Spread spectrum ghost imaging

Jinfen Liu; Jinfen Liu; Le Wang; Shengmei Zhao; Shengmei Zhao

doi:10.1364/OE.442390

1. Introduction

Ghost imaging (GI), by using the intensity correlations between the different optical fields to reconstruct the target object, is known now as an attractive imaging technique [1], and has received more and more attention recently [2–14]. In the conventional GI system, there are two optical beams. One is called a signal beam, where the beam illuminates the object, and then is collected by a bucket detector. The other one is named a reference beam, whose light intensity is measured by a resolving resolution detector, such as the charge-coupled device (CCD) or the complementary metal oxide semiconductor (CMOS) camera. Sooner, computational ghost imaging (CGI), was proposed [5,6] to simplify GI’s configuration, where the light speckle patterns in the reference beam are not by detection but by calculation [15]. GI offers great promise for its better robustness against harsh environment [3,4] and multiple types of information processing [5,6].

However, there are still some obstacles to prevent its practical applications [16–27]. One of them is that the time varying noise from environment, which significantly degrades the imaging quality. Thus, a lot of methods have been presented to overcome the obstacle. For example, the interpolation computational ghost imaging (ICGI) was proposed to estimate the noise by using a specific assay pattern during periodic illuminations to correct the bucket detector signal [24], and the results were demonstrated to attenuate the interference from background light noise greatly. In [25], Soltanlou et al. demonstrated a method that could remove the time varying noise and constant background noise by normalization and DC blocking, respectively. Lately, instant ghost imaging (IGI) was presented to eliminate the spatiotemporally varying optical background noise, by using the difference between two consecutive signal measurements [26]. Besides, T. Torii et al. proposed a novel scheme named time division ghost imaging (TDGI), by dividing the total number of patterns in the calculation process of the correlation function into the sub-units with fewer illuminated patterns, to improve the quality of the image in noisy environment [27].

On the other context, the spread spectrum technique has a great advantage to withstand both intentional and unintentional interference in a communication system [28–32]. Spread spectrum is a means of transmission, the band spread is accomplished by means of a code which is independent of the data, and a synchronized reception with the code at the receiver is used for despreading and subsequent data recovery. Some benefits can accrue simultaneously if the spread spectrum technique is done properly, such as, antijamming, antiinterference, low probability of intercept, etc [28]. There are plenty of practical methods, including frequency modulation (FM), direct sequence modulation, frequency hopping [29,30], and various interference rejection filters [31,32], to suppress interference in a spread spectrum communication system.

In the paper, we propose a novel GI scheme based on the direct sequence spread spectrum technique to overcome the time varying noise in GI’s system, named spread spectrum ghost imaging (SSGI). In the scheme, GI is regarded as a communication system, where the light source is regarded as the transmitter, the illumination process is considered as the transmission, and the bucket detector is treated as the receiver. The source is used to produce the light beam, whose light intensity varies according to the designed direct sequence code, and the digital mirror device (DMD) is later utilized to modulate the light beam with a spatial distribution. Then the bucket detector is used to collect the light transmitted through or reflected from the object, and the resultant bucket value is despreaded by the same direct sequence code, called computed bucket value. Finally the object can be recovered by utilizing the speckle patterns and corresponding computed bucket values. The interference suppression of the proposed scheme on the time varying noise is discussed by simulations and experiments.

The contribution of this paper is that the direct sequence spread spectrum technique is adopted in the GI system to present the stronger noise resistance capability for the first time, where the GI system is regarded as a communication system. The usage of the effective signal processing methods developed in communication systems to the GI system provides a way for new GI schemes. The results show that the ability of the proposed scheme to withstand interference under the lower signal-to-noise ratio (SNR) could present the great potential of GI in practical applications.

The organization of the paper is as follows. In Section 2, the spread spectrum ghost imaging scheme is proposed and its theoretical analysis is also presented. In Section 3, the performance of SSGI is discussed by numerical simulations and experiments. Finally, Section 4 summarizes the paper.

