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Abstract
Due to the characteristics of photon-counting LIDAR, there exists range walk error (RWE) when the intensity of the signal fluctuates. In this paper, an effective method to rectify underwater RWE was proposed. The method allows the separation of signal detections from noise detections, and based on a prior model, the method can compensate for RWE. An underwater experiment verified its feasibility and results showed RWE of three parts in a plane was reduced from 75mm to 7mm, from 45mm to 3mm and from 5mm to 0mm, respectively, even when the rate of backscatter photons reached 4.8MHz. The proposed correction method is suitable for high precision underwater photon-counting 3D imaging application, especially when the signal intensity varies sharply.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Photon-counting LIDAR system with time-correlated single-photon counting (TCSPC) technique has obtained intensive research interests for its high sensitivity and time resolution since its first prototype appeared [1–3]. It has been reported in many imaging applications such as long-range imaging [4–6], low light level imaging [7–9] and obscurants imaging [10]. These researches revealed that photon-counting LIDAR has many unprecedentedly excellent characteristics over traditional LIDAR.
However, due to its unique avalanche and quench process, photon-counting LIDAR suffers range walk error (RWE) problem which can be briefly interpreted that signal range position would walk as intensity varies. Many previous researches have explored RWE reduction method to improve imaging precision. Oh et al. proposed a method which conducted two measurements to calculate noise count rate and detection probability respectively and then based on Poisson statistics model to compensate for RWE [11]; Xu et al. adopted a recursive method to infer the signal photon distribution and used Gaussian functions to fit the distribution to correct RWE [12]; He et al. introduced a method which investigated the relationship between the range walk error and the laser pulse response rate and then used this relationship to fix RME [13]; Ye et al. and Ma et al. established RME correction methods by utilizing several detectors at the same time [14,15]. Nonetheless, these researches only focused on photon-counting LIDAR used in atmosphere and no researches pay attention to RWE problem for underwater photon-counting LIDAR where backscatter noise is highly populous.
Very recently, photon-counting LIDAR has been applied in underwater environment [16,17]. This is a very novel work to widen photon-counting LIDAR’s application scope and experiment results demonstrated intensity and depth images with good quality could be acquired in this manner. It indicates photon-counting LIDAR with TCSPC method is one of the most promising technique for obtaining 3D underwater images. Nonetheless, RWE problem also exists in underwater photon-counting LIDAR and this problem is even more serious than that in atmosphere for laser pulse width would be broaden during the propagation in water [18]. Besides, removing RWE in underwater environment is more challenging because noise photons, especially from backscatters, are much stronger and could degrade signal detections [19]. The main purpose of this paper is to introduce a method to correct RWE in underwater photon-counting imaging application and with our method, image precision is expected to increase significantly.
In this paper, we firstly analyzed theoretical model of RWE. Then we introduced our underwater photon-counting LIDAR experiment setup and demonstrated data acquisition process and data format. Afterwards we proposed our RWE correction method through 1) separation of signal detections from noise detections, 2) calculation of measured range time and signal primary electrons (PEs) number, 3) establishment of prior model between signal PEs and RWE and 4) compensation for RWE. Eventually, experiment result and analysis were presented that our method allowed RWE reduction from 75mm to 7m even when backscatter intensity was as high as 4.80MHz (4.80M backscatter photons per second). Conclusion and our future work were also given in the end.
2. Theoretical model of RWE
2.1. Typical RWE model
As a core component of the photon-counting LIDAR, laser emitter ejects beam pulses periodically. For each single pulse we define ${S_e}(t )$ as its emitting instantaneous photon flux rate whose waveform is supposed to be Gaussian as ${S_e}(t )= S{\textstyle{1 \over {\sqrt {2\pi } \sigma }}}\exp \left( { - {\textstyle{{{t^2}} \over {2{\sigma^2}}}}} \right)$. S is the total number of photons of each pulse and$\sigma$is root mean square (RMS) pulse width. Returned signal photon flux rate is thusly expressed as ${S_r}(t )= \alpha {S_e}({t - {\tau_{t\arg et}}} )= \alpha {S_e}\left( {t - {\textstyle{{2d} \over c}}} \right)$, where d is the range of target, c is the velocity of light and thusly ${\tau _{t\arg et}} = {{2d} / c}$. $\alpha$is a constant less than 1 which means signal photon flux rate waveform remains consistent with only intensity modulated by$\alpha$. Many target parameters could influence $\alpha$, such as reflectivity, range and target surface orientation. Then the rate function of the signal pulse for the mean number of PEs is [11]
2.2 New features of the underwater RWE model
When a photon-counting LIDAR is used for underwater imaging, there would be some new features for underwater RWE. Among others, there are two most significant features. The first is that jitter in Eq. (1) would be larger because pulse width would be broadened through the propagation in water [18]. This means ${S_{PE}}(t )$ would be wider and occupy more time bins and consequently more vulner${n_w}$able to RWE. The second is that the existence of backscatter would to some degree deteriorate detection probability [19]. Additionally, velocity of light c needs to be replaced with velocity of light in water as ${c_w} = {c / {{n_w}}}$, where is the refractive index of water which is usually 1.33 for 532nm wavelength [22].
