Analysis and Simulation of Space-Based LM-APD 3D Imaging

Abstract

The linear mode avalanche photodiode (LM-APD) array has the capability of real-time 3D imaging for moving targets, which is a promising 3D imaging means in space. The main system parameters of the LM-APD array 3D imaging system, the characteristics of the space target itself, and the relative positional relationship between them will affect the 3D imaging results at the same time, and there is a need for an appropriate simulation method to describe the space target point cloud acquired by the LM-APD array 3D imaging system under different conditions. We propose a simulation method for the 3D imaging of space targets with LM-APD arrays, which takes the characteristics of the space targets and the relative position into consideration, and build a link from the laser to the receiving system to simulate the echo waveform of each pixel in the LM-APD array. The experiment results under different conditions show that the proposed simulation method can accurately describe the imaging results of the LM-APD array 3D imaging system for space targets with different shapes, materials, and motion states, which providing theoretical and data support for the design of LM-APD array 3D imaging systems.

1. Introduction

With the booming space industry and the increasing number of spacecrafts in space, there is an urgent need for effective means to observe the condition of a spacecraft in orbit for the purpose of fault diagnosis and feature identification. Most of the current optical means of observing space targets rely on spectral features or two-dimensional imaging, missing the important feature of the three-dimensional structure of the target. The 3D imaging of space targets is important for the maintenance of spacecraft in orbit and for the stability and safety of the space environment, but there are more challenges. At present, the space-based space target 3D imaging methods are mainly binocular vision, inverse synthetic aperture, and range-based laser three-dimensional imaging. Binocular vision is affected by light, and the distance resolution decays with distance. Inverse synthetic aperture has certain requirements for flight trajectory. Laser 3D imaging uses laser active illumination, which is less affected by light, and is combined with distance selection to filter out noise interference, with high range accuracy and long working distance, which can well meet the requirements of space targets for 3D imaging. Limited by the imaging system and modulation frequency, the longest working distance of space-based phase continuous wave laser 3D imaging is only 40 m, which cannot meet the demand for 3D imaging of non-cooperative targets in space. But the pulse-ranging laser 3D imaging obtains the relative distance through the time of flight of photons, the distance resolution is usually on the decimeter scale, which can simultaneously meet the working distance range and accuracy requirements of space target 3D imaging.

The avalanche photodiode (APD) array detector has an all-solid-state structure, high quantum efficiency, and low operating voltage, making it very suitable for satellite platforms in terms of size, weight, and performance. APD arrays have two operating modes, Geiger and linear [1], of which the linear mode avalanche photodiode (LM-APD) is capable of real-time 3D imaging of moving targets and is mainly used in space-based pulsed flash laser 3D imaging systems: Some Lidars, such as DragonEYE [2,3] and STORRM VNS [4], have already started in-orbit experiments and are becoming practical. The LM-APD array is a promising imaging system for space target 3D imaging applications, but conducting experiments in orbit is difficult, and there is still a lack of corresponding hardware experimental conditions. The system of the LM-APD includes the optical part and the electronic part, the main system parameters of the LM-APD array 3D imaging system, the characteristics of the space target itself, and the relative poses between them would simultaneously affect the 3D imaging results. Simulation must fully reflect the influence of system parameters, relative pose, and other factors on the final 3D imaging point cloud. But it is difficult to establish corresponding simulation conditions on the ground for space-based imaging applications, mainly for the following two reasons: First, only a few research institutions have LM-APD with the number of pixels larger than 128 × 128, and three-dimensional imaging can hardly be completed when the number of pixels is smaller than this number. Second, it is very difficult to simulate a large dynamic range of distance and attitude change in ground. The system design of LM-APD array 3D imaging and the research of point cloud processing methods, such as multi-view point cloud 3D reconstruction, are in urgent need of corresponding simulation methods to verify system design parameters and obtain point clouds from a flexible perspective, the study of LM-APD array 3D imaging characteristics and simulation methods can provide theoretical support and data support for system design and point cloud data processing, which is of great value. In paper [5], the influence of the SNR of the LM-APD array 3D imaging system and the SNR of the threshold on the imaging results was analyzed under the assumption that the target is a planar Lambertian body. The paper [6] simulated the imaging results of the array laser 3D imaging system for space targets, but only considered the target occlusion and distance factors, and added noise directly to the target by experience after performing field-of-view occlusion and fading processing, ignoring the role of the laser echo waveform and energy on the noise, and similar problems exist in other papers [7]. The paper [8] considered the effect of the target shape on the waveform and simplified the target to a Lambertian body, ignoring the effect of the target material on the waveform, but the space target does not fully conform to the characteristics of a Lambertian body. The papers [9,10] analyzed the effect of target material and attitude on Lidar echoes with single unit detector but did not extend to the array detector. Wang [11] and Chen [12] have also investigated space target 3D imaging simulations for position measurement. Zhang [13] used multivariate models to describe the optical scattering characteristics of the satellite, added specular reflective surface elements to the satellite body, and obtained results that are more consistent with the measured data in simulating the satellite’s unique photometric characteristics. Using the Monte Carlo method, the paper [14] discussed the simulation method of the GM-APD array applied to a single photon counting 3D imaging, which can simulate the imaging process of Lidar well, but the Monte Carlo method has high computational complexity for the simulating LM-APD. The paper [15] simulated the received beam broadening and action distance of Lidar with a diffractive optical system based on the Lidar action distance equation, taking into account the influence of analog-to-digital conversion quantization, etc., which is a useful method for the electronic simulation. The simulation of LM-APD array 3D imaging not only needs to consider the target occlusion, as the distance and attitude between the target and the imaging system will have a significant effect on the laser echo power and waveform, and the difficulty of LM-APD array 3D imaging simulation is to describe the effect of complex structured space targets with different positions on the laser echo.

Based on this requirement, we have carried out a study on the simulation method of LM-APD array 3D imaging system. The difficulty of the simulation lies in how to completely simulate the whole process of laser energy emission and reception and represent it in waveforms. This is because only by simulating the accurate echo waveform with energy change process can we accurately get the accurate point cloud characteristics. In the process of solving this problem, it is necessary to consider the difference between LM-APD as an array detector and APD unit for imaging, such as crosstalk, noise, etc. It is necessary to consider the obscuration caused by the relative positional change between the space target and the imaging system, the modulation of the waveform by reflection, etc. Our main innovation lies in the complete analysis of the propagation process of the echo energy, obtaining the characteristics of the temporal distribution of the echo energy, and realizing the circuit simulation of the LM-APD, thus being able to simulate the photoelectric conversion and the process of the electrical signal to the distance value. We first analyze the main performance parameters of each device in the imaging chain of the LM-APD array 3D imaging that affect the 3D imaging results, and then analyze the changes of relative position between the space target and the imaging system during the 3D imaging process, and then comprehensively analyze the statistical characteristics of LM-APD array 3D imaging, such as point cloud noise, etc. On this basis, the simulation method for the 3D imaging of space target LM-APD array 3D imaging is proposed.