2. Theory

A schematic diagram of the SSGI scheme is shown in Fig. 1. The whole imaging process is viewed as a communication system. Here, a light emitting diode (LED) is regarded as the source, which is used to produce a light beam, whose light intensity is first modulated by the designed direct sequence code, $c(t)$, described as [30],

(1)$$c(t)=c_{k},kT_{c} \le t< (k+1)T_{c}$$

(2)$$c_{k}\in \left \{{-}1,1 \right \}$$

where $\left \{ c_{k} \right \}$ is a pseudonoise (PN) sequence, $T_{c}$ refers to the PN sequence bit time interval, and $k$ is an integer. Here,

(3)$$N=T/T_{c}$$

is the length of the direct sequence code, where $T$ is the duration of a speckle pattern. For simplicity, we assume that $T=1$.

Fig. 1. A schematic diagram of the SSGI scheme. LED: Light emitting diode. DMD: Digital mirror device. T: Duration of a speckle pattern. c(t): Waveform of direct sequence code.

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Then the light beam is modulated by the DMD to obtain the temporal-spatial speckle patterns, $S_{i}(x,y,t)$,

(4)$$S_{i}(x,y,t)=I_{i}(x,y)c(t)$$

where $I_{i}(x,y)$ refers to the $i$th speckle pattern.

Let $J_{i}(x,y,t)$ be the time varying noise, whose energy or average intensity varies with time, such as, the noise caused by the experimental environment, the sinusoidal noise generated by the fluorescent lamp [27,33]. For simplicity and generality, the time varying noise $J_{i}(x,y,t)$ is given as,

(5)$$J_{i}(x,y,t)=J_{i}(x,y)n(t)$$

where $J_{i}(x,y)$, $n(t)$ represent the spatial distribution and time distribution of the noise, respectively. Hence, the speckle pattern illuminating on the object becomes,

(6)$$Z_{i}(x,y,t)=S_{i}(x,y,t)+J_{i}(x,y,t)$$

At the receiver, the bucket detector is used to collect the light transmitted through or reflected from the object, denotes as $b_{i}(t)$,

(7)$$b_{i}(t)=\int _{A}Z_{i}(x,y,t)T(x,y)dxdy$$

where $T(x,y)$ is the reflection/transmission function of the object, $A$ is the illuminated region by $Z_{i}(x,y,t)$.

Afterwards, the bucket detection value is despreaded by the same direct sequence code waveform, named computed bucket value $B_{i}$, which can be achieved as,

(8)$$\begin{aligned}B_{i}&=\int_{0}^{T}b_{i}(t)c(t)dt=\int_{0}^{T} \left [ \int _{A}Z_{i}(x,y,t)T(x,y)dxdy \right ]c(t)dt\\ &=\int_{A}I_{i}(x,y)T(x,y)dxdy\int_{0}^{T}c(t)c(t)dt+\int_{A}J_{i}(x,y)T(x,y)dxdy\int_{0}^{T}n(t)c(t)dt\\ &=\int_{A}I_{i}(x,y)T(x,y)dxdy+\int_{A}J_{i}(x,y)T(x,y)dxdy\int_{0}^{T}n(t)c(t)dt\\ &=B_{i}^{s}+B_{i}^{J}\int_{0}^{T}n(t)c(t)dt\\ &=B_{i}^{s}+\varepsilon B_{i}^{J}\end{aligned}$$

where $B_{i}^{s}$ represents the true bucket values, $B_{i}^{J}$ refers to the noise items to the imaging process, and $\varepsilon$ is a factor. Finally, the object can be reconstructed according to the second-order correlation algorithm [9] as,

(9)$$\begin{aligned}O_{SSGI}(x,y)&=\frac{1}{P}\sum_{i=1}^{P}\left ( B_{i} -\left \langle B \right \rangle\right )I_{i}(x,y)\\ &=\frac{1}{P}\sum_{i=1}^{P} ( B_{i}^{s} +\varepsilon B_{i}^{J}-\left \langle B^{s} \right \rangle-\varepsilon \left \langle B^{J} \right \rangle)I_{i}(x,y)\\ &=\frac{1}{P}\sum_{i=1}^{P}(B_{i}^{s}-\left \langle B^{s} \right \rangle)I_{i}(x,y)+\frac{1}{P}\sum_{i=1}^{P}(\varepsilon B_{i}^{J}-\varepsilon \left \langle B^{J} \right \rangle)I_{i}(x,y)\\ &=O^{T}(x,y)+\varepsilon O^{J}(x,y)\end{aligned}$$

where $\left \langle \cdot \right \rangle$ denotes the assemble average, $\left \langle B^{s} \right \rangle =\frac {1}{P}\sum _{i=1}^{P}B_{i}^{s}$, $\varepsilon \left \langle B^{J} \right \rangle = \varepsilon \frac {1}{P}\sum _{i=1}^{P}B_{i}^{J}$, and $O^{T}(x,y)$ represents the true reconstructed results, $O^{J}(x,y)$ denotes the influence on the reconstructed results generated by the noise, $P$ is the number of the bucket detection results, $N$ is the length of the direct sequence code (the number of measurement results for each detection in TGI). Obviously, when $\varepsilon$ is close to zero, the object should be perfectly reconstructed.