3. Experiment setup and data acquisition
3.1. Experiment setup
The experimental setup used to collect the photon arrival data is shown in Fig. 1.
Fig. 1. Underwater photon-counting LIDAR experimental setup. (a) Schematic of the system, which comprises a Nd:YAG 532 nm laser, a PIN module, a time delay module, a Gm-APD module, a TCSPC module, a collimation lens (CL), a beam splitter (BS), two focus lenses FL1 and FL2), a beam expander package (BEP), a mirror with a centered hole (MCH), a narrow-bandwidth filter (NBF), two galvo scanning mirrors (GSM1 and GSM2), and a tube. (b) Picture of experimental setup
Our underwater photon-counting LIDAR system is within dashed line in Fig. 1(a). Nd:YAG 532nm laser emits beam pulses periodically through the fiber and then the beam is collimated by CL. BS splits beam into two portion with 99:1 ratio and the 1% would travel through FL1 and fiber to trigger PIN module to generate start signal. Time delay module receives start signal and after delay of a preset time it activates Gm-APD through gate input. 99% portion continues passing through BEP, which is used to lower down beam divergence further. MCH serves as a transmitting-receiving switch, i.e. transmitting beam travels through the center hole while receiving beam is mostly reflected to FCR and then to Gm-APD (Micro Photon Device PDM series PD-050-CTC-FC, quantum efficiency for 532nm is about 42%, dead time is 70ns). In order to suppress out-of-band noise photons and at the same time reserve signal photons, narrow bandwidth filter (NBF) of 1 nm bandwidth is placed before the FL2. Two galvo scanning mirrors (GSM1 and GSM2, Thorlabs GVS012, 10mm aperture) direct beam to raster scan target.
The target is placed at a distance of approximately 8 m in a tube. The tube is full of clear water and some SiO2 spheres and the size of spheres is several micrometers. Adding SiO2 spheres could improve the intensity of Mie scattering and accordingly backscattering effect is enhanced. In this manner we simulated real underwater environment where backscattering is severe. There exists a window at the front of tube to let laser beam in and signal photons out. The target is a plane submerged in this mixture of water and SiO2 with three different reflectivities parts. Higher reflectivity part is supposed to reflect more photons back than lower reflectivity part. Due to highly nonlinear characteristics of Gm-APD (Gm-APD could only respond to the absence or presence of photons but could not respond to the number of them and after each response, Gm-APD is disabled for a finite time, or dead time, during which all incident photons would be omitted.), it is a general requirement for photon-counting LIDAR to operate in low photon flux level [23]. Therefore during our experiment we adjust laser pulse energy to maintain the maximum number of signal PEs no more than 5, which is the typical signal photon level for a well-designed photon-counting LIDAR [24].
Coaxial optical configuration used in our system has many advantages while the most significant drawback is that internal back-reflections from optical components and backscatters from water could be critical to reduce detection efficiency [19]. To temporally avoid these unwanted noise photons, a popular gate-mode technique was utilized in our experiment as [16] have done. Gm-APD was gated on, or activated, for a temporal window of 30 ns during which signal photons were expected to return.
3.2. System delay and target range calibration
Since we set the position of window as our origin plane, it is necessary to firstly make a calibration to obtain truth range between target plane and window as well as the time delay between start signal and window signal.
The truth range of the target plane is measured as follows. We firstly put and fastened the target plane in tube without pouring water. Then a commercially available laser range finder with sub-millimeter precision (Deli DL331100B) was utilized to measure target range as Fig. 2(a) demonstrated. With range finder fixed, 50 measurements to target plane were conducted and averaged range was obtained as ${l_1}$. Then we pasted a white paper on window and continue measurements of range to the paper. Similarly 50 measurements were carried out and mean value was ${l_2}$, and the truth range was thusly calculated as ${l_2} - {l_1}$, which is 8.196m.