2. Characterization of LM-APD Arrays for Laser 3D Imaging

2.1. Imaging Chain Analysis

Figure 1 shows the chain of the LM-APD array 3D imaging. After the laser emits laser pulses, it is collimated and expanded by the transmitting optical system, irradiated to the target surface, reflected by the target surface, received by the receiving optical system, and converged to the LM-APD array. The photoelectric signal conversion is completed by each LM-APD pixel in parallel. After amplification, the signal is detected and processed by the time identification module to identify the stopping time and to obtain the photon flight time, which could be converted to the relative distance. Finally, the point cloud of the target is obtained according to the relative distance of each pixel, and the 3D imaging of the space target realized.

2.1.1. Laser

The 3D imaging of space targets usually involves a Q modulated solid-state laser with ${\mathrm{TEM}}_{00}$ mode, whose laser output power waveform $P_0$ in the time domain is shown in Equation (1) [10],

$$P_0(t)=P_{\max }\left(\frac{3.5t}{{\tau }_{\mathrm{t}}}\right)\exp \left(1-\frac{3.5t}{{\tau }_{\mathrm{t}}}\right)$$

(1)

in which $P_{\max }$ is the peak laser output power, ${\tau }_{\mathrm{t}}$ is the full half-peak width of the laser pulse, and exp is the natural logarithm with base e.

The laser has a Gaussian distribution in the vertical plane along the direction of propagation, and the beam divergence angle $\theta {\prime }_0$ is as shown in Equation (2),

$$\theta {\prime }_0=2\lambda /\pi w{\prime }_0$$

(2)

in which $w{\prime }_0$ is the laser beam waist. In order to compress the power density of the laser on the target and to match the received field of view, the output beam of the laser is usually expanded. The divergence angle ${\theta }_0$ after beam expansion with lens of focal length $F$ is shown in Equation (3).

$${\theta }_0=\frac{2\lambda }{\pi }\sqrt{\frac{1}{w{\prime }_0^2}{(1-\frac{l}{f_{\mathrm{e}}})}^2+\frac{1}{f_{\mathrm{e}}^2}{(\frac{\pi w{\prime }_0}{\lambda })}^2}$$

(3)

The relative light intensity distribution $E(r)$ from the point $(x,y)$ to the center of the optical axis in the plane at a distance $z$ is shown in Equation (4),

$$E(x,y,z)=E(0,0,z)\exp [-\frac{x^2+y^2}{w_0^2[1+{(\lambda z/\pi w_0^2)}^2]}]$$

(4)

in which $w_0$ is the beam waist after beam expansion. In order to meet the requirements of uniform distribution of laser power density on the target, the laser usually needs be homogenized. The laser power density $E_{\mathrm{in}}(R_{\mathrm{st}},t)$ in the plane of distance $R_{\mathrm{st}}$ is as shown in Equation (5).

$$E_{\mathrm{in}}(R_{\mathrm{st}},t)=\frac{4P_0\left(t-R_{\mathrm{st}}/c\right){\eta }_{\mathrm{t}}}{\pi {\theta }_0^2{R}_{\mathrm{st}}{}^2}$$

(5)

2.1.2. Space Targets

The laser emits laser pulses to irradiate the target, which are reflected from the target surface to form an echo. The target material and shape affect the reflectivity and the angle between the laser incident direction and the normal vector of the target surface, respectively, which affect the waveform of the echo signal.

The surface of a space target is usually covered with thermally controlled materials or coatings, and the reflection of the laser from these materials is divided into specular and diffuse reflections. The bidirectional reflectance distribution function (BRDF) model can describe the ability of the target surface to reflect laser energy, and its physical meaning is the ratio of the increment of incident laser irradiance to the increment of reflected radiation brightness per unit stereo angle in the specified direction. A four-parameter single-station BRDF model is typically used, defined as in Equation (6),

$${\rho }_{\mathrm{r}}=\frac{A}{{\cos }^6\theta }\cdot \exp (-\frac{{\tan }^2\theta }{s^2})+B{\cos }^m(\theta )$$

(6)

in which $\theta$ is the angle between the laser incident direction and the normal vector of the target surface; ${\rho }_{\mathrm{r}}$ is the BRDF parameter of the target; $A$ is the specular reflection amplitude component of the target; $B$ is the diffuse reflection amplitude component of the target; $s$ is the specular reflection coefficient of the target; and $m$ is the diffuse reflection coefficient of the target. The BRDF parameters of an ideal fully diffuse reflecting material, a fully specular reflecting material, and several commonly used materials [16,17] on the satellite surface are shown in

Table 1. Ideal material and typical space target surface material BRDF parameters.

Material Type	Completely Diffuse Reflection	Complete Specular Reflection	Windsurfing	Coating 1	Coating 2
A	0	1	0.5	0.55	0.65
B	1	0	0.03	0.05	0.04
m	1	0	4	2	3
s	1	0.25	0.05	0.25	0.15

.

Figure 2 shows the BRDF parameter curves of various materials when the angle $\theta$ between the laser incidence direction and the normal vector of the target surface varies from 0° to 90°. The BRDF of the surface material varies while the incidence angle is different.

In summary, the target material and the laser incidence angle jointly affect the laser echo, and the 3D imaging simulation model must consider these two factors simultaneously.

2.1.3. LM-APD Array

(1)

How LM-APD works

The APD is a device that converts optical signals into electrical signals using the photoelectric effect, i.e., the incident light energy is absorbed by a reverse-biased PN junction and converted into photocurrent. Figure 3 shows the working principle of a typical ${\mathrm{N}}^{+}\mathrm{P}\pi {\mathrm{P}}^{+}$ structure APD. The high reverse bias voltage forms a high electric field region, so the photogenerated carriers are accelerated by the electric field to obtain high kinetic energy, and when the photogenerated carriers collide with valence band electrons in motion, more electrons are excited to form electron-hole pairs, and the newly generated electrons are again excited by collisions under the action of the electric field to form an avalanche multiplication effect, which greatly improves the detection sensitivity.

APDs are primarily made of Si, Ge, and InGaAs. Different materials not only have different response bands, but also have differences in gain, dark current, and rise time, as shown in

Table 2. The main features of APDs made of several typical materials are.

Parameters	Si	Ge	InGaAs
Wavelength of response/nm	400–1000	800–1650	1100–1700
Gain	20–400	50–200	10–40
Dark current/nA	0.1–1	50–500	10–50
Rise time/ns	0.1–2	0.5–0.8	0.1–0.5

.

The reverse bias voltage of the LM-APD is lower than its breakdown voltage, and the APD plays a linear amplification role for the incident photoelectrons. In linear mode, the higher the reverse bias voltage, the higher the multiplication factor $M_{APD}$, which is generally $10~{10}^3$ depending on the reverse bias voltage. At this point, the APD amplifies the input photoelectrons and other gains to form a continuous current, forming a continuous laser echo signal.

(2)

Structure of the array

Multiple APD units are fabricated by molecular beam epitaxy on the same substrate, and APD arrays are obtained by combining the corresponding parallel readout circuits. Compared with APD units, the structural characteristics of APD arrays have a certain impact on 3D imaging, mainly in three aspects: inter-element gain inconsistency, crosstalk, and time delay inconsistency.