We further analyse the noise items. In SSGI scheme, the undesired bucket value caused by the noise could be reduced to its $1/N$.

The detailed proof is shown as follows. When the time function $n(t)$ of the noise is multiplied by the PN waveform $c(t)$, the resulting noise output is

(10) $$j(t)=n(t)c(t)$$

Considering that the time function $n(t)$ of the noise is a stationary random process with autocorrelation

(11)$$R_{n}(\tau )=E\left \{ n(t+\tau )n(t ) \right \}$$

where $E$ represents the mathematical expectation, and its power spectral density is defined as $S_{n}(f)$.

Additionally, the PN waveform $c(t)$ is also stationary, whose autocorrelation is

(12)$$R_{c}(\tau )=E\left \{ c(t+\tau )c(t ) \right\}=\left\{ \begin{array}{ll} 1-\frac{|\tau|}{T_{c}}, &|\tau| \le T_{c}\\ 0, & |\tau| > T_{c} \end{array} \right.$$

and its power spectral density is defined as $S_{c}(f)$,

(13)$$S_{c}(f)=T_{c}\left [ \frac{sin\pi fT_{c}}{\pi fT_{c}} \right ]^{2}$$

Since $n(t)$ and $c(t)$ are independent, the autocorrelation of $j(t)$ is

(14)$$\begin{aligned}R_{j}(\tau )&=E\left \{ j(t+\tau )j(t) \right \}\\ &=E\left \{ n(t+\tau )n(t) \right \}E\left \{ c(t+\tau )c(t) \right \}\\ &=R_{n}(\tau )R_{c}(\tau )\end{aligned}$$

which has the power spectral density as,

(15)$$S_{j}(f)=S_{n}(f)\ast S_{c}(f)$$

where $\ast$ indicates the convolution operation. $S_{n}(f)$ is arbitrary and $S_{c}(f)$ is a broad $sin^{2}x/x^{2}$ spectrum of bandwidth roughly $1/T_{c}$ according to Eq. (13). The resultant spectrum $S_{j}(f=0)$ is

(16)$$S_{j}(0)=\int_{-\infty }^{\infty }S_{n}(f)S_{c}(f)df\leq S_{c}(0)\int_{-\infty }^{\infty }S_{n}(f)df= S_{c}(0)B_{i}^J$$

and $B_{i}^J$ is the total noise intensity detected by the bucket detector from $i$th detection. Since

(17)$$S_{c}(f)\leq S_{c}(0)=\int_{-\infty }^{\infty }R_{c}(\tau )d\tau =\int_{{-}T_{c}}^{T_{c}}(1-\frac{\left | \tau \right |}{T_{c}})d\tau =T_{c}$$

Thus

(18)$$S_{j}(0)\leq B_{i}^J T_{c}=\frac{B_{i}^J}{N}$$

Note that the equivalent noise intensity bounded by $\frac {B_{i}^J}{N}$ represents the resulting noise intensity computed from $i$th despreading operation. Therefore, it is theoretically deduced that the interference of the intensity fluctuations of the time varying noise, measured by the detector, is reduced to $\frac {B_{i}^J}{N}$ after $i$th despreading operation, that is, $\varepsilon =\frac {1}{N}$. Hence, Eq. (8) can be rewritten as,

(19)$$B_{i}=B_{i}^{s}+\frac{1}{N} B_{i}^{J}$$

3. Numerical simulations and experimental results

In order to evaluate the performance of SSGI scheme, the mean square error (MSE) [34] is introduced to estimate the imaging quality, which is defined as,

(20)$$MSE=\frac{1}{L} \sum_{x,y} (O(x,y)-T(x,y))^2$$

where $O(x,y)$ and $T(x,y)$ describe the original and the reconstructed image respectively. $L$ is the number of pixels of the reconstructed image.

