Single-photon counting lidar exhibits distinct advantages in detection sensitivity and spatiotemporal resolution, making it crucial for precision target detection. However, laser point clouds are susceptible to noise interference due to the high sensitivity of single-photon detectors (SPDs), and achieving stable target tracking still requires further research. This paper proposes a dual-branch tracking method (DBTM) for single-photon lidar target detection. First, a multi-layer denoising approach is performed in a coarse-to-fine manner to filter out noise point-clouds, separating the echo data of the moving target from the background interference. Subsequently, a dual-branch bidirectional guidance technique is designed, with a parameter estimation method integrated into the adaptive kernel Kalman filtering algorithm to estimate the target position information. This strategy fully utilizes both target intensity and depth information under typical water mist interference conditions. Simulation results show that the DBTM can still track the target effectively and accurately even under a photon noise rate of 1 MHz. Field experiments demonstrate the real-time ranging and tracking of a ship model at a distance of 84 m under water mist interference conditions, with the positioning error of 0.161 m, which verifies the effectiveness of DBTM. Performance analyses confirm that the DBTM outperforms typical conventional approaches in trajectory fitting, error control, and positioning accuracy. Moreover, the dual-branch structure supports adaptation to diverse feature scenarios and holds significant practical value for the precise positioning of small moving objects at a long range in complex environments.

The challenges of optical interference and attacks are crucial in light detection and ranging (LiDAR) sensors. Random modulation of transmitted laser can effectively suppress interference and counter attacks. The robustness of LiDAR sensors to interference and attacks completely depends on the use of high-quality random numbers. Quantum random number generation (QRNG) provides an excellent solution for true random number generation due to the inherent probability properties of quantum processes. Using the built-in single-photon avalanche diode (SPAD) array of LiDAR to obtain random numbers has become an efficient QRNG solution. However, the current QRNG method only utilizes the temporal information of single-photon detection, resulting in low QRNG rates, especially under low light flux conditions. To this end, we propose a method to improve the QRNG rate, which generates quantum random numbers through pixel grouping in the spatial dimension and arrival time difference in the temporal dimension (GpDiff-QRNG). Experimental results indicate that the GpDiff-QRNG method significantly enhances the QRNG rate. In scenarios with a photon event probability of 1%, the QRNG rate of the GpDiff-QRNG method is 37 times higher than that of the traditional QRNG by difference of arrival time (Diff-QRNG) method. Furthermore, the random sequences generated by the GpDiff-QRNG method exhibit no periodicity and belong to pure random sequences of white noise. Our method greatly expands the application of QRNG for LiDAR anti-interference and anti-attack.

Single-photon light detection and ranging (LiDAR) is constrained by the first photon bias effect, which requires the use of low photon flux to avoid signal distortion. This results in longer detection times, and requirements for rapid detection and imaging are not met. In this study, we propose a single-photon array imaging reconstruction algorithm based on photon waveform recovery. By leveraging the principles of high-flux photon signal distortion and incorporating a high-flux photon detection probability model, we effectively mitigate the imaging distortion issues typically encountered in highflux single-photon LiDAR. Simulations demonstrate the algorithm's capability to extract highly accurate distance and intensity information. Our analysis reveals that even when the signal flux is increased from the traditional 0.05 to 7 photons, the distance error remains < 1 cm, while the intensity error remains < 0.13. Furthermore, experiments on high-flux array imaging reveal that at a flux level of 3, the average distance error is 0.476 cm and the intensity error is 0.081, achieving a 5-fold improvement in intensity dynamic range. The proposed algorithm eliminates the dependency on low flux for photon imaging, effectively enabling precise imaging with single-photon LiDAR within complex scenes featuring multiple reflectivity levels and targets at various depths.