• Gain inconsistency. The thickness of the multiplication layer of each APD pixel in the APD array cannot be uniform, which affects the gain consistency of the APD arrays. The gain consistency difference is a systematic error that can be corrected by calibration [18].
• Crosstalk. The crosstalk of the APD array mainly has two kinds of photon scattering and photogenerated carrier diffusion. Photon scattering refers to the scattering of photons emitted from the multiplication area to the adjacent APD pixels when the avalanche multiplication of photoelectrons occurs. When the number of incident photons is the same between adjacent pixels of the APD array, the effect of photon scattering on the SNR of the LM-APD array is smaller than ${10}^{-6}$ which can be neglected. Photogenerated carrier diffusion refers to the formation of photogenerated carriers by photon irradiation of APD pixels, and some of the photogenerated carriers diffuse through the substrate in the APD array to the surrounding pixels causing crosstalk [19]. The effect of photogenerated carrier diffusion is inversely proportional to the image element spacing and the image element size, and according to the experiment, when the image element size is 205 × 205 μm and the image element spacing is 115 μm, the effect of photogenerated carrier diffusion on the signal-to-noise ratio is lower than ${10}^{-5}$ and can be negligible when the number of incident photons between adjacent pixels of the LM-APD array is almost the same. Therefore, the effect of inter-image crosstalk on 3D imaging can be neglected.
• Time delay inconsistency. The time delay inconsistency is mainly due to the inconsistency of the flight time caused by the beam divergence angle and the inconsistency of the corresponding circuit of each pixel. The inconsistency of the time delay due to the beam divergence angle is the time delay between the photon flight time of each pixel corresponding to the field of view and the photon flight time of the center of the field of view due to the spatial distribution of the beam in the far field, and the time delay due to the beam reflection angle can be corrected by the angular value between the center axis of the field of view corresponding to each APD pixel and the center axis of the APD array and the relative distance value of the target [20]. The time delay due to circuit inconsistency can be neglected by proper circuit design and calibration.

It can be seen that each APD pixel in the APD array can be considered as a separate imaging unit with an independent, instantaneous field of view in 3D imaging thanks to a proper circuit design and calibration method.

(3)

Response to light

After the return of the laser echo reflected from the target is received by the receiving optical system, the field of view of the space target laser 3D imaging system is small, and the effect of lens distortion on imaging can be ignored. The field of view of the receiving optical system is generally consistent with the laser beam divergence angle, and the optical axis is parallel. In order to fully utilize the laser energy, the received laser echo converges on the LM-APD array located in the image plane, and the LM-APD array realizes the photoelectric conversion of the laser echo, converting the laser echo power into signal photocurrent.

For the LM-APD pixel $(m,n)$, the relationship between the signal photocurrent $i_s(m,n,t)$ at the time $t$ and the laser power $P_r(m,n,t)$ received by the pixel is given by Equation (7),

$$i_s(m,n,t)=P_r(m,n,t)R_{\mathrm{resp}}$$

(7)

in which $R_{\mathrm{resp}}$ is the short-circuit photocurrent generated by the LM-APD imaging element per unit of optical power as shown in Equation (8),

$$R_{\mathrm{resp}}=\frac{{\eta }_{\mathrm{eff}}{e}_{\mathrm{elec}}}{h_{\mathrm{P}}{v}_{\mathrm{freq}}}{M}_{\mathrm{APD}}$$

(8)

in which ${\eta }_{\mathrm{eff}}$ is the quantum efficiency of the LM-APD, generally 30–90%, and approximately 60% for Si-based LM-APD; $e_{\mathrm{elec}}$ is the electron charge number; $h_{\mathrm{P}}$ is the Planck constant; $v_{\mathrm{freq}}$ is the photon frequency; and $M_{\mathrm{APD}}$ is the multiplication factor of the LM-APD, as shown in Equation (9),

$$M_{\mathrm{APD}}=\frac{1}{1-{(V_{\mathrm{Bias}}/V_{\mathrm{B}})}^{n_{\mathrm{APD}\_\mathrm{M}}}}$$

(9)

in which $V_{\mathrm{Bias}}$ is the bias voltage, $V_{\mathrm{B}}$ is the breakdown voltage, $n_{\mathrm{APD}\_\mathrm{M}}$ is the coefficient related to the fabrication material, Si-based LM-APD is generally 1.8~4, and Ge-based LM-APD is generally 2.8~8. $V_{\mathrm{B}}$ and $n_{\mathrm{APD}\_\mathrm{M}}$ are affected by the electron hole ionization rate and are therefore temperature dependent. When using LM-APD arrays for 3D imaging, it is necessary to design temperature compensation circuits to adaptively change according to the temperature, adjusting the bias voltage to ensure the stable output of signal photocurrent.

(4)

Noise characteristics

The space-based laser 3D imaging system needs to avoid the appearance of celestial bodies, such as the sun and the moon, in the field of view, and the background light noise can be neglected. The main noise in the LM-APD output current is signal light scattering particle noise, dark current noise, and thermal noise.

Signal light scattering noise $i_{\mathrm{ns}}$ is involved in the multiplication process, and its current spectral density is shown in Equation (10),

$$i_{\mathrm{ns}}(m,n,t)=\sqrt{2e_{\mathrm{elec}}{B}_{\mathrm{b}\mathrm{w}}{P}_{\mathrm{r}}(m,n,t)R_{\mathrm{resp}}{M}_{\mathrm{APD}}\Delta F}$$

(10)

in which $B_{\mathrm{b}\mathrm{w}}$ is the system equivalent band width. $\Delta F$ is the excess noise factor as shown in Equation (11).

$$\Delta F=M_{\mathrm{APD}}\left[1-(1-k_{\Delta F}){(M_{\mathrm{APD}}-1)}^2/M_{\mathrm{APD}}^2\right]$$

(11)

in which $k_{\Delta F}$ is the hole electron ionization ratio, which is related to the material of LM-APD, and $k_{\Delta F}\approx 0.02$ for the Si LM-APD.

Dark current noise $i_{\mathrm{n}\_\mathrm{d}}$ is caused by breeding dark currents in the LM-APD circuit, whose current spectral density is shown in Equation (12),

$$i_{\mathrm{n}\_\mathrm{d}}=\sqrt{2e_{\mathrm{elec}}(i_{\text{n\_ds}}+i_{\text{n\_db}}{M}_{\mathrm{APD}}^2\Delta F)B_{\mathrm{bw}}}$$

(12)

in which $i_{\text{n\_ds}}$ is the LM-APD surface leakage current and $i_{\text{n\_db}}$ is the LM-APD body leakage current.

The thermal noise $i_{\mathrm{n}\_\mathrm{t}}$ is caused by the thermal motion of electrons with the current spectral density as shown in Equation (13),

$$i_{\mathrm{n}\_\mathrm{t}}=\sqrt{4k_{\mathrm{B}}{T}_{\mathrm{K}\_\mathrm{APD}}{B}_{\mathrm{bw}}/R_{\mathrm{L}}}$$

(13)

in which $T_{\mathrm{K}\_\mathrm{APD}}$ is the absolute temperature of the LM-APD pixel and $R_{\mathrm{L}}$ is the load impedance of the LM-APD pixel.

