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Abstract
Indirect methods based on intensity for time-of-flight measurement have attracted considerable research interest in recent years because they can provide high spatial resolution in 3D imaging. However, the majority of indirect methods are inapplicable when echo signals are small (e.g., less than one photon). We propose a novel polarisation-modulated photon-counting 3D imaging method based on a negative parabolic pulse model (NPPM) to solve this problem. We measure weak signals using the number of received photons after repetitive pulsed laser emission. We establish a computational method by exploring the relationship between photon flight time that corresponds to the polarisation-modulated state of photons controlled by phase shift and calculated photon rates from received photon-counting values based on Poisson negative log-likelihood function to calculate the distance. We specifically utilise the NPPM to estimate distribution of echo signals and reduce ranging error given that echo signals are constantly time-varying. We build the first experimental system for polarisation-modulated photon-counting 3D imaging for verification by integrating it with a dual-axis galvo scanning device. Experimental results demonstrate that the proposed method can achieve ranging accuracy at the millimeter-level and exhibit superior 3D imaging performance even when the average received number of echo signals per pulsed laser emission is smaller than 0.05.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Three-dimensional (3D) photon counting-based imaging (3DPI) has attracted considerable research interest in fields of microscopy, biological imaging, fluorescence and remote sensing because it aims to capture 3D structures of targets at extremely low light levels [1–3]. Roberts et al. [4] used a multiwavelength single photon-counting scanning microscopy system to perform multidimensional imaging of 3D space, time, spectrum and fluorescence lifetime on biological tissues. Cho et al. [5] proposed a new passive imaging technique in scattering media via photon-counting modelling to overcome the challenge of dynamic imaging through the scattering of media under natural light. Qiu et al. [6] used a time division-multiplexed single photon-counting system to calculate linear depolarisation ratio of urban aerosol and estimate air pollution index.
Geiger-mode avalanche photo diode (Gm-APD), which relies on electron avalanche multiplication to generate a large voltage signal from a single photon [7], plays an important role in 3DPI. Gm-APD requires a high operating voltage for the avalanche effect, resulting in a large detector size with a low spatial resolution and a high power dissipation. Recently, a novel photon-counting detector (PCD) called quanta image sensor (QIS) has attracted considerable research interests due to its advantages of small pixel size, low dark current and high quantum efficiency [8–11].
Although the development of PCD becomes gradually mature, ranging the target is another important factor in 3DPI. Time-of-flight (TOF) technology is mainly used to calculate the distance by obtaining the flight time of light via direct or indirect methods. Time-to-digital converter (TDC) is necessarily integrated with PCDs in direct methods [12]. The difficulty in integrating PCDs and TDC is a serious problem in designing 3DPI with high spatial resolution. Compared with direct methods, indirect methods based on intensity can provide higher spatial resolution. Indirect methods mainly include time slicing, intensity correlation, and gain modulation [13]. The time slicing method needs to acquire a large number of images to analyse time information [14,15]. For example, Morimoto et al. [15] recently proposed a time-gated technology that can achieve $1$-megapixel 3DPI. However, the complicated time-gated technique that requires high-speed data transmission for hardware control (e.g. a detection of $1.5$ m requires a maximum data flow of $25$ Gbps [15]) limits the detection range and time resolution. Intensity correlation is a special case of time slicing that achieves fast acquisition because it requires only three images for 3D reconstruction. However, rectangle-shaped pulses are necessary to produce a trapezoid-shaped range intensity image and the gated time must be twice as long as the pulsed time [16]. By comparison, the gain-modulated method is a pulse-shape-free method with flexible application because it generally requires only two gain-modulated images [17–19]. For example, Chen et al. [13,20,21] utilised an electro-optic phase modulator (EOM) to perform time-correlated polarisation modulation on echo beam. The distance is calculated from the flight time derived from the intensity of the interference light at two polarisation angles. However, gain modulation methods using intensified or electron multiplying charge-coupled devices for signal detection fail to respond to weak signals of less than one photon. Therefore, these methods are limited when echo signals are extremely small.