Photon counting lidar has emerged as a strong candidate technology for active detection applications because of its advantages of single photon sensitivity and high ranging accuracy. The timing histogram of a single pixel for photon counting lidar contains the target’s range information, while the laser echo of full-waveform lidar contains abundant structure and reflection information of the target. Based on the previous work of full waveform correction for stationary target, we propose a new method of the full waveform recovery for moving target, aiming at the issue of obtaining the characteristics of ultra-long-range moving targets under high-flux conditions. Our method achieves full-waveform recovery by means of data preprocessing, motion compensation, and photon waveform correction. Through simulation calculations, we analyze and compare the effectiveness of each step of the method. Compared with the raw histogram, the Normalized Root Mean Square Error (NRMSE) of the recovery full waveform and the ideal waveform is reduced from 0.137 to 0.032. Furthermore, we validate the algorithm’s robustness. As the speed increases from 5 to 340m/s, the NRMSE is always less than 0.04. The results indicate that the recovery waveform of targets hardly vary with changes in velocity. For an accumulation of 200 pulses, when the signal photons is 0.019 − 3 and the signal-to-noise ratio is below 0.033, the algorithm consistently exhibits excellent performance. Besides, we have demonstrated that for single-layer moving targets, multi-layer moving targets, and round-trip moving targets, the algorithm has good performance on the recovery of the targets’ full-waveform, and the NRMSE is less than 0.0054. This provides a new idea for obtaining the shape of targets with variable speeds at a single pixel, and provides exciting news for applications such as detection and recognition of ultra-long-range aerial targets and detection of space debris.

Single-photon lidar stands out as a promising technology for long-distance lidar applications, owing to its attributes of single-photon sensitivity and high repetition rate. Existing single-photon lidar systems typically rely on single-point scanning for positioning and tracking, necessitating intricate and precise scanning control. In pursuit of a more concise and efficient positioning, we incorporate the four-quadrant theory to articulate the signal formula of photon detection, and propose a novel single-photon four-quadrant positioning method. Our method, which includes signal preprocessing, compensation for longitudinal motion, extraction of pixel intensity, and acquisition of lateral motion, facilitates motion acquisition and positioning for targets. Through simulation calculations, we analyze and compare the effectiveness of each step of the method. With longitudinal and lateral speeds of 100 m/s and 50 m/s, respectively, the trajectory error is 1.7%, and the average speed error is 1.8%. Moreover, for various verification experiments, the trajectory errors are all below 4.2%, and the average speed errors remain under 5.4%, effectively verifying the validity of our method in acquiring the motion information and positioning of targets. It provides an excellent option for acquiring motion information and tracking small moving targets over long distances.

Photon counting lidar has revolutionized the field of lidar technology with its exceptional single-photon sensitivity and picosecond-level time resolution. It is particularly effective for detecting ultra-long-range targets and measuring global ecosystems. Timing histograms and waveforms play vital roles in these applications, as they contain rich structural information about the targets. To systematically explore the performance boundary model for full-waveform applications in photon counting lidar, we have developed a model based on underlying theory and feasibility, which overcomes the limitation of detecting fewer than 5% of illumination cycles. By using the cumulative emission pulse number as the objective function, we establish the performance boundary model for full-waveform in photon counting lidar, revealing the relationship between the accuracy of the full-waveform and system parameters. The model’s accuracy is verified through theoretical analysis and experimental validation. Subsequently, we utilize Pareto Optimality to determine the optimal parameters for the full-waveform performance boundary model. Experimental data indicates that, to ensure a normalized root mean square error ( nRMSE ) of less than 0.03 between full-waveform and ideal waveform, the performance boundaries are as follows: the optimal time bin width is 256ps, the signal intensity falls within [0.8, 1.6], the tolerable noise is [0, 0.63M]Hz, and the minimum cumulative pulses required is between [282, 319], given that the echo width is 5ns. Finally, we discuss the practical application of the full-waveform performance boundary in photon counting lidar for complex target detection scenarios. Under the optimal parameter configuration, the R-Square ( R 2 ) between the full-waveform and the ideal waveform consistently exceeds 90%. This work not only expands the range of applications for photon counting lidar in the field of full-waveform, but also establishes a strong connection with full-waveform processing algorithms.

