Computational imaging based on time-correlated

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Computational imaging based on time-correlated single-photon-counting technique at low light level YING YANG,1 JIANHONG SHI,1,* FEI CAO,1 JINYE PENG,2

AND

GUIHUA ZENG1,2,3

1

State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Key Laboratory on Navigation and Location-Based Service and Center of Quantum Information Sensing and Processing, Shanghai Jiao Tong University, Shanghai 200240, China 2 College of Information Science and Technology, Northwest University, Xi’an 710127, Shaanxi, China 3 e-mail: [email protected] *Corresponding author: [email protected] Received 14 August 2015; revised 5 October 2015; accepted 5 October 2015; posted 6 October 2015 (Doc. ID 247972); published 28 October 2015

Imaging at low light levels has drawn much attention. In this paper, a method is experimentally demonstrated to realize computational imaging under weak illumination conditions. In our experiment, only one single-photon detector was used to capture the photons. With the time-correlated single-photon-counting technique, photons at a quite low level can be recorded and the time distribution histograms were constructed. The intensity of the light can be estimated from the histograms. The detection model was discussed, and clear images were obtained through a ghost-imaging algorithm. In addition, we propose a modified algorithm for the conventional ghostimaging method that works more efficiently than the traditional ghost-imaging algorithm. Moreover, this method provides a solution for three-dimensional imaging combining with the time of flight of the photons. © 2015 Optical Society of America OCIS codes: (030.5260) Photon counting; (040.3780) Low light level; (100.3010) Image reconstruction techniques. http://dx.doi.org/10.1364/AO.54.009277

1. INTRODUCTION Imaging under weak illumination has drawn much attention in many fields, ranging from biological sciences to astronomy. It’s hard to record the intensity of light even to the single-photon level using the normal methods. Usually, the photon-counting technique is used to detect weak signals in an optical imaging system. Under weak-illumination conditions, detectors with high sensitivity and quantum efficiency are usually required. There are several kinds of detectors with high sensitivity, such as the electron-multiplying CCD (EMCCD), intensified CCD (ICCD), photon multiplier tube (PMT), and avalanche photodiode (APD). Both EMCCDs and ICCDs are array detectors that can achieve limited spatial resolution due to their limited pixels. APDs are a kind of point detector and generally have higher quantum efficiency than PMTs. Single Photon Avalanche Diodes (SPADs, also known as Geiger-mode APDs) are a kind of single-photon detector with high detection efficiency, high time resolution, and low dark counts. They can detect low-intensity signals even to the single-photon level by registering the arrival times of the photons. The singlephoton detector has drawn much attention due to its particular ability to capture photons in ultra-dark environment. It has 1559-128X/15/319277-07$15/0$15.00 © 2015 Optical Society of America

been widely used in a variety of fields such as high-resolution spectral measurement, rigorous analysis, fluorescence lifetime microscopy, quantum information applications [1], ladar ranging systems, and so on. As the point detector has no spatial resolution, its application is limited in many imaging systems. However, ghost imaging provides an intriguing imaging technique and can retrieve the image of the object by using only one point detector. Ghost imaging (GI) [2–9] has generated considerable and increasing interest due to its novel feature. In GI, the image of an unknown object can be retrieved through the intensity correlation between two spatially separated light beams—the object beam and the reference beam. The object light has been transmitted through or reflected from an object and is collected by a bucket detector which has no spatial resolution. The bucket detector serves as a point detector. The reference light, which never interacts with the object, is collected by a detector with high spatial resolution. Correlating the measurements of both detectors can produce an image of the object; however, neither detector’s output alone can yield an image. Computational ghost imaging (CGI) [10], in which the light beam that illuminated the high-spatial-resolution detector