Although a variety of TOF techniques have been investigated in 3DPI during the past few decades, 3D imaging with large spatial resolution under low-light conditions is still challenging due to the following reasons. (i) The difficulty of hardware integration in direct methods limits the increase of spatial resolution in 3DPI. (ii) The zero response of intensity in the majority of indirect methods works poorly when echo signals are less than one photon. (iii) Existing gain modulation methods consider echo signals with narrow pulse width and therefore assume that the response of echo signals is an impulse. However, these modulation methods have large measurement errors when the repetition period is comparable to the pulse width of echo signals under fast 3D imaging application.
We propose a polarisation-modulated photon-counting 3D imaging method based on a negative parabolic pulse model (NPPM) to solve these problems. We suggest counting the number of received photons via repetitive pulsed laser measurement to solve the difficulty in measuring the intensity of extremely weak echo signals. We estimate their time distribution using the NPPM because of the time-varying nature of echo signals. We establish a computational method for calculating the distance by exploring the relationship between the photon flight time that corresponds to the polarisation-modulated state of photons controlled by phase shift and calculated photon rates from received photon-counting values based on the Poisson negative log-likelihood function. Lastly, polarisation-modulated photon-counting 3D imaging is carried out by integrating it with a dual-axis galvo scanning device. It is noted that the proposed method can be directly applied to the array-based imaging system with the QIS.
The main contributions of this study are summarised as follows:
- – We describe echo signals as the detection probability of received photons given that echo signals are less than one photon. Traditional wave theory is inapplicable to this situation. We utilise quantized electric field theory to describe the physical detection process accurately and construct our computational method.
- – The NPPM is adopted to estimate the time-varying characteristic of echo signals in the real situation. Consequently, the ranging accuracy increases.
- – We build the first experimental system of polarisation-modulated photon-counting 3D imaging to verify the efficiency of the proposed method in real experiments. To the best of our knowledge, such a system remains unverified.
2. Proposed method
The diagram of the proposed polarisation-modulated photon-counting 3D imaging method is shown in Fig. 1. A repetitive pulsed laser illuminates the target through two-axis galvo scanning, and echo signals are detected by the PCD after passing through the specially designed polarisation-modulated ranging module. Each spatial point is illuminated by one measurement with $N$ pulses. We can utilise the number of received photons to calculate the reflectivity of the target even when echo signals are extremely weak (e.g. less than one photon). Lastly, we can generate the desired 3D image after depths of all spatial points are calculated using the polarisation-modulated ranging module.
Figure 2 shows the principle of polarisation-modulated photon-ranging module. We utilise a quantized electric field to describe the polarisation state of photons [22]. Some photons pass through a compensator, followed by an LP with an angle of $\theta$. We describe the effects of the compensator by introducing phase delays $\alpha _x$ and $\alpha _y$ in the $x$ and $y$ components, respectively, of the field operator $\widehat {\mathbf {E}}^{(+)}$. The photon counting rate received in the PCD can be described as follows:
Figure 3 presents the schematic of the proposed polarisation-modulated photon-counting ranging technique. We use a periodically pulsed laser with the repetition period $T_r$ to detect the target. An EOM with period $T_G$ is utilised to generate a periodical compensator correlated with time. We suppose the distance between the target and the detector is $R$. We use $T_G>2\frac {R}{c}$ and $T_r=T_G$, where $c$ is the speed of light to avoid aliasing. A controller is used to synchronise states of the pulse laser and EOM. LP1 is adopted to filter photons in the fixed polarisation state. We suppose that its polarisation direction is along the vertical direction, as shown in Fig. 3. The following relationships are demonstrated: $G_{xx}^{(1)}=G_{yy}^{(1)}=\frac {1}{2}$, $\left |g_{xy}^{(1)}\right |=1$, and $\beta _{xy}=0$ after photons pass through the LP1. The EOM will subsequently introduce a linear phase shift $\alpha =\frac {\pi }{T_G}t$ when the voltage that linearly increases with time is applied to the EOM [13]. Therefore, if LP2 with the angle $\theta$ is placed behind the EOM, then (1) can be simplified as follows:
The superscript $v$ indicates that the polarisation state of input photons for the EOM is along the vertical direction.