反射层析激光雷达是一种新型成像探测技术手段，探测系统发射脉冲激光束对目标进行全覆盖，通过其与目标之间的相对运动，多角度准确获取包含目标表面反射分布信息的回波数据，并解算得到反射系数投影分布。根据傅里叶切片定理，投影数据的一维傅里叶变换和目标投影图像的二维傅里叶变换相等，利用专用算法进行重构处理，从而实现激光垂轴方向的目标投影成像。该成像方式在激光回波信噪比足够大的情况下，成像分辨率与探测距离、系统孔径无关，而主要与激光脉冲宽度、探测电路带宽和采集系统采样率有关。

The relationship between the axial structures of three-dimensional (3-D) targets and the signal of the remote-sensing techniques, such as the modulated echo waveform of the full-waveform light detection and ranging (FW-LiDAR), has been well established. However, the relationship between the lateral structures on modulated echo waveform has not been exploited in detail. For FW-LiDAR, the peak intensity, rather than the shape and width, of the echo waveform reflects lateral structural information. Using four typical two-dimensional shapes to approach the lateral structures, a mathematical formula bridging the peak intensity and the lateral structures is derived. Based upon the echo waveform simulated using the formula, modulations of the lateral structural information on the peak intensity are examined to establish a database of target properties, including shape, size and position. The physics regarding peak intensity dependence on target position is used for facile shape recognition. The modulation of the lateral shapes of ground and aerial targets on the FW-LiDAR can be assessed extensively following the procedure demonstrated in this paper. Owing to the conciseness, the peak intensity formula enables the retrieval of lateral structural information by inversion, promoting the applications of FW-LiDAR in domains including topological mapping, ecological monitoring and space debris detection and removal.

When detecting moving targets via photon counting Lidar, the target information contained in the echo photon statistical histogram is distorted, because the target position in a cumulative time changes. To solve the above distortion, this work proposed a method of acquiring moving target structural characteristics from the photon echo statistical histogram via waveform processing. Firstly, the probability distribution model of photon detection echo corresponding to a moving target was established. Then, the mathematical expressions of the laser radar cross section (LRCS) and depth structure corresponding to a moving target were derived by utilizing the photon waveform correction and waveform fitting filtering. Finally, the structural characteristics of a multi-layer moving target with a speed of 20m/s at 10km were obtained. Under the condition of SNR (signal-to-noise ratio) being 1.48, to detect a multi-layer moving target composed of two sub-targets with 0.5×1m, between which the distance was 0.5m, when the detection time was 0.01s (i.e., the cumulative number was 300), the consequential LRCS was 1.009m², and the ratio of LRCS within the moving target was 0.967:1. Meanwhile, the depth within the sub-targets was 0.493m, whose error was less than 0.7%. The proposed method in this work provided theoretical support for the acquisition of moving target detail information and the recognition of moving targets.

In remote sensing domains, it is difficult to evaluate the lateral structures using the current remote sensing techniques. The mathematical peak intensity formula of the echo waveform modulated by the lateral structures establishes a quantitative yet concise relationship between the peak intensity and the lateral structures, enabling the retrieval of lateral structural details in terms of inverting the formula. The process of the retrieval includes: 1) mathematical formula derivation; 2) target shape discrimination; and 3) mathematical formula inversion. Using the sizes estimated from the simulated echo waveforms, this study demonstrates how the estimated lateral structures are affected by the number of lateral structural parameters to be solved, instrument noise, movement direction, target shape, and target size. The results reveal that for unknown target size and lateral structures, the averaged size errors are 0.56% and 4.30%, respectively. When the instrument noise is absent and only the target size is unknown, the size error averaged over four shapes is 0.3%, and the size error averaged over the square, circle, and triangle is 0.04%. When only the size is unknown, the size errors of the rectangle, square, circle, and triangle estimated by fitting the experimental peak intensity with the formula are 2.41%, 3.47%, 0.89%, and 1.42%, respectively. The small size errors prove the possibility of retrieving the lateral sizes at a centimeter-level resolution and a distance of hundreds of kilometers, which is of great practical significance in precisely mapping the lateral structures of 3-D targets using full-waveform light detection and ranging (FW-LiDAR).