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was unnecessary as its illumination patterns were precomputed, was proposed in 2008. A proof-of-principle experiment for CGI was quickly accomplished in 2009 [11]. The experimental setup was simplified with only a single detector used in the object beam. Over the past few years, there has also been much analysis concerning the SNR [12] and visibility [13,14] of the ghost image. Many improved algorithms were proposed, such as compressive sensing [15,16] and normalized ghost imaging [17]. In recent years, the focus of GI has turned to its practical applications in fluorescence microscopy [18,19], optical encryption [20,21], remote sensing [22], three-dimensional imaging [23], and so on. In these applications, imaging at low light level is an important issue that has actually been investigated over the past few years [24–27]. Most of these researches used an entangled photon source, which is difficult to manipulate in real application cases. In this paper, we focus on realizing GI based on photon counting in very weak illumination cases using a pulsed laser. Considering the basic idea of CGI, it is possible to obtain the image of an object with only a pinhole detector in the object beam. The SPAD can serve as the pinhole detector to provide measurements of the light intensity. SPADs are often used in time-correlated single-photon-counting (TCSPC) applications. The TCSPC technique has been proven to be an enabling optical detection technique for a wide range of applications, including time-correlated fluorescence spectroscopy [28], range finding, depth imaging, and so on. It is a statistical sampling method based on the registration of a single photon repeatedly with respect to a reference signal. The reference signal is usually provided by the corresponding repetitive laser pulse and serves as a synchronization signal. In general, it is necessary to require a high repetitive source to accumulate a large enough number of photon events for a high precision. This technique can achieve high time resolution, which may be ps level, and high sensitivity even to the single-photon level. In this paper, we implement an experiment to realize computational imaging based on photon counting in a weak-illumination environment. Section 2 simply demonstrates the setup of the experiment and the technique used in the experiment. The photons reflected from the object were captured by a SPAD. Combined with the TCSPC technique, the time distribution histogram of the photons was constructed and the intensity was estimated. Section 3 focuses on the reconstruction method of the image. The photon-detection model of the SPAD is discussed and a modified algorithm for GI is proposed. This algorithm can improve the contrast and SNR of images compared with the conventional GI algorithm. 2. EXPERIMENT A. Setup

We implemented the experiment with a supercontinuum pulsed laser with a repetition rate of 5 MHz and a wide range of wavelength (450–2000 nm). The experimental setup is shown in Fig. 1. The pulsed light was irradiated onto the digital micromirror device (DMD) after an attenuation plane (A) and then a beam expander (E). The DMD is an array of micromirrors, and consists of 1024 × 768 independent addressable units.

Fig. 1. Experimental setup. A, attenuation plane; E, beam expander; DMD, digital micromirror device; (L1, L), lens; SPAD, single-photon avalanche diode. TCSPC module: Picoquant HydraHarp 400 (H).

Each unit is a micromirror of 13.68 μm × 13.68 μm with an adjustable angle of
12°. During operation, the DMD controller loaded each underlying memory cell with values of “1” or “0” that were computed offline. As a result, the light projected onto the DMD was modulated to create a random binary speckle field. After modulation, the light could be regarded as a series of random binary speckle patterns with a blackto-white ratio of about 1∶1. These random patterns were projected to the object by a lens (L1) with focal length f 1  50 mm. The distance from DMD to the L1 was about Z 1  6 cm. The object (I) in this experiment was a symbol π made of white paper about 1.5 cm × 1.5 cm in size. It was pasted onto a black paper board at a distance about Z  39 cm from the DMD to satisfy the Gaussian thin lens equa1 . The light reflected from the object was tion f1 ≈ Z11  Z −Z 1 1 first collected by a lens (L) with focal length f  200 mm, and then focused onto a Geiger-mode SPAD. Then the output of the SPAD was input into the TCSPC module [Picoquant HydraHarp 400 (H)]. The synchronous signals of the laser pulses were input into the SYNC IN channel of the HydraHarp 400 and the synchronous signals of the DMD patterns to the M1 channel to maintain the synchronization of the whole system. For every pattern, the corresponding photons reflected from the object were recorded by the SPAD and the arrival times were used to construct a statistical histogram. The intensity of the reflected light could be estimated from the histogram. As has been discussed in [11], given by Van Cittert–Zernike theorem, the transverse resolution δx is determined by the speckle size at the object plane. When it comes to CGI based on DMD, the transverse resolution is actually determined by the minimum size of the speckle in the precomputed pattern and the magnification coefficient of the projecting lens. In our experiment, we chose 8 × 8 micromirrors as the minimum independent unit. Then the side length of the unit is a  8 × 13.68  109.44 μm. Thus, the transverse resolution δx in our experiment is expressed as δx  a × Z − Z 1 ∕Z 1 ≈ 601.92 μm. B. Method

