{2}is reduced to

*N*by conjecture in [24], the $m\text{'}$ measurements required for reconstruction is much larger than the

*m*measurements required in our method.

There are spatial overlap between the windows of different $P({x}_{2},{y}_{2})$, which means the intensity signal at each spatial point on the reference detector is reorganized and reutilized in different sensing matrix **A*** _{p}*. It is important to point out that this spatial multiplexing can further improve sampling efficiency. The mutual coherence of the sensing matrix in our method is smaller than the case of traditional single point method (detailed in Appendix). According to the theory of compressive sensing, the smaller the mutual coherence, the less the required measurements. Therefore, better image quality is foreseeable under the same measurements compared with traditional methods, and Fourier-transform ghost imaging with just a few measurements can be expected.

Using similar methods adopted in other GISC applications [14, 16, 18, 24, 28], the signal **X** is reconstructed by solving the following optimization

**X**into an image according to Eq. (12). And the spectrum can be transformed to the spatial frequency domain by using the factor $\lambda {d}_{22}$ in coordinates. The standard hybrid input-output (HIO) retrieval algorithm [30] is adopted in this paper to recover the object’s transmittance in the spatial domain.

## 3. Experimental results

The scheme of our Fourier-transform GISC experiments is showed in Fig. 1. The pseudo-thermal light was generated by illuminating a rotating ground glass with a 532*nm* laser source, and the diameter of the laser beam was 5*mm*. The distances in our experiments were ${d}_{21}=32.5cm$, ${d}_{22}=10cm$,${d}_{1}=42.5cm$. The pixel size of the CCD cameras was $\Delta {x}_{CCD}=4.6\mu m$. The coherence size on the CCD plane was 46*μm* which was obtained by calculating the autocorrelation of the intensity distributions recorded by CCD-1, and it is consistent with the theoretical value 45.2*μm* derived according to Van Cittert-Zernike theorem.

In the first experiment, we used a double-slit sample. The width of each slit was 90*μm* and the distance between the slits was 215*μm*. Thus, the diffraction peak spacing should be 247.4*μm* theoretically. Figure 2 shows the Fourier spectrums of the double slits obtained with different reconstruction methods. Figure 2(a) is the result of traditional intensity correlation reconstruction, Fig. 2(b) is the Fourier spectrum retrieved using standard single point reconstruction, and Fig. 2(c) is the Fourier spectrum retrieved using spatial multiplexing reconstruction of 40 points. The measurements to obtain the results in (a)(b)(c) are the same number 800. It is almost unable to recognize the Fourier spectrum in Fig. 2(a), and it is hardly to distinguish the diffraction peaks in Fig. 2(b). While in Fig. 2(c), we see the spectrum of the double slits clearly, and the diffraction peak spacing is 248.4*μm*, which is close to the theoretical value. The peak spacing varies slightly with different random points, and the results converge when sufficient amount of points are multiplexed. Thus, the method of spatial multiplexing reconstruction dramatically improve the image quality compared with previous two methods. Beyond this, we can achieve high quality Fourier spectrum with fewer measurements by means of utilizing more points in spatial multiplexing as demonstrated in Fig. 2(d). The measurements is only 40, which is much less than the measurements required in those traditional methods. The points used for spatial multiplexing is 1000. It is obviously that the spectrum in Fig. 2(d) is as clear as the result in Fig. 2(c). Therefore, by increasing the number of multiplexed points, we can reduce the number of measurements required in Fourier-transform GISC while maintaining the image quality.

In the second experiment, we used a 1.42 × 1.42*mm*^{2} object with letter ‘GI’ in center. The distance parameters are the same as those in the first experiment. The number of measurement is 1200. Figure 3 shows the result Fourier spectrums reconstructed in different ways and the object recovered from corresponding spectrums. Figure 3(a) and Fig. 3(c) are the results of spatial multiplexing reconstruction of 100 points, while Fig. 3(b) and Fig. 3(d) are the results obtained using standard single point reconstruction. The spatial frequency interval in Fig. 3(a) and Fig. 3(b) is ${f}_{0}=\Delta {x}_{CCD}/(\lambda {d}_{22})=86.5{m}^{-1}$, and the dimension of the Fourier spectrums used in the phase retrieval is 1000 × 1000 pixels, so the pixel size in Fig. 3(c) and Fig. 3(d) is $1/{f}_{\mathrm{max}}=1/(500{f}_{0})=23.1\mu m$. It is obviously that the spectrum of spatial multiplexing reconstruction is better and the recovered letters are more legible.