Waveform decomposition is needed as a first step in the extraction of various types of geometric and spectral information from hyperspectral full-waveform light detection and ranging (LiDAR) echoes. We present a new approach to deal with the “pseudo-monopulse” waveform formed by the overlapped waveforms from multitargets when they are very close. We use one single skew-normal distribution (SND) model to fit waveforms of all spectral channels first and count the geometric center position distribution of the echoes to decide whether it contains multitargets. The geometric center position distribution of the “pseudo-monopulse” presents aggregation and asymmetry with the change of wavelength, while such an asymmetric phenomenon cannot be found from the echoes of the single target. Both theoretical and experimental data verify the point. Based on such observation, we further propose a hyperspectral waveform decomposition method utilizing the SND mixture model with: 1) initializing new waveform component parameters and their ranges based on the distinction of the three characteristics (geometric center position, pulsewidth, and skew coefficient) between the echo and fit SND waveform; 2) conducting single-channel waveform decomposition (SCWD) for all channels; 3) setting thresholds to find outlier channels based on statistical parameters of all single-channel decomposition results (the standard deviation and the means of geometric center position); and 4) reconducting SCWD for these outlier channels. The proposed method significantly improves the range resolution from 60 to 5 cm at most for a 4-ns width laser pulse and represents the state-of-the-art in “pseudo-monopulse” waveform decomposition.

Objective Photon counting LiDAR is widely used in target ranging, three- dimensional imaging, and other fields, owing to the advantage of high sensitivity. The return echo data are obtained in the photon counting LiDAR by recording the presence or absence of photon events at the corresponding time, resulting in the inability to acquire the target echo waveform in one detection. The cumulative histogram of the photon count is obtained by the accumulation of multiple detections. The probability histogram of the photon count is regarded as the photon return detection probability waveform, which is closely related to the true return waveform of the target. In traditional LiDAR, the distance of a target can be determined by calculating the centroid of the return signal. However, in photon counting radar, the detection probability waveform of the photon echo is significantly distorted relative to the target waveform owing to the long response dead time of the detector, which significantly affects the accuracy of the photon ranging and the effective acquisition of the target information. Most researchers recover photon echo information based on the detection probability function with a large data error under low signal- to-noise ratio (SNR), making it difficult to obtain the target echo waveform information. Therefore, we discuss the photon echo correction method for the photon counting signal with a low SNR in this paper. Methods A photon detection echo model is discussed based on the LiDAR detection equation and probability response of photon detection. Combined with the simulated annealing algorithm, the particle swarm optimization algorithm is modified to estimate the photon echo parameters, including the echo signal strength, signal pulse width, peak position of the signal, and average photon noise intensity. The simulated annealing algorithm makes the swarm particles jump out of the local optimal position and effectively improves the global solution search ability. However, to avoid losing the possible dominant particle population, only a few particles are randomly selected for simulated annealing. The consistency between the recovery signal and target true return signal is evaluated by defining the evaluation function. The algorithm's accuracy is evaluated by calculating the difference between the real target location and location information determined by peak method for the recovered target echo signal. Results and Discussions The algorithm proposed in this study can achieve fine signal recovery results at a low SNR, whereas the iterative solution based on the photon detection probability has a severe signal recovery distortion ( Fig. 2). When the total number of noise photons increases from 0.5 to 5.0, the difference between the recovered signal recovered by the iterative method and true signal increases from 0.007 to 0.061, and the ranging error oscillates from 5.5 cm to 13.7 cm. The difference between the recovered signal acquired by our algorithm and the real signal is always below 0.005, and the ranging error is below 3.4 cm ( Fig. 3). The signal recovered by our algorithm, which still maintains good performance in the case of high noise, is closer to the real target echo signal. The photon detection experiments are conducted on a deep plane target with a distance of 120 cm from the front to the back. Using our method, the recovered distance of the two target signals is 123.45 cm and the error is 3.45 cm. The corresponding distance of the two target signal peaks obtained by the iterative recovery algorithm is 130.32 cm and the error is 10.32 cm (Fig. 6). The method proposed in this study can better extract the target information with the depth structure. Conclusions The simulation and experimental results show that the target signal recovery algorithm based on partial annealing particle swarm optimization can obtain stable target echo signal recovery results under the condition of a low SNR. Compared with the existing iterative method, it improves the effectiveness of signal recovery under the condition of a low SNR and avoids the increase of the error caused by iterative accumulation. Furthermore, the algorithm proposed in this paper has better performance in recovering the depth information of the target.