Fig. 3. Schematic of the proposed polarisation-modulated photon-ranging technique. LP1 and LP2, linear polarisers (polarisation direction of LP1 is vertical); EOM, electro-optic phase modulator.
We can use an inhomogeneous Poisson process with the time-varying rate $\eta \Re ^{v}_{\theta ,\alpha }s(t)+d$ to describe photon detections produced by the PCD when only a single pulse is transmitted given the low-flux condition considered in this study. $d$ represents the rate of an independent homogeneous Poisson process generated by detector dark counts and $\eta$ is the quantum efficiency of the PCD. This condition indicates that the probability mass function for the number of photons detected in response to a single modulation period in time interval $(0,T_G]$ is expressed as follows:
where In (4), $s(t)$ represents the photon-flux waveform of echo signals measured in counts/seconds. $s(t)$ is assumed a constant in [13,20,21] because the width of echo signals is relatively small compared with the long repetition period (e.g. $1$ s) of the laser. A pulsed laser with high frequency (e.g. $10$ MHz) is required to generate the measurement of photon counting in our method. Consequently, the time-varying characteristic of $s(t)$ must be considered. We replace the commonly used Gaussian pulse model with NPPM [23] for describing echo signals to simplify the calculation process of (4) as follows:(i) We rotate the LP2 to the vertical direction ($\theta =0^{\circ }$) to obtain
(ii) We rotate the LP2 by $45^{\circ }$ to construct the following relationship:
(iii) We delay the working voltage of the EOM by half a period to transform the relationship between the modulation phase and the photon flight time as follows:
The negative log-likelihood function $\mathcal {L}$ of $m_0$ given count data $k_0$ is computed on the basis of (11) as follows [26]:
We obtain the following equation by maximising the negative log-likelihood function $\mathcal {L}$ in (12): Similarly, we have and The distance $R$ can be computed on the basis of (7), (8), (10), (13), (14), and (15) as3. Experimental results
3.1 Ranging experiment
We construct an experimental system in an indoor environment to verify the ranging performance of the proposed method. The pulsed laser (Picoquant PDL 800-D and LDH-D-C-640) demonstrates a wavelength of $640$ nm with a working frequency of $20$ MHz. The EOM (Thorlabs EO-PM-R) exhibits a working frequency of $20$ MHz and a half-wave voltage of $7$ V. The PCD (Thorlabs SPCM20A/M) is used to obtain photon-counting values. The detector has a quantum efficiency of $23\%$ at a wavelength of $640$ nm, an effective area of $180$ $\mathrm{\mu}$m and approximately $25$ dark counts. We adjust $N_s$ using the attenuator. Echo photons are detected by the PCD after passing through the EOM and the linear polariser (LP2). The range of the detection distance in this experiment is within $[0, 7.5]$ m and can be extended by increasing the modulation period. Figure 4(a) shows the results of detecting a target at $4.5$ m under different $N$ and $N_s$ values. Each experiment is repeated $1,000$ times to reduce the influence of environmental noises and generate the average results for analysis. The absolute value of the difference between the final distance and the real value is computed as the ranging error. Figure 4(a) shows that the ranging error gradually decreases with the increase of $N$ and $N_s$ and stabilises when $N$ becomes extremely large. The maximum error in Fig. 4(a) is $11$ mm when $N=5,000$ and $N_s=0.1$ because the noise will increase uncertainty when $N$ is a small value and lead to the increase of random errors in the computation process. Meanwhile, large $N_s$ values improve the SNR of the system and reduce the influence of noise. Therefore, the increase of both $N$ and $N_s$ can help reduce ranging errors.
Fig. 4. Experimental results of the ranging performance. (a) Experimental results at different $N$ and $N_s$ values. (b) Comparison of IGM and the proposed method.