In our experiment, a TCSPC module (HydraHarp 400) was used to record photon events combined with a SPAD. The timing circuits of this module allow high measurement rate—up to

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3. IMAGE RECONSTRUCTION

Fig. 2. Basic principle of TCSPC.

12.5 million counts per second (Mcps)—and provide a time resolution of 1 ps. Its basic principle is shown in Fig. 2. The TCSPC technique is a sampling method to record the photons. It can measure the time delays between the synchronization signal and the arrival times of the individual photons. In its most common form, the technique is used in a weak illumination condition to maintain single-photon statistics so that the probability to register more than one photon per cycle is low and there will actually be no photons registered during many cycles. To obtain the time distribution histogram of the photon events, a stream of photons are recorded; every photon with respect to a synchronous signal generated by the repeated laser pulse. The single-photon events are sorted into different histogram channels according to different time delays. Then a timing histogram of the photons can be constructed by serially incrementing those histogram bins according to the time delays. With a large number of photon events, a temporal distribution histogram of the photons can be accumulated, and this produces a highly accurate representation of photon distribution. Based on the distribution, we can estimate the intensity of the light by simply summing up all of the photon numbers during the distance gate. In the experiment, we attenuated the light to maintain the photon counts to a level 1%–5% of the laser repetition as the TCSPC technique usually required. With a laser of high repetition rate (5 MHz), we got the sampling data quickly. The DMD patterns were set to change every other 10 ms. Consequently, there were 50,000 pulses corresponding with each DMD pattern. We obtained the dark count of the SPAD, which was about 240 counts/s, by turning off the laser and all the background light. We mainly implemented the experiment under three different illuminations, where the average photon counts recorded by the software of the HydraHarp 400 were about (a) 6.81 × 104 ∕s, (b) 9.78 × 104 ∕s, and (c) 2.68 × 105 ∕s. Then the probability to detect one photon during a pulse was about (a) 0.01362, (b) 0.01956, and (c) 0.0536, respectively.

As a Geiger-mode APD was used in our experiment, we need to discuss the model of the detector. It can be modeled as a twopart device as addressed in [29]. The first part is assumed to be a linear photon detection device with unit gain and quantum efficiency ηq. Primary electrons were inspired after this part. The second part will amplify the primary electrons to an extremely high level, and then the signal can be detected. We assume that the coefficient of amplification is ηa . The detection efficiency can be expressed as η  ηq · ηa . The detection efficiency of the SPAD used in our experiments is up to 50% at 550 nm. As had been analyzed in [30,31], in a direct detection laser radar, the number of receiver-integrated photoelectrons from a diffuse target follows a negative-binomial distribution. In typical laser radar applications, the Poisson distribution is a good approximation of the negative-binomial distribution. In a Geiger-mode avalanche detector, the noise mainly comes from the background light and dark counts. The noise electrons also follow Poisson distribution. In a Geiger-mode detector, following every detection event, the device is blind for a period of time called dead time (t dead ). The detector can detect at most one event per dead time. In the following analysis, it is assumed that the period of the laser is T t dead < T  and the width of the laser pulse is ττ ≪ t dead . In fact, in our experiment, the period T is 200 ns and the dead time is about 80 ns. In the following analysis, it is assumed that the mean value of the incident photons is N s and the mean value of the noise photons is N n , so the probability of detecting k events during a pulse can be denoted as ηN sn k e −ηN sn  ; (1) k! where N sn  N s  N n is the mean number of signal plus noise photons. Table 1 shows the probability of different values of k. Note that the probability of detecting more than one event is denoted as Pk 