The space environment is becoming increasingly complicated; therefore, precise detection and identification of space targets are critical to preserving space security. Compared to optical and radar imaging, obtaining space target information using laser echo waveform is more efficient for detection. Based on the skew-normal distribution decomposition, the connection between sub-echoes and target scale following the decomposition of the space target pulse laser echo waveform is explored, and an inversion approach is suggested to identify the scale information of the solar panel utilizing the intersection of skewness and kurtosis contours in the coordinate system of the variables to be solved, based on the high-order moment characteristics of waveforms such as skewness and kurtosis. Using the proposed method, we realized the scale inversion of a 60-cm solar panel of a cube satellite inclined at 45 deg, with the turntable and the detection system placed 80 m apart. The results show that the skewness and kurtosis of the decomposed echo can represent the target size information, and the approach used here can successfully extract the solar panel scale information of a typical satellite, providing methods and data support for space target detection and classification. (C) 2022 Society of Photo-Optical Instrumentation Engineers (SPIE)

The 101-channel full-waveform hyperspectral LiDAR (FWHSL) is able to simultaneously obtain geometric and spectral information of the target, and it is widely applied in 3D point cloud terrain generation and classification, vegetation detection, automatic driving, and other fields. Currently, most waveform data processing methods are mainly aimed at single or several wavelengths. Hidden components are revealed mainly through optimization algorithms and comparisons of neighbor distance in different wavelengths. The same target may be misjudged as different ones when dealing with 101 channels. However, using the gain decomposition method with dozens of wavelengths will change the spectral intensity and affect the classification. In this paper, for hundred-channel FWHSL data, we propose a method that can detect and re-decompose the channels with outliers by checking neighbor distances and selecting specific wavelengths to compose a characteristic spectrum by performing PCA and clustering on the decomposition results for object identification. The experimental results show that compared with the conventional single channel waveform decomposition method, the average accuracy is increased by 20.1%, the average relative error of adjacent target distance is reduced from 0.1253 to 0.0037, and the degree of distance dispersion is reduced by 95.36%. The extracted spectrum can effectively characterize and distinguish the target and contains commonly used wavelengths that make up the vegetation index (e.g., 670 nm, 784 nm, etc.).

The photon counting Lidar enhances the signal-to-noise ratio of the echo signal and reduces the number of photons required for signal analysis, thereby improving the detection range and measurement accuracy. At present, the photon counting Lidar is mainly used to detect stationary targets, and the mechanism of the influence of long-distance target motion characteristics on the photon echo probability distribution is still unclear. Therefore, it is urgent to study the photon ranging performance of long-distance moving targets.In this paper, the probability distribution model of photon detection echo of moving targets is established, and a Monte Carlo model for photon detection of arbitrary targets is given. Through experimental comparison, the correctness of the Monte Carlo simulation model is verified. Furthermore, the probability distribution characteristics of the laser echo and photon echo of a small rectangular target in translation within a detection period are compared. And the variation law of the probability distribution of photon detection under different translational speeds is analyzed. In addition, the relationship between the photon ranging error and the translational speed of the target is discussed.The results show that the photon echo probability distribution of the translational target is more forward and the width is narrower than the laser pulse echo probability distribution. Compared with the extended target, the detection probability of the translational small target is significantly reduced, and the maximum average echo photon number is \begin{document}$ 1/10 $\end{document} times that of the extended target, as a result, the photon detection of the translational target requires higher laser pulse energy. When the length of target is 1m, the range walk error reaches a maximum value at a speed of \begin{document}$25\;{\text{m/s}}$\end{document}, i.e. \begin{document}$6.72\;{\text{ cm}}$\end{document}, which is \begin{document}$ 1/2 $\end{document} times that of the extended target. With the increase of the translational speed, the range walk error first increases and then turns stable with the light spot acting as the boundary.The method proposed in this paper can be further extended to photon detection and ranging of targets with other shapes, materials and attitudes. The research results provide a theoretical basis for the correction and performance improvement of the photon ranging of moving target. Furthermore, it lays the foundation for the detection of moving targets and accurate acquisition of information by photon counting Lidar.