Signal light scattering noise, background light scattering noise, dark current noise, and thermal noise are independent of each other, and the signal-to-noise ratio of the LM-APD output current signal is shown in Equation (14).

$$SN{R}_{\mathrm{APD}}=\frac{i_{\mathrm{s}}(m,n,t)}{\sqrt{i_{\mathrm{ns}}^2(m,n,t)+i_{\mathrm{nd}}^2+i_{\mathrm{nt}}^2}}$$

(14)

2.1.4. Amplifiers

The output current value of LM-APD is on the microamperes scale, and a transimpedance amplifier is usually used to convert the output current signal into a voltage signal and amplify it. In order to meet the signal voltage range requirements of the time-discriminating circuit, it is necessary to cascade an automatic gain amplifier after the transimpedance amplifier to amplify again. While amplifying a signal, the amplifier also amplifies noise and introduces the inherent noise of the amplifier. In terms of practical application, the selection of amplifier needs to consider input voltage range, gain bandwidth, voltage-swing rate, and equivalent input voltage noise spectral density [20]. The effect of the amplifier on 3D imaging is mainly reflected in the effect on the signal-to-noise ratio, which can be described by the noise factor of the amplifier $F_{\mathrm{A}}$, as shown in Equation (15).

$$F_{\mathrm{A}}=\frac{SN{R}_{\mathrm{APD}}^2}{SN{R}_{\mathrm{A}}^2}$$

(15)

in which $SN{R}_{\mathrm{A}}$ is the output signal-to-noise ratio of the amplifier, $F_{\mathrm{A}}=1$ in an ideal amplifier, and in practice, $F_{\mathrm{A}}>1$. The larger $F_{\mathrm{A}}$ is, the more serious the influence of amplifier on the signal to noise ratio of laser echo signal is. The noise factor is related to the amplifier’s own characteristics and working conditions, as shown in Equation (16),

$$F_{\mathrm{A}}=1+\frac{E_{\mathrm{n}}^2+{(I_{\mathrm{n}}{R}_{\mathrm{s}})}^2}{4KT{R}_{\mathrm{s}}\Delta f}$$

(16)

in which the equivalent noise voltage $E_{\mathrm{n}}$ and the equivalent noise current $I_{\mathrm{n}}$ are inherent properties of the amplifier, factors, e.g., source load resistance $R_{\mathrm{s}}$ and circuit $\Delta f$ also remain constant, i.e., $F_{\mathrm{A}}$ is not affected by the signal power.

In the process of 3D imaging of the space target, the change of the relative pose between space target and imaging system will lead to a wide range of laser echo power changes. In order to ensure the working distance and dynamic range of 3D imaging, it is usually necessary to amplify the echo signal in multiple stages. The gain of $N$ amplifiers $G_{N\mathrm{A}}$ and the noise factor $F_{N\mathrm{A}}$ are shown in Equation (17),

$$\{\begin{array}{l}{G}_{N\mathrm{A}}=G_1{G}_2\dots G_n\hfill \\ F_{N\mathrm{A}}=F_1+\frac{F_2-1}{G_1}+\frac{F_3-1}{G_1{G}_2}\dots +\frac{F_n-1}{G_1{G}_2\dots G_n}\hfill \end{array}$$

(17)

in which $G_n$ and $F_n$ are the noise factor and power gain of each amplifier stage, respectively. The final signal-to-noise ratio of the signal output from the multi-stage amplifier is shown in Equation (18).

$$SN{R}_{N\mathrm{A}}=\sqrt{F_{N\mathrm{A}}SN{R}_{\mathrm{APD}}^2}$$

(18)

As Equations (14) and (18) show, the longer the working distance of LM-APD array during 3D imaging is, the larger the distance error will be when the relative attitude remains unchanged.

2.1.5. Time Identification

Time identification refers to extracting the echo cutoff time as the timing stop signal of the laser echo flight time, which mainly includes the frontier threshold method, high-pass resistance method, and constant-ratio timing method. The constant ratio timing method uses the constant trigger ratio to avoid the detection error caused by the amplitude change of echo signal, which is more commonly used in 3D imaging. Figure 4 shows its schematic. The constant ratio timing circuit needs to add a comparator with a threshold voltage of $V_{\mathrm{T}}$ to control its working ability to avoid being triggered by noise. The value of $V_{\mathrm{T}}$ will affect the system’s false alarm probability, and its performance in 3D imaging is outlier noise in point cloud.

Constant ratio timing divides the signal into delay and attenuation channels and calculates the difference. The moment when the two channels are equal is the timing moment, as shown in Equation (19),

$$V_{\mathrm{AMP}}{f}_{\mathrm{AMP}}(t-t_{\mathrm{delay}})=V_{\mathrm{AMP}}{k}_{\mathrm{attenuat}}{f}_{\mathrm{AMP}}(t_{\mathrm{T}})$$

(19)

in which $t_{\mathrm{delay}}$ is the delay of the first way, $k_{\mathrm{attenuat}}$ is the attenuation coefficient of the second way, usually $k_{\mathrm{attenuat}}=0.2~0.5$. The time identification of the constant ratio timing method can avoid the influence of the timing error of laser echo amplitude change but cannot avoid the influence of laser echo waveform change.

2.2. Motion State Analysis of Space Targets

Space targets usually cannot provide a stable and cooperative attitude, and there will be relative motion during the imaging process. The relative motion will have an impact on the results of the 3D imaging, which needs to be analyzed and considered.

2.2.1. Spin State

Space targets need to maintain attitude stability, which is mainly divided into two types: spin stabilization and three-axis stabilization, as shown in Figure 5. Spin stabilization relies on the rotational momentum generated by rapid rotation around the spin axis to keep the spin axis pointing stable in inertial space, while three-axis stabilization relies on the active attitude control system to keep the three orthogonal axes pointing stable, and the specific pointing of each target is related to its mission objective. Three-axis stabilization has high attitude control accuracy and is the most used attitude control method today.

2.2.2. Relative Position

A space-based laser 3D imaging system needs to control the flight mode of the imaging satellite to keep the target satellite within the working range. The flight mode of the imaging satellite relative to the target satellite is mainly divided into three kinds, namely follow-fly mode, fly-around mode, and rendezvous mode. In practice, several flight models are usually used in combination.

Follow-fly mode means that the imaging satellite and the target satellite maintain the same orbit. In this mode, the relative angular velocity is small, the relative distance is fixed, and the attitude is fixed. Fly-around mode means that the imaging satellite moves in a circular motion with the space target as the center in a horizontal or spatial circle. In the fly-around mode, the relative angular velocity is small, the relative distance is fixed, the relative attitude varies periodically, and the imaging angle is large. The rendezvous model refers to the process of approaching and moving away of the imaging satellite and a section of the target satellite orbit that are close to each other. The rendezvous model has a large relative angular velocity, a large relative distance variation, and a large relative attitude variation.