Pk ≥ 2  1 − P 0 − P 1  1 − e −ηN sn − ηN sn e −ηN sn :

(2)

The SPAD actually operated as a binary device that could not determine the number of the photons. There are two possible outcomes, “photons detected” and “no photons detected,” and the photon events can be denoted as “click” and “no click.” The probability can be denoted as Pnoclick  P 0  e −ηN sn and Pclick  1 − P 0  1 − e −ηN sn . Such a detector registers only whether photons are detected or not but cannot determine the number of the photons. It will give out a response if it detects one or more photons, but has no way of distinguishing two or more photons from one photon. As a result, the detector is a nonlinear device. If the probability of detecting more than one photon during a pulse (Pk ≥ 2) is almost zero, the SPAD Table 1. Probability of Detecting k Events Probability

k
0

k
1

k≥2

Pk

e −ηN sn

ηN sn e −ηN sn

1 − P0 − P1

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can be regarded as a linear device and the detection of the photons is not affected by the dead time. Letting η  50% and Pk ≥ 2  0, we can obtain the value of N sn , which is about 0.024 according to Eq. (2). However, if Pk ≥ 2 is much greater than zero, the effect of dead time cannot be neglected and the SPAD works in a lossy way. Figure 3 shows the probability of detecting an event under different values of N sn . It is obvious that ηN sn ≈ 1 − e −ηN sn when ηN sn is small. The probability that the SPAD gives out a response is roughly linear to the number of the photons at low light level. The detector behaves quasi-linearly when photon flux is low, so the intensity of the light can be estimated by summing up all the detected photons. Then the total number of the detected photons over M pulses can be expressed as Num  M · 1 − e −ηN sn :

(3)

The intensity of the light can be stated as I ∝ Num  M · 1 − e −ηN sn :

(4)

However, the method of estimating the intensity by summing up all the counts is not precise when the photon flux is high. As the SPAD cannot resolve the number of the photons, estimating the intensity by summing up the detected photon counts works in a lossy way. It will cause measurement inaccuracy in many applications such as time-resolved fluorescence spectroscopy. As an iterative technique, the image of the object is constructed by averaging the correlation of the measured intensities and the incident patterns over many iterations. For the ith iteration, the spatial distribution of the speckle pattern is R i x; y, actually a two-dimensional matrix of randomly distributed “1” and “0.” Correspondingly, the total intensity of the light reflected from the object is S i . Assuming that the reflection or transmission function of the object is Ox; y, the total intensity S i could be simply described as ZZ R i x; yOx; ydxdy: (5) Si 

Fig. 3. Probability of detecting a photon event under different values of N sn . N sn denotes the average value of signal plus noise electrons during a pulse.

In the experiment, the total intensity S i was obtained from the histogram corresponding to the ith pattern. Then the ghost imaging of the object with total iteration time of N could be achieved by correlation arithmetic: GI  hΔSΔRi  hS i − hS i iR i x; y − hR i x; yii; (6) P where h·i  N1 · denotes an ensemble average for N iterations, hS i i denotes the average value of the measured intensities in the object light over total N measurements, and hR i x; yi denotes the mean distribution of the precomputed patterns. The time distribution histograms of captured photons corresponding to one DMD pattern are depicted in the first row in Fig. 4. The unit of time interval is 10 ps. Considering that the intensity of the light is roughly proportional to the counts of the photons when the light is weak, we estimated the intensity (S i ) of the light by summing up all the photon events during a time gate. The time gate was selected as 3700–3820, as shown in Fig. 4. By setting a time gate, some noise events were filtered out. In addition, the noise count during the gate is quite low and can be neglected. According to the arithmetic of ghost imaging [Eq. (6)], we finally obtained images under different light intensities, as shown in Fig. 4. In the case where the average photon count is 6.81 × 104 ∕s, we obtained the average number of photon counts corresponding with one pattern: about 744 counts. As mentioned before, there were 50,000 pulses interacting with the object. The photons were accumulated over all these pulses. It is clear that there were actually no photons detected over many cycles. As a result, the probability of capturing photons during a pulse is about 744∕50; 000  0.01488. This indicates that there is a small probability of capturing a photon during a laser pulse. We can get the time distribution of the photons through many cycles’ accumulation when the light is weak.