Fig. 2. Calibration process. (a) Truth range measurement (b) Time delay measurement, dashed box represents our photon-counting LIDAR
The time delay between start signal and window signal were calibrated similarly. Paper was pasted on window and we adjusted laser pulse energy to extremely low level and conducted one million detections and data was transferred to quTAG time to digital converter (TDC) to generate a histogram. And then the peak position of histogram was designated as time delay between start signal and window signal, which is 1.968ns. For origin plane is set on window plane, time tag data hereafter would minus this offset value, i.e. the zero time tag corresponds to the position of window.
After calibrations, we poured water and SiO2 in with photon-counting LIDAR and target plane unmoved.
3.3. Data acquisition and data format
We raster scanned target plane over x*y pixels with GSM1 and GSM2. Acquisition time ${T_a}$for each pixel was equal and thusly the number of accumulated pulses is$N = {{{T_a}} / {{T_r}}}$, where ${T_r}$ is the repetition period of laser which is 100us (namely repetition frequency is 10K) for our laser. We recorded histogram of the observed photon detections with format as$N_{}^{i,j} = \{{n_1^{i,j},n_2^{i,j}, \cdots ,n_m^{i,j}} \}$. $n_k^{i,j}$ means the number of detections in k-th time bin at $(i,j)$pixel and $m = ceil({GateTime/\Delta } )$ is the total number of time bins(ceil() rounds the element to the nearest integer greater than or equal to that element). Time bin resolution $\Delta $is configurable for our quTAG TDC and in our experiment we set it as 8ps. And hence m = ceil(30ns/8ps) = 3750. Image size and accumulated pulses number are set as x*y=64*64 and N=10K respectively. The optical power of laser is adjustable and maximum power can be up to 30mw. An example of acquired histogram data was plotted in Fig. 3, where we adjust laser energy to maintain signal PEs about 0.1 per pulse and acquisition time is 1s (10K accumulations). As we can notice, noise detections are much more populous than those in atmosphere [13].
Fig. 3. Acquired detection histogram. With our method which would be demonstrated later in Section.4, the estimated number of signal PEs is 0.0858 per pulse.
3.4. Backscattering intensity evaluation
After imaging experiment, we evaluated the amount of backscattering in the tube to demonstrate backscatter intensity in which our system works. Apart from water quality(turbidity), backscattering intensity is dependent on both pulse energy and distance [19]. Thusly, to quantify the amount of backscattering noise, pulse energy and distance should be considered. We adjusted average laser energy as 32.17microwatt to maintain its consistency with that used in our imaging experiment(described in Sec.5). The target plane was removed and 19 gates, each lasting 5ns, were added to divide the whole range into different pieces to evaluate backscattering noise levels with different distance as Fig. 4 illustrated.
For each individual gate, 20s (namely 200K accumulations) acquisition was conducted and the number of detections was recorded. For photon detections in gates are all due to backscatter noise, we can express detection probability for each gate as a function of backscatter intensity as
Here ${T_g}$is the temporal duration of gate and B is the backscatter intensity whose unit is Hz. We made an assumption that in a short time the change of backscatter intensity was tiny and therefore was supposed to be constant. Detection probability is estimated as the ratio of the number of detections and total emitted pulses which is 200K. Consequently, backscatter intensity is estimated as $\hat{B} ={-} \frac{{\ln ({1 - \hat{P}} )}}{{{T_g}}}$, where $\hat{P}$is the estimated detection probability.The result of estimated backscatter intensities is depicted below.
From Fig. 5 we can notice for the 15th gate whose distance is 8.177m, the backscatter intensity is 2.014MHz(2.014M PEs per second) and with the account of quantum efficiency(42%), the actual backscatter intensity is 2.014M/42%=4.80MHz(4.80M backscatter photons per second), which is the backscatter level for our underwater imaging experiment.
4. RWE reduction method
We propose an RWE correction method for underwater photon-counting LIDAR in this section. Now that we have known the fluctuation of intensity of signal photons results in RWE, it is intuitive to eliminate RWE by estimating intensity of signal photons firstly [11]. However, backscatter noise photons cannot be ignored in our underwater environment unlike those in [11–15]. Therefore, the initial step of our method is to separate signal detections from noise detections. Then using separated signal detections, we calculate target range time and the number of signal PEs. A prior model which demonstrated the relationship between the number of signal PEs and RWE is also established and compensation time is obtained eventually. Detailed steps are presented below.