The imaging capabilities and angles in the fly-around mode and rendezvous mode are more suitable for the 3D imaging of space targets. Among them, the fly-around mode can realize the 360° three-dimensional imaging of space targets and obtain complete information. In practical applications, the space target can be orbited by other observation means to obtain the orbital element of the target satellite. The orbital element of the imaging satellite is designed according to the orbital element of the target satellite, and when the target satellite enters the working area of the imaging system, the target satellite is tracked by the tracking system so that the center of the field of view of the 3D imaging system is always pointed at the target, and the target is continuously imaged during the flight.

According to the orbital element of the target satellite and the orbital element of the designed imaging satellite, the relative position of the two is simulated. It is assumed that the imaging system is designed to fly around with a radius of 200 m, and

Table 3. Orbital elements of target satellite and imaging satellite.

Parameters	Half-Length Axle/km	Eccentricity	Track Inclination/°	Perigee Amplitude/°	Ascending Intersection Equator/°	Average Point Angle/°
Target satellite	7073.22	0.001521	98.105	60.047	314.880	164.242
Imaging satellite	7073.22	0.001535	98.104	59.904	314.881	164.385

shows the corresponding orbital elements of the two.

Add a sensor to the imaging satellite as a Lidar and set its $z$ axis direction to always point to the center $o_{\mathrm{t}}$ of the target satellite coordinate system. As shown in Figure 6, the body coordinate system of imaging satellite is $o_{\mathrm{o}}-x_{\mathrm{o}}{y}_{\mathrm{o}}{z}_{\mathrm{o}}$, the Lidar coordinate system is $o_{\mathrm{s}}-x_{\mathrm{s}}{y}_{\mathrm{s}}{z}_{\mathrm{s}}$, and the space target body coordinate system is $o_{\mathrm{t}}-x_{\mathrm{t}}{y}_{\mathrm{t}}{z}_{\mathrm{t}}$. Hence, the target satellite’s positional relationship in the Lidar coordinate system can be described by the positional transformation relationship between $o_{\mathrm{t}}-x_{\mathrm{t}}{y}_{\mathrm{t}}{z}_{\mathrm{t}}$ and $o_{\mathrm{s}}-x_{\mathrm{s}}{y}_{\mathrm{s}}{z}_{\mathrm{s}}$, i.e., the translation vector is ${\mathit{t}}_{\mathrm{s}-\mathrm{t}}$ and the rotation vector is $(r_x,r_y,r_z)$.

The target satellite in the simulation is three-axis stabilized with the z-axis pointing to the center of the Earth, the orbital period duration is 5925 s, and the simulation time length is 3 orbital periods. The relative attitude and distance between the imaging satellite and the target satellite with time obtained from the simulation are shown in Figure 7.

Figure 7a shows that the relative position of the two has changed continuously during the imaging process, and the change of the relative position will have a significant effect on the laser echo. Moreover, the 3D imaging simulation must consider the relative position relationship at the moment of imaging. Figure 7b shows that the relative distance between the two is 200 m on average, but it does not always remain 200 m and fluctuates within a certain range. The space-based 3D imaging system usually uses a fixed-focus lens, and the fluctuation of the distance value will affect both the echo power and the point cloud density of the 3D imaging of the space target.

2.2.3. Analysis of the Effect of Relative Motion on 3D Imaging

(1)

Drift in Pixel ${\omega }_{\mathrm{t}}$

Due to the fixed position between the pixel, the spatial error only needs to consider the drift in pixel caused by the relative motion of the LM-APD array and the space target. During the interaction between the laser pulse and the space target surface, both the spin and the relative motion cause a shift of the echo in the direction of the line of sight, which is called the drift in pixel. The severity of the effect of the drift in pixel on imaging can be evaluated by the ratio of the relative offset angle ${\delta }_{\mathrm{l}}$ of the optical axis pointing before and after the drift to the field of view angle in a pulse, and the smaller the ratio to the field of view angle, the smaller the effect of the drift on imaging, as shown in Equation (20),

$${\delta }_{\mathrm{l}}=\frac{\tau ({\omega }_{\mathrm{t}}+v_{\mathrm{st}\_\mathrm{h}}/R_{\mathrm{st}}+{\sigma }_{\mathrm{s}})n_{\mathrm{APD}}}{{\theta }_{\mathrm{FOV}}}$$

(20)

in which ${\theta }_{\mathrm{FOV}}$ is the receiving optical system field of view, $\tau$ is the pulse width, ${\omega }_{\mathrm{t}}$ is the spin angular velocity of the target relative to the imaging satellite, $v_{\mathrm{st}\_\mathrm{h}}$ is the spatial motion velocity of the target relative to the imaging satellite, ${\sigma }_{\mathrm{s}}$ is the jitter of the imaging satellite itself, and $n_{\mathrm{APD}}$ is the number of LM-APD pixels. The effects of several factors on the drift are expressed as the sum of vectors, which are simplified to scalar summation for the convenience of analysis. Limited by the single pulse energy and detector performance of Lidar, the usual working distance of 3D imaging is in the range of 20–500 m. Figure 8 shows the drift under typical conditions when ${\omega }_{\mathrm{t}}=100 \mathrm{mrad}/s$, and it can be seen that the drift in pixel is below 1 pixel and can be neglected.

(2)

Distance measurement error

In the rendezvous mode, the relative velocity is large, and assuming that the velocity component of the target relative to the imaging satellite in the distance direction during a single laser pulse time is $v_{\mathrm{st}\_\mathrm{v}}$, the average range error caused is shown in Equation (21).

$$\Delta e_v=-v_{\mathrm{st}\_\mathrm{v}}\tau /2$$

(21)

The maximum linear velocity of the vehicle operating in the Earth’s gravitational field is less than the second cosmic velocity of 11.2 km/s. In 3D imaging, a laser with a pulse width of less than 10 ns is typically used, and when the relative velocity between the space target and the imaging system is 22.4 km/s, the limiting range error is still less than 0.1 mm, which can be ignored.

In summary, due to the short duration of the laser pulse, the relative motion during the single imaging process has a negligible effect on the three-dimensional imaging.

2.3. Analysis of the 3D Imaging Performance of LM-APD Arrays

The 3D imaging performance of the LM-APD array is affected by the shape, material, and attitude of the space target, and it is difficult to make an accurate quantitative evaluation. This section analyzes the statistical characteristics of each key index based on the relevant influencing factors.

(1)

Field of view

The 3D imaging field of view is determined by the field of view of the receiving optics system ${\theta }_{\mathrm{FOV}}$, as shown in Equation (22).

$${\theta }_{\mathrm{FOV}}=D_{\mathrm{APD}}/f_{\mathrm{r}}$$

(22)

where $D_{\mathrm{APD}}$ is the physical size of the LM-APD array and $f_{\mathrm{r}}$ is the equivalent focal length of the receiving optical system. The 3D imaging field of view at a distance of $R_{\mathrm{st}}$ from the system is ${\theta }_{\mathrm{FOV}}^2{R}_{\mathrm{st}}^2$, and normally ${\theta }_{\mathrm{FOV}}^2{R}_{\mathrm{st}}^2$ should be larger than the projected area of the target in the field of view axially to ensure the target integrity in the field of view.