Fig. 4. Results under different light illuminations with laser repetition rate at 5 MHz. The photon counts are shown as percentages (a) 1.362%, (b) 1.956%, and (c) 5.36% compared with the repetition rate of the laser. Actually, the photon counts are about (a) 6.81 × 104 ∕s, (b) 9.78 × 104 ∕s, and (c) 2.68 × 105 ∕s, respectively. The horizontal axis represents the arrival time and the unit is 10 ps. N denotes the total number of the patterns used to construct an image.

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Inspired by [32], we propose an algorithm to modify the just-discussed GI algorithm: (a) calculate the average value hS i i of the total N measured intensities and the average distribution of the corresponding N patterns hR i x; yi. (b) Re-sort the N measured intensities from largest to smallest, and the reference patterns are sorted accordingly. (c) Modify the GI algorithm, K 1X S − hS i iR i x; y − hR i x; yi; (7) K i1 i P where h·i  N1 · denotes an ensemble average for N iterations and K K ≤ N  denotes the selected number from the sorted sequence of the measured intensities. In Eq. (7), both the average values of the object intensity and reference patterns are calculated over the total N iterations. However, we can use only part of the total N measurements in the final correlation. Actually, the selection of K affects the result a lot. Assuming that there are N 1 measurements bigger than hS i i and N − N 1 smaller than hS i i in the re-sorted measurements, if we select the first K measurements (K ≤ N 1 ), we can obtain bigger fluctuations than by selecting K measurements from the original measurements directly. The images obtained through these two algorithms with the same selected measurements are shown in Fig. 5. To compare these two algorithms in a more obvious way, we calculated the contrast and SNR of the images obtained with the same number of iterations. The contrast and SNR of the image are denoted as    I obj − I bg  ;  (8) C  I I 

MGI 

obj

bg

I obj ; SNR  I bg

(9)

where I obj indicates the average intensity of the object and I bg represents the average intensity of the background area.

Fig. 5. Images obtained by GI and MGI algorithms. K denotes the number of iterations used for correlation.

Fig. 6. Comparison of the contrast and SNR of images obtained through GI and MGI algorithms. N shows the number of iterations. Blue dots, GI; red stars, MGI.

The result is shown in Fig. 6. We can easily see that the MGI algorithm works better than GI: it achieves better contrast and SNR with the same number of iterations. As we have known, the fluctuation of the signal light affects the image contrast a lot. Bigger fluctuations contribute more to the acquisition of the image formation than smaller ones. In the MGI algorithm, the intensities are sorted and, as a result, the bigger fluctuations are sorted as well. By selecting some of the sorted measured intensities and the reference patterns, an image with good quality was obtained. As a result, the MGI algorithm can achieve better results than the GI algorithm with the same calculation of correlations. We used a server (64 GB RAM and CPU Clock Speed at 3.10 GHz) to process the data; to get an image of the same contrast, it needs about 20,000 patterns for the correlation in MGI compared to about 80,000 patterns in GI. In our calculation environment, the MGI algorithm takes about 10.12 s (including sorting time) and the GI algorithm takes about 21.13 s, so the MGI algorithm saves about 52% of the time of the GI algorithm. From Fig. 4, it is clear that the quality of the image becomes better with more patterns. In addition, we analyzed the effect of the sampling time of each pattern. The longer the sampling