We construct a computational strategy from the intensity-based gain modulation method (IGM) to demonstrate the efficiency of the adopted NPPM that considers the time-varying characteristic of echo signals by replacing the measurement of intensity in [13], by the number of received photons. We test a plane mirror target within the range of $[0.5, 6.5]$ m. Each measurement includes $10,000$ laser pulses. We use an attenuator to control $N_s$ of each measurement at $0.5$. The experimental results are shown in Fig. 4(b). The proposed method significantly reduces the ranging error with the introduction of NPPM. For example, IGM demonstrates the best ranging performance within the range of $[2,5]$ m, with a ranging error close to $5$ mm, whilst the ranging error of the proposed method can be reduced by approximately $40\%$, thereby indicating the importance of considering the time-varying characteristic of echo signals.
3.2 Fast 3D imaging experiment
We further utilise the dual-axis galvo scanning device (Thorlabs GSV002) to implement a scanning-based photon-counting imaging system for experiment and demonstrate the performance of the proposed method in fast 3D imaging [3,26]. The experimental results are shown in Fig. 5. We utilise two metal triangle plates that are $10$ mm apart and $2$ m away from the imaging system. These true distances are measured by a handheld laser rangefinder. We set $T_G=50$ ns and $N=10,000$ in this experiment. The imaging frequency of a single pixel for one measurement is equal to $2$ kHz given that the working frequency of the pulsed laser is $20$ MHz. This condition implies the possibility of fast-imaging speed for an array-based photon-counting imaging system if the proposed technique is adopted. $N_s\approx 0.5$ in this situation, thereby indicating that average echo signals are extremely small (less than one photon for a single pulse). The red frame of the intensity image in Fig. 5(a) denotes the scanning area in the experiment. The triangular metal plates cannot be distinguished from the photon-counting image on the basis of the reflectivity of the target given the similar reflectivity of the triangle plates, as shown in Fig. 5(b). The distance of a single measurement can be calculated on the basis of the computation from the photon-counting images of $k_0$, $k_1$, and $k_2$ in a single measurement (Figs. 5(c-e)). The results are illustrated in Fig. 5(f). We notice that the triangle plates can be distinguished from a certain distance. Average distances of the right and left triangle plates are $2,003$ and $2,014$ mm, respectively. Average ranging errors for the right and left triangle plates are $3$ and $4$ mm, respectively. We can calculate the distance using $10$ measurements to reduce ranging errors and decelerate the imaging speed of a single pixel to $200$ Hz. As shown in Fig. 5(g), the distance calculated by multiple measurements can reduce the ranging errors. For example, average ranging errors of the right and left triangle plates are reduced to $2$ and $3$ mm, respectively.
Fig. 5. Experiments of fast 3D imaging with two metal triangle plates. (a) Intensity image. (the red frame denotes the scanning area). (b) Photon-counting image based on the reflectivity of the target. (c)-(e) Photon-counting images of $k_0$, $k_1$ and $k_2$ in a single measurement. (f) Distance calculated from a single measurement. (g) Distance calculated by averaging $10$ measurements.
3.3 Experiment of 3D imaging under extremely weak echo signals
We use a ball with the shape of a standard sphere (radius: $90$ mm) for experiment to demonstrated the 3D imaging under extremely weak echo signals. The ball is placed $2$ m away from the imaging system. $N_s\in [0.039,0.051]$ are very small and different for various pixels in this experiment due to the 3D structure of the target that increases the difficulty in detecting. Therefore, we set $N=100,000$ to reduce ranging errors in this experiment. The experimental results are shown in Fig. 6. Figure 6(c) presents the depth (distance) calculated by single and multiple measurements. The 3D spherical surface of the target can be further extracted from the depth. Meanwhile, we adopt the median filter to reduce the influence of the noise. The 3D view of the target shown in Fig. 6(d) demonstrates that the 3D spherical surface of the target can be properly reconstructed. Compared with a single measurement, multiple measurements can efficiently reduce the absolute errors between the calculated 3D spherical surface and real values, as shown in Fig. 6(e). For example, the maximum ranging error of the entire spherical surface does not exceed $8$ mm. Compared with the edge region, the centre region of the ball has a larger $N_s$. As a result, small errors exist in the centre region of the ball. For example, errors of the central region for single and multiple measurements are approximately $3$ mm and $1$ mm, respectively.