(2)

Angular resolution

The angular resolution of the 3D imaging system, i.e., the corresponding instantaneous field-of-view angle of the LM-APD pixel, is shown in Equation (23).

$${\theta }_{\mathrm{IFOV}}=d_{\mathrm{APD}}/f_{\mathrm{r}}$$

(23)

where $d_{\mathrm{APD}}$ is the size of LM-APD and the instantaneous field of view of each pixel at a distance of $R_{\mathrm{st}}$ is $R_{\mathrm{s}-\mathrm{t}}{\theta }_{\mathrm{r}\_\mathrm{s}}$.

(3)

Ranging error

The ranging error is expressed as noise in the point cloud in 3D imaging and is shown in Equation (24) under the condition that errors, such as fixed delay (caused by delays in hardware circuits), have been corrected in advance.

$$e_{\mathrm{d}}=\Delta R_{\tau }+\epsilon \sqrt{{\sigma }_{\mathrm{CFD}}^2+{\sigma }_{\mathrm{t}}^2}$$

(24)

where $\Delta R_{\tau }$ is the error caused by the variation of the echo waveform, which cannot be described by statistical features [18,20] and cannot be corrected by multiple measurements, and there is no suitable correction method. $\epsilon$ is the random error confidence coefficient, ${\sigma }_{\mathrm{CFD}}^{}$ is the moment identification error of the constant ratio timing circuit, and ${\sigma }_{\mathrm{t}}^{}$ is the timing circuit accuracy error. ${\sigma }_{\mathrm{CFD}}^{}$ and ${\sigma }_{\mathrm{t}}^{}$ are both random errors.

${\sigma }_{\mathrm{CFD}}^{}$ is the timing drift due to the noise superimposed on the amplifier output signal, the rising edge of the echo signal is approximated as a Gaussian signal. According to analysis of Equation (23), the moment discrimination error ${\sigma }_{\mathrm{CFD}}^{}$ is determined by the signal signal-to-noise ratio and the signal pulse width as shown in Equation (25) [20].

$${\sigma }_{\mathrm{CFD}}=\frac{c{\tau }_{\mathrm{r}}\sqrt{1+k_{\mathrm{attenuat}}^2}}{SN{R}_{N\mathrm{A}}(t_{\mathrm{T}})\cdot (1-k_{\mathrm{attenuat}})}$$

(25)

where ${\tau }_{\mathrm{r}}$ is the rising edge width of the echo signal; $k_{\mathrm{attenuat}}$ is the timing ratio which usually taken as 0.4~0.5; $SN{R}_{N\mathrm{A}}$ is the signal-to-noise ratio of the signal input to the constant ratio timing circuit, i.e., the signal-to-noise ratio of the output signal of the multi-stage amplifier.

The timing circuit accuracy error ${\sigma }_{\mathrm{t}}^{}$ is the error caused by the fact that the stop signal may appear at any position of the minimum cycle of timing accuracy due to the limited accuracy of the timing circuit. Both start timing and stop timing introduce errors, and they are independently distributed [20], so the timing circuit accuracy error is shown in Equation (26).

$${\sigma }_{\mathrm{t}}=\frac{c{T}_{\mathrm{t}}}{2\sqrt{6}}$$

(26)

where $T_{\mathrm{t}}$ is the minimum time resolution of the timing circuit, typically better than 100 ps. It can be seen that ${\sigma }_{\mathrm{CFD}}^{}$ is the main component of the random error.

(4)

False alarm rate

The false alarm rate is mainly related to the number of outlier points in the 3D imaging point cloud. The false alarm probability is influenced by the threshold comparison voltage $V_{\mathrm{T}}$ in the constant ratio timing circuit, and when the noise is greater than $V_{\mathrm{T}}$, the constant ratio timing circuit will be false triggered, which is manifested as outliers in the point cloud in 3D imaging.

When the noise is approximated as a Gaussian white noise with root mean square ${\sigma }_{\mathrm{n}}$, the detection probability is shown in Equation (27) [20,21],

$$P_{\mathrm{d}}=\frac{1}{2}+\frac{1}{2}\mathrm{erf}(\frac{SN{R}_{N\mathrm{A}}-TNR}{\sqrt{2}})$$

(27)

in which $TNR$ is the threshold noise ratio which equal to $V_{\mathrm{T}}/{\sigma }_{\mathrm{n}}$. When within the distance gate width time of the 3D imaging, a false alarm signal is formed if the noise voltage is higher than $TNR$, and the corresponding false alarm rate $\overline{FAR}$ is shown in Equation (28),

$$\overline{FAR}=2.2\frac{2R_{\max }}{c}\Delta f[\frac{1}{2}-\frac{1}{2}\mathrm{erf}(\frac{TNR}{2})]$$

(28)

in which $R_{\max }$ is the distance selective pass gate width of the system. When $R_{\max }=1 \mathrm{km}$, $\Delta f=12 \mathrm{MHz}$, the detection probability and false alarm rate are related to $TNR$ as shown in Figure 9.

Normally, the detection rate needs to be higher than 99%, and the value of $TNR$ cannot be too high in order to ensure the detection rate when the signal-to-noise ratio of the laser echo signal is low. However, Figure 9 shows that when $TNR$ is too low, the false alarm rate, which is the proportion of outlier points in the 3D imaging point cloud, will increase accordingly. The value of $TNR$ is usually between 3 and 6, and $TNR$ should be greater than or equal to 5 under the simulation conditions.

In summary, the noise in the point cloud is determined by the range error and the number of outliers is determined by the false alarm rate. The RMS of the point cloud noise is related to factors such as target distance, laser pulse width, and timing accuracy, and the number of outliers in the point cloud is related to factors, such as the $TNR$ of the constant ratio timing circuit.

3. LM-APD Array Laser 3D Imaging Simulation Method

The simulation of laser echo is realized by considering the comprehensive space target shape, material, and real-time position relationship with the system. Then, according to the echo power and signal-to-noise ratio of each pixel in LM-APD, the current distance value, range error, and false alarm rate are calculated to realize the simulation of 3D imaging, and the main process of simulation is shown in Figure 10. The simulation ignores the correctable system errors, such as response inconsistency, pixel response time, circuit delay, and so on.

The coordinates and attitude of the target in the imaging satellite coordinate system are obtained based on the orbital element of the space target and the imaging satellite, and then the target is converted from the imaging satellite coordinate system to the laser 3D imaging system coordinate system to obtain the coordinates and attitude of the target in the 3D imaging system coordinate system. The 3D model used in the simulation is NASA’s Mars Reconnaissance Orbiter (MRO) [22], and Figure 11 shows the three views of the model.

The focal length of the receiving optical system and the relative attitude of the Lidar are used to calculate the field of view at the distance of $R_{\mathrm{st}}$. The visible area of the target at the current attitude is obtained by fading. The visible area $Q$ is discretized into dense surface elements. The number of surface elements depends on the required accuracy of the simulation. The greater the number of surface elements, the higher the accuracy of the simulation, but the complexity also increases accordingly. The visible area $Q$ is illuminated to obtain the material of each surface element and the angle of laser incidence and normal vector angle. Figure 12 shows the MRO model after illumination processing when the relative pitch angle of the target is 0°, 5°, 10°, and 30°. The color information of each face element is used to mark the corresponding materials of different face elements: white, yellow, and gray correspond to the antenna, coating, and solar panel on the MRO, respectively. According to the material of each surface element and the corresponding laser incidence angle, the BRDF of each surface element can be obtained from Equation (6).