The experimental setup for the SSGI scheme system is shown in Fig. 2. Two red LEDs (LED1 and LED2) are controlled by an Arbitrary Waveform Generator (Keysight 33612A) with a waveform $c(t)$ and $n(t)$, where $c(t)$ is a PN binary waveform, whose frequency is 2.02kHz and amplitude is 10vpp. $n(t)$ is a random noise waveform built-in the Arbitrary Waveform Generator, whose frequency and amplitude are variable. DMD (ViALUX V-7001) is driven by a computer to modulate the light beam from LED1 to generate speckle patterns. The light beam from LED2 illuminates directly onto the object, and the photodetector (Thorlabs PDA100A-EC) is used to collect the total transmission light intensity. Then the measured results $b_{i}(t)$ are recorded via an analogue-to-digital converter (NI USB-6351). Thus, the computed bucket value $B_{i}$ can be achieved by despreading the bucket detector values with the same direct sequence code. Finally we can reconstruct the image of the object by using the modulated speckle patterns and corresponding computed bucket values.

Fig. 2. An experimental setup for the SSGI scheme system. LED: Light emitting diode. DMD: Digital mirror device. L: Lens. PD: Photodetector. ADC: Analog-to-digital converter.

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Here, SNR is used to evaluate the ratio of the signal power to the noise power

(21)$$SNR(dB)=10 log_{10} \frac{P_{s}}{P_{n}}$$

where $P_{s}$ and $P_{n}$ are the signal power and the noise power respectively. $P_{s}$ ($P_{n}$) is the average of bucket values detected by the photodetector when there are only speckle patterns (noise).

We first verify the feasibility of SSGI by numerical simulations in Fig. 3, where a grayscale image (’Circle’) and a binary image (’NUPT’) with $64\times 64$ pixels are selected as the objects. Here, both binary random speckle patterns and Walsh-Hadamard speckle patterns are adopted as the illuminating speckle patterns, and the SNR of the random time varying noise is −3dB. To provide a fair comparison, we present the TGI with the $1 \times P$, and $N \times P$ measurements together. For TGI ($N = 100$), the images are reconstructed by $P$ detection results according to $P$ speckle patterns, where each detection result is averaged over $N$ measurement results. For TGI ($N = 1$), the images are recovered with $P$ detection results, each detection result is one measurement result. For SSGI ($N = 100$), the images are reconstructed by $P$ detection results, where each detection result is despreaded from $N$ measurement results by the spread spectrum code. The results show that the performance of TGI $(N=1)$ and TGI $(N=100)$ are almost the same. Their MSE values are very closer, while the proposed scheme has a better performance. It is also indicated that the advantage of our proposed scheme is based on the spread spectrum technique. In addition, we can see that as the number of speckle patterns increases (binary random speckle patterns), the imaging quality improves gradually for the proposed SSGI schemes. Compared with TGI ($N = 100$), the reconstructed images by SSGI are more clearer and their corresponding MSEs are more smaller. In addition, the images reconstructed by SSGI by using $4096$ Walsh-Hadamard speckle patterns are better than those using $20000$ random speckle patterns, the corresponding MSEs have one order of magnitude in comparison with those using $20000$ random speckle patterns. Simultaneously, the corresponding MSEs have two order of magnitude in comparison with those of TGI ($N = 100$), while there is little MSEs variation for TGI($N = 100$) with binary random speckle patterns and Walsh-Hadamard speckle patterns. Therefore, we do the simulations and experiments with Walsh-Hadamard speckle patterns at later.

Fig. 3. The simulated results by TGI($N = 1$), TGI ($N = 100$) and SSGI($N = 100$) , where SNR = −3dB.

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Fig. 4 shows the simulations and experimental results by SSGI and TGI schemes under different SNRs. Here, the length of the direct sequence code is $N = 100$, the size of the object is $64\times 64$ pixels. The frequency of the noise $n(t)$ is set as 25Hz, the amplitude of the noise $n(t)$ is setup at a range from $1$ to $9$ vpp. Both simulations and experimental results show that as the SNR increases, both schemes has a better performance. Compared with TGI scheme, SSGI scheme could achieve better images and lower MSEs values for each SNR. When the images reconstructed by TGI are completely blurry, say SNR = −2.12dB, one can have very clearer images by using SSGI. It is indicated that the proposed SSGI has a better anti-noise performance.