Fig. 6. Experiment of 3D imaging under extremely weak echo signals. (a) Intensity image of a ball with the shape of a standard sphere (radius: 90 mm). (b) Photon-counting image based on the reflectivity of the target. (c) Depth (distances) calculated by single measurement and multiple measurements. (d) 3D view of target shapes after median filter. (e) Difference in the result of real values evaluated using absolute errors.
3.4 Experiment of the robustness of the proposed method
We carry out the following experiments to verify the robustness of the proposed method by varying the width of echo signals from (i) changing the width of emitted pulsed laser (the width of echo signals is correlated with the width of the emitted pulsed laser) (ii) using different roughness targets and (iii) modifying tile angles of the target.
Firstly, we adopt the pulsed laser with a width $90$ ps and $300$ ps for testing. We set $N=10,000$ and $N_s= 0.5$ in this experiment. The experimental results are shown in Fig. 7. We place nine coins with a thickness of $2$ mm on the light-absorbing plate as the target, and the light-absorbing plate is $4$ m away from the imaging system (Fig. 7(a)). These coins are divided into three groups. The first group has one coin with a height of $2$ mm, the second group has three coins with a total height of $6$ mm and the third group has five coins with a total height of $10$ mm. Amongst them, top coins of the first and the second groups are both circular, and the top coin of the third group is flower-shaped. The 3D imaging results of the IGM method that adopts the pulsed laser with a width of $90$ and $300$ ps are shown in Figs. 7(b) and 7(c), respectively. Although some abnormal values are caused by noise in IGM at a pulsed laser with a width of $90$ ps, the height difference of the three groups of coins can be clearly distinguished. However, when the width of the pulsed laser changes to $300$ ps, depth errors of IGM increase significantly. For example, heights of the first and third groups differ largely from ideal values and RMSE is increased nearly by $45.9\%$ with the increase of the pulse width from $90$ ps to $300$ ps. The proposed method is robust against the variation of the width of the emitted pulsed laser by adopting NPPM, as shown in Figs. 7(d) and 7(e). The influence of the width of the pulsed laser is small on the imaging performance of the proposed method because the time-varying characteristic is considered in the proposed method. Consequently, the increase of the pulse width from $90$ ps to $300$ ps only leads to an approximate $7.7\%$ increase of RMSE. Widths of echo signals depend not only on the width of the pulsed laser but also relate to the roughness of the target and the incident angle of the laser. As described in [24,25], the rough target surface and the large laser incident angle will lead to echo signals with a large width. We conduct experiments on targets with different roughness values and tilt angles in Figs. 8 and 9. As shown in Fig. 8(a), we use planes, including plastic, wooden and carton, as targets with different roughness values. Red frames denote the scanning area ($5$ cm $\times$ $5$ cm) in the experiment. The distance between the target and the imaging system is $3$ m, with $N= 1,000$ and $N_s= 0.5$. The distribution of the depth image obtained by the IGM method is shown in Fig. 8(b). Mean values of the depth gradually deviate from the true depth of $3$ m with the increase of target roughness. Compared with the plastic target, RMSE of wooden and carton targets increase by $50\%$ and $98.6\%$, respectively. Hence, the performance of the IGM method degrades due to the high-variance ranging errors of the imaging of the high-roughness target. The increasing noise in the depth with the increase of surface roughness indicates that the uncertainty of the depth measurement is high for the rough target. The distribution of the depth image obtained by the proposed method is shown in Fig. 8(c). The depth image becomes slightly noisy as the target roughness increases. Compared with the IGM method, RMSEs of the proposed method for the three targets reduce by $64.3\%$, $72.4\%$ and $74.1\%$, thereby proving its robustness to the target roughness.
Fig. 7. Comparison of 3D imaging experiments by adopting the pulsed laser with different widths. (a) Frontal and side views of the target. Targets are nine coins divided into three groups. The number of coins in the three groups is $1$, $3$ and $5$. The thickness of each coin is $2$ mm. (b)-(e) IGM and the proposed method with the pulsed laser width of $90$ and $300$ ps, respectively.