4.1. Separation of signal and noise
Although backscatter intensity changes along the propagation, it can be viewed as constant during a small temporal of gate time [18]. Statistically, noise detections have large variance while signal detections would cluster to a rather short period [20,21]. Exploiting this statistical property, we firstly use clustering algorithm to distinguish signal detections from noise detections. More exactly, we adopt the idea of density-based spatial clustering algorithm to design our separation process to discover clusters (signal detections) against anomalies (noise detections) in our raw data [25,26]. The process is as follows: we firstly set neighborhood bound$\varepsilon$and threshold value$\mu$, both of which are positive integers. Then a rectangle window whose width is$\varepsilon$slides along each time bin and if the number of detections within this rectangle window is larger than$\mu$, the center bin of the rectangle window is deemed as signal bin. The two parameters$\varepsilon$and$\mu$actually jointly describe what kind of dense cluster we aim to find. With properly setting$\varepsilon$and$\mu$, signal detections can be extracted from overall detections because the data structure, i.e. signal detections cluster with each other while noise detection distribute randomly, is ideal for this density-based separation process. After separation, photon detections are grouped into two classes as signal detections and noise detections.
Apparently, $N_{}^{i,j} = N_{signal}^{i,j} + N_{noise}^{i,j}$.4.2. Calculation of target range time and photon number
Accordingly the measured target range time is
For $N_{signal}^{i,j}$, most of its elements would be zero with only a few time bins having counts which are estimated as target bins. On the contrary elements of $N_{noise}^{i,j}$ would obviously be zero in target bins. We define the first non-zero time bin index of $N_{signal}^{i,j}$ as $l$which also can be interpreted as the start of target bins. Noise detection probability before$l$is written as
${P_{nd}}$ evaluates the probability that noise detections would block succeeding signal detections. Combining ${P_{sd}}$ and ${P_{nd}}$, signal detection probability without the influence of noise detections is recovered asAnd according to Poisson distribution, the number of signal PEs can be assessed as
Thusly, we could estimate the number of signal PEs from detection histogram. The next step is to establish the prior relationship between the number of signal PEs and RWE.4.3. Prior model establishment
For a fixed pulse width and system jitter, RWE is the function of signal PEs number. Instrument Response Function (IRF) summarizes overall characteristics of pulse width and jitter. Therefore, we firstly measure IRF and then based on IRF we calculate prior model which conveys the relationship between RWE and PEs number.
The IRF was obtained by replacing target plane with Lambertian reference scatterer which was placed at normal incidence to beam with highly attenuated laser energy and binning the photon detections to generate a histogram. To obtain an accurate IRF, accumulation number was set as N=1M. Using our separation process, signal was separated from noise as Fig. 6 displays. Here we set and $\varepsilon = 2\quad \textrm{and} \quad \mu = 10$.
Fig. 6. IRF histogram (a) Original IRF detection histogram (b) IRF signal detections separated from noise detections with Gaussian curve fitting, where estimated Gaussian RMS width is 0.7ns
We used Gaussian curve to fit the IRF to obtain its RMS width which is estimated as ${S_{PE}}(t )$ width. The fitted RMS width is 0.7ns and using Eqs. (1)–(3) prior relationship between RWE and photon number is established.
Fig. 7. Prior model between the number of signal PEs and RWE. Some discrete points are calculated and polynomial curve with the order of 2 is used to fit this prior model with p1 = 0.0093, p2 = -0.1945 and p3 = -0.0103.
Discrete signal PEs points from 0.1 to 10 are calculated according to Eqs. (1)–(3) and a second order polynomial curve with p1=0.0093, p2 = -0.1945 and p3 = -0.0103 is applied to fit the prior model, as Fig. 7 demonstrated. R-square is extremely close to 1 which indicates the second order polynomial allows a very good fitting to represent this prior model.
Once established, prior model can be utilized without any further updates or calibrations since system parameters are fixed. However, different water turbidities would cause different overall jitter and thenceforth prior model actually needs calibrations for various turbidities. It is necessary to re-establish prior model when turbidity varies sharply, e.g. from coastal water area to remote ocean area.