The incident and reflect of laser pulse is a continuous process, and the duration is the laser pulse width. At any moment, the LM-APD pixel corresponds to the reflection of laser energy by each surface element on the target surface in the field of view, which jointly determines the return energy. The phase information carried by each surface element is lost in LM-APD array 3D imaging, and the joint action of each surface element is expressed as a direct superposition of the return power of each surface element. As shown in Figure 13, $Q$ is the visible area of the target surface in the field of view of the LM-APD array, and $q_0{q}_1\dots q_n$ is the surface element of $Q$ after discretization. At the time of $t$, the LM-APD array will receive the echo energy of the laser energy reflected from the surface element in $Q$ at a distance of $2R_{\mathrm{st}}/c$ from the imaging system at the time of $t-2R_{\mathrm{st}}/c$, as well as from any other surface element in $Q$ at a distance of $2(R_{\mathrm{st}}+z_n)/c$ at the time of $t-2(R_{\mathrm{st}}+z_n)/c$. The echo energy values of these surface elements $P_r(t)$ are superimposed together to form the echo power received by the LM-APD array at the time of $t$.

The subdivided individual surface elements can be considered to have the same material and laser incidence angle, i.e., the reflection of laser energy from the visible region of the target surface $Q$ in the surface element at $t$ can be considered as a linear system with the unit impulse response, as shown in Equation (29).

$$h(t)={\displaystyle {\iint }_Q{\rho }_{\mathrm{r}}(x,y)\delta \left\{t-[R_{\mathrm{st}}+z(x,y)]/c\right\}\mathrm{d}x\mathrm{d}y}$$

(29)

where $z(x,y)$ is the surface equation of each surface element in the visible region $Q$ with $R_{\mathrm{st}}$ as the center. The power $P_{\mathrm{r}}(t)$ received by the LM-APD array is the convolution $E\left(R_{\mathrm{st}},t\right)$$\ast h(t)$ of the unit impulse response $h(t)$ and the laser irradiance distribution $E\left(R_{\mathrm{st}},t\right)$ in the visible region $Q$ of the target surface. Based on the pixel distribution of the LM-APD array, the corresponding partition of each pixel in the visible region $Q$ can be obtained from $Q(m,n)$. Based on the corresponding ${\rho }_{\mathrm{r}}(x,y)$ and surface element distance of each surface element in $Q(m,n)$, the return power received by the pixel $P_{\mathrm{r}}(m,n,t)$ with coordinates $(m,n)$ in the LM-APD array at $t$ can be obtained from Equation (29), as shown in Equation (30).

$$P_{\mathrm{r}}(m,n,t)=\frac{{\eta }_{\mathrm{r}}{A}_{\mathrm{r}}}{4\pi R_{\mathrm{st}}^2}{\displaystyle {\iint }_{Q(m,n)}E\left\{R_{\mathrm{st}}^2,t-[R_{\mathrm{st}}+z(x,y)]/c\right\}}{\rho }_{\mathrm{r}}(x,y)\mathrm{d}x\mathrm{d}y$$

(30)

where ${\eta }_{\mathrm{r}}$ is the receiving optical system transmittance and $A_{\mathrm{r}}$ is the receiving optical system area. Equation (30) shows that the surface affects both the amplitude and waveform of the laser echo.

When the laser pulse width is 10 ns and the incident angle is 0, the laser waveform and the echoes from different parts of the target after normalization are shown in Figure 14. Figure 14a shows that the laser echoes obtained with the simulation present a double peak phenomenon due to the existence of a certain longitudinal depth between the mirror barrel part and the satellite body part. Figure 14b shows that the simulated laser echoes show the spreading phenomenon because the antenna is curved, and the simulation results are consistent with the actual situation.

According to the CFD principle in Equation (19), for the pixel with the coordinates of $(m,n)$, the target distance value of the corresponding region is shown in Equation (31).

$$R(m,n)=\frac{c}{2}\underset{t}{\arg }\left\{P_0(m,n,t)=\frac{1}{2}\max [P_0(m,n,t)]\right\}-\frac{c}{2}\underset{t}{\arg }\left\{P_{\mathrm{r}}(m,n,t)=\frac{1}{2}\max [P_{\mathrm{r}}(m,n,t)]\right\}$$

(31)

where $\underset{t}{\arg }$ is the value of $t$ that satisfies the function requirement. In cases where the hardware parameters of the constant ratio timing circuit are known, the extraction of the target distance value can also be implemented by building the corresponding analog circuit in Simulink according to the literature [23], in order to accurately characterize the effects of noise and device parameters on the moment identification accuracy. Figure 15a shows the circuit diagram of the constant ratio timing circuit for signal attenuation with delay to generate the over-zero signal. Figure 15b shows the analog echo signal and the converted over-zero signal, and the position of the over-zero signal is the position of the stopping time.

Based on the power value of the laser echo, the SNR of the signal passing through the LM-APD pixel and amplifier is calculated, and the standard deviation in noise of the point cloud is obtained from Equation (24). Then, the number of outlier points of the point cloud is obtained from Equation (28) based on the $TNR$. The noise is added as additive noise to the coordinates of each point in the point cloud obtained by Equation (31). The final simulation results are obtained by adding the outlier points according to the outlier number calculated by the selected pass gate width. The difference between the simulated and real values of 3D laser imaging in this paper mainly exist in the statistical noise model, because the statistical model can only describe the data statistically and cannot accurately describe each imaging process.

4. Results and Discussion

The setting of system parameters mainly considers two application scenarios of simulation: one is to analyze the influence of different system parameters on the final imaging results during system design; the other is to simulate the existing system to obtain the imaging results of the system for different targets under different conditions for data analysis and point cloud processing algorithm research. For the first application scenario, the parameters can be set according to the system design considerations. For the second application scenario, the system parameters, such as dispersion angle, pulse width, etc., can be obtained by referring to the system data sheet or measurement, and the target parameters are determined according to the simulation requirements. The main parameters of simulation in the following charter are listed in

Table 4. Main parameters of LM-APD array 3D imaging simulation.

Model	Part	Parameters
Lasers	Dispersion angle/mrad	2
	Pulse energy/mJ	5
	Pulse width/ns	5
Emit optics system	Beam expansion ratio	1/15
	Lens transmittance	0.95
Target	Size/m	15 × 10.25 × 5.5
	Distance/m	500
Timing Circuit	Timing accuracy/ps	100
Receiving optics System	Field of view/mrad	30
	Lens aperture/m	0.5
Si-LM-APD Array	Number of pixels	256 × 256
	Image size/μm	150 × 150
	Sensitivity A/W	25
	Bandwidth/MHz	120
	Temperature/K	298.15
	Load impedance/Ω	50
Amplifier	TIA noise factor	1.05
	AGC noise factor	1.15
	Number of AGC levels	2

.