Fig. 8. Experiment of 3D imaging on targets with different roughness values. (a) Different roughness planes as targets, including plastic, wooden and carton, with increasing roughness. (b) Depth distribution and images obtained using the IGM method. (c) Depth distribution and images obtained using the proposed method.
Fig. 9. Experiment of 3D imaging on targets with different tilt angles. (a) Targets with four tilt angles of $0^{\circ }$, $20^{\circ }$, $40^{\circ }$ and $60^{\circ }$. (b) Depth images obtained using the IGM (left) and the proposed (right) methods. (c) Residual images obtained using the IGM (left) and proposed (right) methods.
We use plastic planes with different tilt angles as targets to verify the influence of the laser incident angle. The experimental results are shown in Fig. 9. We rotate the plastic plane by $0^{\circ }$, $20^{\circ }$, $40^{\circ }$ and $60^{\circ }$ on the central axis in Fig. 9(a). The scanning area is a rectangular region ($5$ cm $\times$ $5$ cm) in the centre of the plane. Other conditions are the same as the above experiment. Depth images obtained by the IGM (left) and proposed (right) methods are shown in Fig. 9(b). The imaging results obtained by the IGM method demonstrated sharper noise fluctuations as the tilt angle of the target surface increases, whilst those obtained by the proposed method are relatively stable. Residual images obtained by the IGM (left) and proposed (right) methods are shown in Fig. 9(c). Compared with the tilt angle of $0^{\circ }$, RMSE of the target with a tilt angle of $20^{\circ }$, $40^{\circ }$ and $60^{\circ }$ increases by $35.6\%$, $69.9\%$ and $111.0\%$, respectively, and further affects the range accuracy and depth resolution of complex target imaging when the IGM method is used. Compared with the IGM method, RMSEs of the proposed method for the four tilt angles of the target surface reduces by $63\%$, $67.7\%$, $69.4\%$ and $66.9\%$. Hence, the proposed method is robust to the laser incident angle. Therefore, the proposed method has more potential in the reconstruction of 3D shapes of targets with complicated textures. Experiments in Figs. 8 and 9 indicate that the performance of the IGM method is significantly reduced for the target with high roughness or large tilt angle because the width of the echo signal is significantly broadened. The proposed method has a superior performance as the time-varying characteristic of echo signals are fully considered in the NPPM.
4. Conclusion
A novel polarisation-modulated photon-counting 3D imaging method was proposed in this study. To the best of our knowledge, this work provides a new 3DPI technique that can achieve large spatial resolution under low-light conditions based on the polarisation-modulated ranging method. We established a computational method for distance measurement by exploring the relationship between the photon flight time corresponding to the polarisation-modulated state of photons controlled by the phase shift and calculated photon rates from received photon-counting values based on the Poisson negative log-likelihood function. We described the echo signal via the NPPM and solved the error caused by the echo pulse width to optimise the ranging accuracy. We conducted ranging experiments to analyse the performance of the proposed method, influence of main parameters on the ranging accuracy and the optimisation of the ranging performance relative to the IGM. We performed fast 3D imaging experiments and showed reliable 3D imaging results under extremely weak echo signals, with a maximum ranging error of $8$ mm. We also conducted 3D imaging experiments to compare the imaging performance with the IGM method, and the results verified that the proposed method is more robust under different conditions of laser emission pulse width, surface roughness and tilt angle. In the future, the following studies are our next work: 1) The proposed method is only suitable for a single echo signal in each detection. Therefore, multiple echo signals will introduce some measurement errors to the proposed method. How to apply the proposed method in the environment of multiple echo signals is one of our future research interests. 2) We will design a system for obtaining measurements in a parallel way to improve the imaging speed. 3) We will focus on the reconstruction of 3D shapes of targets with complicated textures in future investigations.
Funding
National Natural Science Foundation of China (61971315); Natural Science Foundation of Hubei Province (2018CFB435).
Acknowledgments
The authors wish to thank the editor and the anonymous reviewers for their valuable suggestions.
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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