In order to evaluate the image quality quantitatively, a function indicating the visibility of an image is defined as $\delta =\frac{1}{{N}_{x}{N}_{y}}{\displaystyle \sum _{i,j=1}^{{N}_{x}{N}_{y}}\frac{I{(i,j)}_{\mathrm{max}}-I(i,j)}{I{(i,j)}_{\mathrm{min}}+I(i,j)}}$, where *I(i,j)* is the intensity at pixel *(i,j)* and *I(i,j) _{max}* is the maximum intensity in the image [31].

Another way to estimate the image quality is the signal-to-noise ratio (SNR). Figure 4 shows the statistical distributions of the Fourier spectrums in Fig. 3. The full width at half maximum (FWHM) presents the level of noise, and the difference between the maximum intensity and the background intensity provides the signal information. The blue curve has a narrower peak and a larger maximum value than the red one, which indicates better performance in SNR. The parameter to describe the performance can be defined as $\alpha =({I}_{\mathrm{max}}-{I}_{background})/FWHM$. The results of these two kinds of evaluation are demonstrated in Table 1. It can be found that the spectrum reconstructed via spatial multiplexing has better visibility and SNR, which guarantees the successful recovery of the object in spatial domain.

## 4. Conclusion

We propose a spatial multiplexing reconstruction method which can be applied in Fourier-transform ghost imaging to obtain high quality Fourier spectrums of samples. We establish the sensing equation of Fourier-transform GISC by recombining and reutilizing the correlated intensity information of light fields. It is theoretically proved that the scale of the sensing matrix can be reduced dramatically. The column dimension of the sensing matrix in our method is equal to the image size, while previously it is equal to the square of the image size. Theoretical analysis and experimental results show that the sampling efficiency of Fourier-transform ghost imaging is greatly improved by adopting spatial multiplexing. High quality spectrum of a double slits is obtained with only 40 measurements, which is much less than the measurements required by traditional methods. While under the same number of measurements, the Fourier spectrums reconstructed via spatial multiplexing have better visibility and signal-to-noise ratio than traditional reconstructions. A sample’s recovered image in spatial domain with better image quality has also been obtained from its spectrum of spatial multiplexing reconstruction. This technology has significant applications in Fourier-transform ghost imaging, due to its potential for reducing x-ray radiation damage and achieving high quality images in x-ray microscopy, and it can also be used in high-speed remote sensing with visible light.

## Appendix

The definition of the mutual coherence of a matrix [32] is

**a**

*is the column vector in matrix $A\in {R}^{m\times n}$, angel brackets represent the inner product, and ${\Vert \xb7\Vert}_{2}$represents the*

_{i}*l*

_{2}-norm. Without loss of generality, we set the column vector

**a**

*as normalized vector.*

_{i}We use the notation **A*** _{s}* to represent the sensing matrix of single point, because the entries of the sensing matrix are non-negative, we have

**A**

*represents the sensing matrix of multiple points, then*

_{m}*P*arises as a normalized factor and

**a**

*indicates*

_{is}*i*-th column of single point’s sensing matrix

**A**

*. The inequality sign is because we change the order of sum function and maximize function. The equation above gives the upper bound of the mutual coherence of a sensing matrix of multiple points. Physically, the mutual coherence of different points should be invariant, then the right-hand side of the equation tends to be the mutual coherence of single point. Therefore, the mutual coherence of the sensing matrix of multiple points is smaller than that of single point.*

_{s}## Funding

National Natural Science Foundation of China (NSFC) No. 11627811; National Key Research and Development Program of China No. 2017YFB0503303.

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