Objective Photon detection technology is an effective method for studying far-away small moving targets, and photon echoes are considerably affected by the attitude change of the target in motion. Because the target motion influences echo parameters such as delay, broadening and energy attenuation, they reflect the change in echo waveforms. Therefore, in addition to acquiring the target postural information such as speed, direction, distance and height difference, the acquisition of the laser incidence angle and target attitude angle is crucial for air target detection. Therefore, by collecting and analysing the photon echo at multiple attitude angles of the target, we can compare different echo characteristics of different targets caused by changes in situations. Methods To describe the attitude sensitivity of the photon detection echo of an airborne target during a change in its attitude, the attitude influence level of photon echoes is defined based on the Hellinger distance and Canberra distance theory, which can be used as a quantitative analysis tool that affects the photon detection and identification performance of different targets. Based on the simulation of the photon echo waveform difference degree of three air targets in a certain range of attitude change, the law and difference between targets are analysed and two aircraft models are used for experimental verification. Results and Discussions Figure 5 shows the horizontal distribution function curve of pose effect based on waveform difference, which shows that the influence of attitude angle change on the photon echo waveform of F22 is high, and the effect of attitude angle change the photon echo waveform of F18 or F35 is low. This suggests that the attitude sensitivity of both targets is low, but the attitude sensitivity of F18 is overall higher than that of F35. The kurtosis and skewness of F35 are slightly affected by changes in the attitude (less than 0.1 and 0.15, respectively). Moreover, the echo difference degree within the full attitude angle interval is somewhat symmetric at similar to 180 degrees (Figs. 7 and 8), reaching a minimum value at approximately 90 degrees, 180 degrees and 270 degrees. The influence level induced by the kurtosis and skewness is lower than that induced by F18 and F22, which is easily distinguishable. The horizontal distribution of the attitude influence for F18 and F22 is similar, and their distribution curves are crossed; however, the peak value of kurtosis and skewness difference of F18 is larger than that of F22. Furthermore, the horizontal distribution function of the attitude influence for F18 shows a nearly linear distribution. In the comparison of the influence levels of the target photon detection echo attitude reflected by the echo center moment, it is still the F22 echo waveform that is most affected by the difference in center distance, followed by F18 and F35, respectively. Consistent with the horizontal distribution characteristics of the attitude sensitivity of the echo waveform. The experimental results show that for both targets with a similar appearance (Fig. 10), although the overall trend of the attitude influence level distribution of the echo waveform difference degree is similar, the waveform difference degree of SU-35 does not exceed 0.5 and ranges from 0.45 to 0.5, and the waveform difference of J-15 ranges from 0.5 to 0.6 [Fig. 11(a)]. Targets are distinguishable based on considerable differences in the horizontal distribution of the attitude influence between both kurtosis and skewness difference degrees [Figs. 11(c) and (d) ]. The target echo divergence of SU-35 is relatively small (less than 0.5), whereas that of J-15 is relatively large (more than 0.5). The experimental results agree well with the simulation results, thereby proving the feasibility of discriminating targets based on the influence level of the photon echo attitude. Conclusions The echo complexity of different air targets at different angles is determined by the influence of the photon detection mechanism and the diversity of the target structure and material. Using the photon detection echo theory, this study proposed the concept of photon echo posture affect levels to describe the photon echo of target attitude sensitivity. This method was verified using the modelling and simulation data, quantitative research on the susceptibility of three typical air moving target attitudes, comparison between their characteristics and differences, and experiments. The results show that the obvious difference in the influence level of the photon echo attitude of similar and dissimilar air targets can be used to distinguish the target, providing a new method and data support for target recognition with high theoretical and application values. The other angle range and specific application conditions of the theory for characterising air target movements need to be further discussed.

For detecting long-distance moving aerial targets, in order to solve the problem of low accumulation times and weak echo signal, this work proposes a multi-beam staring photon detection method. Firstly, the photon waveform expression of multi-beam staring photon detection is deduced. Then, the relationships between divergence angle, pulse width, single pulse energy, laser repetition frequency and photon probability distortion are discussed. Finally, the method of calculating the system transmitter parameters is obtained. The results show that when the detection target is the F22 flight with a speed of 400m/s at a distance of 10km, the number of beams is set to 40, the launch angle is set to 2mrad, the pulse width is set to 1ns, and the single pulse energy is set to 0.5μJ at the transmitting end. The research results provide a theoretical basis for the system design and realization of long-distance and fast-moving aerial targets.