4.4. Recovered target range time
Incorporate estimated signal photon number $N_{PE}$ with prior model, we can obtain corresponding range walk error $RWE({{N_{PE}}} )= {p_1}N_{PE}^2 + {p_2}{N_{PE}} + {p_3}$. Then the recovered target range time is achieved through measured target range time minus $RWE({{N_{PE}}} )$ compensation as
To sum up, schematic diagram of our correction method is represented in Fig. 8.5. Results and analysis
Laser pulse energy was adjusted to make the mean number of signal PEs at black location about 0.2 per pulse (32.17microwatt average power measured at front of GSM) and accordingly the number of signal PEs at gray or white locations is expected to be much larger than 0.2 per pulse. The acquisition time for each pixel was set as 1s to get 10K accumulations. Our target plane and histograms of corresponding pixels are depicted in Fig. 9 where and $\mu$ are set as 2 and 5 respectively.$\varepsilon$
Fig. 9. (a) Our target plane (b) Histogram of A pixel (c)Histogram of B pixel (d) Histogram of C pixel (e) Detailed histogram of pixel A (f) Detailed histogram of pixel B (g) Detailed histogram of pixel C
As we can notice, the total detection counts of three pixels are apparently different because signal PEs numbers are variant. In fact, detection distribution histograms of three pixels are also different even though it is not evident. After the finish of 64*64 pixels raw data acquisition, we plot measured range image without correction of RWE in Fig. 10(a). Here for each pixel, we calculated the mean of signal detection times as measured range time.
Fig. 10. Correction steps and corresponding images. (a) Original measured range image without correction (b) Estimated signal PEs number image (c) Compensation range image (d) Recovered range image.
As Fig. 10(a) shows, due to distinct reflectivities of three parts, there are obvious steeps even though three parts are at the same flat plane. Now we begin to correct this RWE as Sec.4 has described. Firstly, we calculate the number of signal PEs for every pixel using Eqs. (9)–(12). Then based on our prior model we obtain compensation time of each pixel and we convert this compensation time to compensation range by multiplying . Eventually, recovered range image is acquired through m${{{c_w}} / 2}$easure range minus compensation range. Figure 10 demonstrates corresponding images for each correction method step.
After our proposed RWE correction method, range steeps disappeared significantly and recovered image is much more plate. To analyze the performance of our method, we calculate the average ranges for three parts respectively before and after correction, as Table 1 shows. Average range is defined as
where ${R_i}$means the average range of part i (i=1, 2 and 3; 1=black, 2=gray and 3=white), part(i) is the collection of pixels which belong to part i and num[part(i)] denotes the number of pixels of part i. In particular, to obtain average range before RWE correction, $\tau$would be ${\tau _{mea}}$ whereas to obtain average range after RWE correction, $\tau$would be ${\tau _{re}}$ instead.
Table 1. Ranges analysis for three parts before and after correction
The truth range of the plane is 8.196m. As we can notice, after correction the whole of ${R_1}$, ${R_2}$ and ${R_3}$are more close to truth range (${\Delta _i}$is reduced for each i) and steeps of three parts vanished notably. For example, ${R_3}$ before correction is 8.121m and ${\Delta _3}$is 75mm. After correction, ${R_3}$is recovered to be 8.189 and ${\Delta _3}$ is reduced to 7mm. The experiment results indicate our correction method works well to compensate for underwater RWE.
6. Conclusion
In this paper, we proposed an effective RWE correction method whose application scope is aimed at underwater photon-counting imaging. Due to nonnegligible noise detections in underwater environment, our correction method firstly separates signal detections from noise detections. Then, using this separated data, target range time and signal PEs number are estimated. Since noise detections cannot be ignored, in our method we particularly consider their blocking impact on succeeding signal detections. After getting the measured target range time and signal PEs number, compensation value is acquired through prior model which is established beforehand. Recovered range is obtained through measured range minus compensation. Underwater experiment results showed RWE could be reduced from 75mm to 7mm with our method even when the rate of backscatter noise photons reached 4.8MHz. For high quality 3D underwater imaging application, it is necessary to take RWE into account and we anticipate that the proposed technique can be widely applied to enhance the performance of underwater 3D images that rely on photon-counting LIDAR.
However, since RWE is the error between expectation range and truth range, reducing RWE could only be helpful to rectify expectation range to be closer to truth range while repeatable accuracy or standard deviation of measured range would remain unchanged. Increasing repeatable accuracy for underwater photon-counting imaging is the main task of our future work.
Funding
National Natural Science Foundation of China (61805249); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2019369).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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