Fig. 4. The simulated and experimental results by SSGI and TGI schemes under different SNRs.

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Later, we analyze the influence of the direct sequence code length on the reconstructed imaging quality by SSGI. Here, the frequency of the noise $n(t)$ is set as 25Hz, and the amplitude of the noise $n(t)$ is setup as $9$vpp. For ’Circle’ image, the SNR is −2.12dB, and for ’NUPT’ image, the SNR is −4.63dB. The length of the direct sequence code is setup to 50 and 100. The experimental results by using SSGI with different direct sequence code lengths, together with the results by using TGI, are shown in Fig. 5, where Fig. 5(a) shows the reconstructed images with the size of $32\times 32$ pixels, and Fig. 5(b) shows the restored images with the size of $64\times 64$ pixels. Fig. 5 shows that the reconstructed images by SSGI are improved when the direct sequence code lengths increase. Furthermore, the reconstructed images are nearly close to the perfect reconstruction when the direct sequence code length is 100. For both binary and grayscale objects, the reconstructed images by SSGI are better than those by TGI, even the direct sequence code length is 50, there are one order of magnitude improvement in MSEs values in comparison with those of TGI.

Fig. 5. The experimental results by using SSGI with different direct sequence code lengths, together with the results by using TGI. (a):32x32 pixels.(b):64x64 pixels.

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Fig. 6 further shows the MSEs curve against different SNRs by using SSGI schemes with different lengths of direct sequence code, together with the MSEs curve against different SNRs by using TGI. The object is a grayscale ’Circle’ image with the size of $64\times 64$ pixels. The results show that the imaging quality of SSGI is always better as the SNR increases. When the direct sequence code length is 100, the performance of SSGI is greatly improved. However, the imaging quality of TGI does not obviously improve even when the SNR is 5.30dB, which hints that SSGI performs much stronger noise resistance in practical environment.

Fig. 6. The experimental results of reconstructed ’Circle’ images’ MSEs as a function of SNR for SSGI schemes with different lengths of direct sequence code, together with the results by using TGI scheme.

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4. Conclusion and discussion

In this paper, we have proposed a novel GI scheme to overcome the time varying noise in practical environment with spread spectrum technique, named spread spectrum ghost imaging, where the GI system has been regarded as a communication system, and the speckle patterns have been modulated by the direct sequence code. The numerical simulations and experimental results have shown that the reconstructed images by using SSGI always have a better performance than those reconstructed by using TGI, and the reconstructed images have a better quality with a longer direct sequence code. For the $64\times 64$ pixels image, the MSEs values calculated by the proposed SSGI are almost unchanged with the growth of SNR when the length of the direct sequence code is 100, indicating our SSGI scheme has a great potential in protection against interfering. Therefore, SSGI will become a promising scheme for the applications of GI, such as remote sensing, optical security and imaging in practical environment.

In addition, the temporal modulation used in the proposed scheme looks like the time modulations in temporal ghost imaging [35–38]. However, for temporal ghost imaging, the aim of the temporal modulation (temporal speckle patterns) is to get the ghost image of temporal object in time domain. In our scheme, the purpose of the temporal modulation is to obtain the temporal-spatial speckle pattern so that the time varying noise can be suppressed by the spread spectrum technique. Of course, it can be used for resolving both spatial and temporal signals in theory, if temporal ghost imaging technique are combined with our proposed scheme, since the space and the time are two dimensions. The spatial modulator (e.g. DMD) can be used to modulate the light field with a spatial speckle pattern, and a temporal modulator can be used to modulate the light field with a temporal speckle pattern, then the temporal-spatial image can be reconstructed according to the spatial and the temporal correlation between the temporal-spatial speckle patterns and the bucket detector results. Moreover, the spread spectrum technique can also be used to modulate the temporal speckle patterns to suppress the time varying noise.

Funding

National Natural Science Foundation of China (61871234, 62001249); The Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX20_0729); Natural Science Research of Jiangsu Higher Education Institutions of China (20KJB510030); Priority Academic Program Development of Jiangsu Higher Education Institutions.

Disclosures

The authors declare that there are no conflicts of interest related to this paper.

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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Spread spectrum ghost imaging

Abstract

1. Introduction

2. Theory

3. Numerical simulations and experimental results

4. Conclusion and discussion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (21)

Optics Express