The relationship between the properties of targets and the features of modulated waveforms is fundamental to remote sensing based on the full-waveform light detection and ranging (LiDAR). Developing a mathematical formula of modulated LiDAR waveforms is of great importance in establishing this relationship. In this study, we derive the mathematical formula of a laser echo waveform modulated by four typical targets: a rectangle, a square, a circle, and an equilateral triangle. By using these formulas, numerical calculations are performed to investigate the relationship between the properties of the targets and the features of the modulated waveform. The results show that, at a rotation angle of 80°, the modulated waveform changes from a Gaussian form to a non-Gaussian form and finally returns to a Gaussian form as the target size increases. When the target center deviates from the laser spot center and the rotation angle increases, the modulated waveform varies from Gaussian to non-Gaussian form, the value of the peak intensity decreases, and the position of the peak intensity shifts. The specific trends of these changes are successfully explained in terms of the geometric characteristics of the target and the spatial intensity distribution of the incident laser. The distinct dependencies of modulated waveforms on geometric shape, size, center position, and rotation angle indicate a convenient method for identity extraction and target recognition in remote sensing using full-waveform LiDAR. This offers exciting implications for applications, such as topological mapping, environment monitoring, and aerial target detection and recognition.

Objective Photon ranging exhibits the advantages of high sensitivity and long-distance detection. Compared with laser ranging in the linear mode, the photon detection exhibits the first photon bias effect owing to the dead-time of the single-photon detector, which results in greater distortion of the probability distribution of photon echo. This distortion is closely related to the intensity and distribution of the laser echo. There is a close relationship between the target shape and posture and the probability distribution of photon echo. As a result, the range errors in photon ranging caused by the target shape and posture cannot be ignored. Most researchers have focused on analyzing the modulation effect of target characteristics on the laser pulse echo. However, there is a lack of research on the range errors of extended targets in the photon-detection mode. Therefore, we discuss the relationship between the target shape and inclination and photon ranging for three typical extended targets. Methods Based on the Poisson probability response model and the traditional laser radar equation, the probability distribution model for the photon detection of an extended target is established herein. Combining this with the coordinate-rotation transformation formula, the general probability distribution equation mixed with spatial and temporal distribution at different inclinations is derived for the three typical extended targets: a plane, a sphere, and an aspheric. Experimental results reveal that the probability distribution of this photon echo is consistent with the numerical results. We then simulate and analyze the differences in the photon echo probability distribution and laser pulse echo characteristics of the three typical extended targets. Finally, the variation between the range errors in photon detection and the types and inclinations of the extended targets is discussed theoretically. Results and Discussions Compared with the laser pulse echo, the probability distribution of the photon echo moves forward as the inclination increases, and the variance decreases. At the same time, the pulse width of the laser echo modulated by the extended plane is wider than those of the extended spherical and aspherical surfaces. Furthermore, the probability distribution of the photon echo of the extended plane moves forward the most. The photon ranging error of the extended targets exponentially increases with the increase in inclination. The average number of echo photons is 3.9, the laser spot radius of the target is 0.2 m, and when the inclination is less than 20, the difference in the photon ranging errors between the three extended targets is less than 1.23 mm. As a result, the range errors in the photon detection caused by different extended target types could be ignored. In addition, when the inclination is greater than 20, the photon ranging accuracy of the extended plane is most affected by the inclination, whereas that of the aspheric surface is the least affected. When the target inclination is 70, the photon ranging errors for the extended plane and spherical and aspheric surfaces are 12.5 cm, 10.6 cm, and 8.9 cm, respectively. Conclusions Based on the center-of-mass detection method, the range errors in the direct detection vary slightly with the inclination of the extended targets, which is negligible compared with those in photon ranging. The range errors in photon detection increase as the inclination of the target increases. The photon-ranging errors for the extension plane are most affected by the inclination. If the inclination was smaller, the photon ranging of the extended target would be almost independent of the shapes of the extended targets. These conclusions provide a theoretical basis for the photon ranging performance and error analysis and provide a reliable information support for range-error correction and performance improvement. Furthermore, the posture information of the extended target can be acquired by combining the equation of photon echo probability distribution with the measurement results of the photon echo.

The first photon bias of photon detection results in distortion of the photon waveform, which seriously affects the accurate acquisition of target information. A rapid universal recursive correction method is proposed, which is suitable for multi-trigger and single-trigger modes of photon detection. The calculation time is 2 to 3 orders of magnitude faster than that of Xu et al.'s method. In the experiment, we have obtained good correction results for area targets and targets with varying depths. When the average number of echo photons is 0.89, the correlation distance of the correction waveform is reduced by 85%.