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Fig. 7. Contrast and SNR of image with different number of pulses for each pattern. Total iterations N  40; 000. Red stars, contrast; blue stars, SNR.

time of each pattern, the more pulses each pattern corresponds with. Figure 7 shows the contrast and SNR of the image under different numbers of pulses that each pattern corresponds with while the total iterations N  40; 000. As the TCSPC is a sampling method, there is statistical noise in the experiment. As the number of pulses corresponding with each pattern increases, the statistical noise decreases. As a result, the quality of the image becomes better with more pulses. 4. DISCUSSION As has been mentioned before, the TCSPC technique is usually used in weak-illumination cases. It is necessary to maintain a low probability of registering more than one photon during a period. If the number of photons during one period was typically greater than 1, the system will often register the first one and miss the others. An effect called “pile-up” will occur and cause serious distortion of the result, such as in fluorescence lifetime imaging. However, when this technique is used to realize ghost imaging, it can be used in bright cases. As mentioned in our paper, the technique is used to measure the intensity of the object light with a SPAD. In a bright case, the number of photons captured has a large distortion from the actual value. The intensity of the light is still positively correlated with the detected photons but not in a linear way. Ghost images can be retrieved under bright illumination as long as the intensity fluctuations can be obtained. One extreme case is that there are always photons detected over each of the laser pulses. Consequently, the total number of the photons will be a constant. Then images cannot be retrieved in such a case because no intensity fluctuations are obtained. Images can still be retrieved in other cases under bright illumination as long as the intensity fluctuations are obtained. Considering our experiment, if the laser’s repetition rate is not high (for example, kHz level), the photon events may be drowned in the noise if the probability of registering one photon is too low. It may cost much time to accumulate a

histogram. We can maintain a high detection probability of photons to implement the experiment. In addition, a binary object which has only two reflectivity values was used for convenience in the experiment. Actually, in the case of a real object, the method in this paper still works. Considering that the reflectivity of a real object is much more complex, more patterns and more sampling time will be necessary to reconstruct an image under the conditions of our experiment. With a laser of higher repetition rate and a DMD that changes faster, we can reconstruct the image of an object with complex reflectivity quickly. Another point worthy to be noted is that the method in this paper can be used for three-dimensional ghost imaging. In three-dimensional scenery, photons with different times of flight correspond with different distance. Through the histograms of the photons, surface images can be retrieved. The depth resolution is limited by the full width at half-maximum of the laser pulse and the time resolution of the TCSPC module. 5. CONCLUSION In conclusion, a method based on photon counting to retrieve the image of an object is experimentally demonstrated in this paper. The experiment was implemented under weak illumination. Only one SPAD with high sensitivity was used to detect the photons. The TCSPC technique was able to register the photons at a quite low level combined with the SPAD. From the histogram of the captured photons, the intensity of the light was estimated. Although the approximation of the intensity is lossy, clear images were obtained through a correlating imaging method. With a modified algorithm, both the contrast and the SNR of the images were improved. Note that the DMD used in the experiment has a wide range of operational bandwidth. Consequently, this method can be used in a wide range of application with little limitation to the wavelengths. The DMD also helps to simplify the experimental setup compared with conventional GI. In addition, this method can be used to realize threedimensional imaging considering the time of flight of photons. Photons reflected from different distances have different times of flight. The time distribution of the reflected photons can be constructed through the TCSPC technique. Images can be obtained through the correlating method of ghost imaging. Funding. Hi-Tech Research and Development Program of China (2013AA122901); National Natural Science Foundation of China (NSFC) (61170228, 61332019, 61471239). REFERENCES 1. R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009). 2. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995). 3. R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-photon” coincidence imaging with a classical source,” Phys. Rev. Lett. 89, 113601 (2002).

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