4.1. Comparative Analysis of the Results of Different Imaging Conditions

4.1.1. Simulation Results of the Target in Different Positions

Under the same conditions of other parameters, when the field of view and the target are perfectly matched, Figure 16 shows the simulation results of the 3D imaging of space targets rotated around the pitch axis at different angles.

Under the same conditions of other parameters, when the field of view and the target are perfectly matched, Figure 17 shows the simulation results of the 3D imaging of space targets rotated around the azimuth axis at different angles.

From Figure 16 and Figure 17, it can be seen that the relative attitude has a great effect on the 3D imaging. For example, regarding the solar sails of the space target, when the azimuth angles are rotated by 60 degrees, there is a serious error in distance due to the weak echo power, which is manifested as noise in the point cloud. From Figure 16 and Figure 17, it can also be seen that different parts of the space target have different reflectivity to the echoes at the same attitude due to different materials, and thus the noise distribution is also different. The simulation results show that the 3D imaging simulation method can reflect the influence of relative attitude and material on the 3D imaging results correctly.

4.1.2. Simulation Results of LM-APD Arrays with Different Number of Pixels at Different Imaging Distances

Under the same conditions of other parameters, Figure 18 shows the simulation results with LM-APD arrays of different number of pixels.

Under the same conditions of other parameters shown in Figure 18, Figure 19 shows the simulation results when the distance is double that presented in Figure 18. The physical scale at the target distance can be calculated from the field of view, so the change of the relative distance of the target will only change the number of sampling points and the echo energy of the point cloud, and not the size of the target.

Figure 18 shows the simulation results of 3D imaging of LM-APD arrays with different numbers of pixels when the distance is changed.

Figure 19 shows that there are large differences in the imaging performance of LM-APD arrays with a different number of pixels. The number of points contained in the space target point cloud obtained from the LM-APD array for the four frame numbers in Figure 18 are 210, 832, 3304, and 13,205, respectively. The simulation uses the ideal case when the target and the field of view are perfectly matched, and the number of points in the actual application will be reduced to account for the variation in distance and attitude. For example, when the relative distance is increased by a factor of 2, as Figure 20 shows, the number of points in the point cloud is 48, 196, 781, and 3149, which basically shows a trend of decreasing the number of points in proportion to the square of the distance. When the number of pixels in the LM-APD array is small, the feature information of the space target is limited. The technical report of TriDAR also pointed out that the point cloud of the space target should contain at least 1000 points to be of greater value.

4.1.3. Comparative Analysis with Real LM-APD 3D Imaging Results

The relevant image data using the LM-APD array 3D imaging system for flight targets and space targets [1,24] are collected to qualitatively compare and analyze the effect of the simulation method.

Figure 21 shows the result of ASC’s DragonEYE 3D imaging of the aircraft, and it shows that the point cloud obtained with the LM-APD array 3D imaging has less noise in the image plane, and the location of points in the point cloud shows an array distribution. We can conclude that the LM-APD array 3D imaging noise is mainly distributed in the distance axis, which is consistent with the simulation results in Figure 16 and Figure 20. The results are consistent with the simulation results of this paper.

Figure 22 shows the result of ASC’s DragonEYE 3D imaging of the ISS on the Space Shuttle at different relative positions, with the ISS docking device shown in the blue box and the ISS surface covered with insulation in the red box. Compared with Figure 22a–c, it can be seen that the point cloud of the ISS has different degrees of noise at different relative positions, and the noise distribution in the area of the docking device is different from that of the insulation material due to the difference in the metal of the docking device and the material of the insulation material. The part of the docking device shown in Figure 22a is less noisy, and the outline of the docking device can be identified. There is more noise in the docking device shown in Figure 22b,c, and the outline of the docking device is no longer recognizable. However, the main area is covered by the insulating material, as shown in Figure 22a–c, which proves that the relative position and the material of the spatial target can jointly affect the 3D imaging results, similar to the noise distribution in Figure 16 and Figure 17.

Qualitative comparison shows that the simulation method in this paper agrees well with the imaging results of the real LM-APD array 3D imaging, and further allows to describe the characteristics of the LM-APD array 3D imaging of space targets.

4.2. Space-Based Imaging Simulation and Analysis

In the case of a fly-around model, the relative positional relationship between the space target and the imaging system is simulated in the orbit calculation software during an orbital cycle, and the 3D imaging simulation is completed based on the relative position, target model, and system simulation parameters.

4.2.1. Simulation Parameters

The LM-APD array 3D imaging system is mounted on the imaging satellite and connected to the imaging satellite via a shortcut link. During the 3D imaging process, the target is kept in the center of the field of view by using auxiliary tracking systems such as infrared cameras. Assuming that the orbital root number of the space target has been obtained by other observation means, the orbital root number of both is shown in Table 3. According to the orbital root number of the space target, the imaging satellite is designed to maintain an average distance of 500 m to fly around it. The attitude and distance variation of the space target with respect to the imaging satellite is shown in Figure 7. The field of view of the imaging satellite is shown in Figure 23, where the purple circle is the visual effect of the imaging field of view of the space target by the sensor, i.e., laser 3D imaging system, on the imaging satellite, the blue coordinate system is the ontological coordinate system of the space target, and the green coordinate system is the coordinate system of the sensor on the imaging satellite. The z-axis of the sensor coordinate system is kept pointing to the origin of the space target body coordinate system to simulate the tracking effect of the sensor on the target. With the sensor coordinate system as the target coordinate system, the relative position of the target in the sensor coordinate system is obtained by computing the quaternion from the space target body coordinate system to the sensor coordinate system. The distance of the space target relative to the imaging satellite can be obtained from the real-time distance between the origin of the space target ontology coordinate system and the origin of the imaging satellite ontology coordinate system.

The main parameters of the space target and LM-APD array 3D imaging system simulation used in the simulation are shown in Table 4 (Pulse energy adjusted to 1 mJ). The simulation time length is one orbital cycle, i.e., 5924.521 s, and the imaging interval of the imaging system for space targets is 300 s, taking into account the field-of-view overlap rate of 3D imaging.

4.2.2. Simulation Results

The space target point clouds of simulation at the moments t = 3000 s, t = 4200, and t = 5400 s are shown in Figure 24.

As can be seen from Figure 24a, the LM-APD array 3D imaging shows a regular array distribution of sampling points with less imaging noise due to the fixed spatial resolution of the LM-APD array 3D imaging. In Figure 24b, the LM-APD array 3D imaging shows more noise in the distance direction, which is especially obvious in the solar sail part. When t = 3000 s, due to the different laser incidence angles and different materials of the main part of the space target, the laser incidence angle at the solar sail is smaller, the echo is weaker, and the echo signal-to-noise ratio is lower, so the noise at the solar sail is more serious than that at the main part of the space target. When t = 4200 s, the solar sail normal vector is perpendicular to the laser incidence angle, so the solar sail part is not visible. When t = 5200 s, the normal vector of the solar sail is close to perpendicular to the laser incidence angle, the return power is lower, and the noise is especially severe. Analysis of Figure 24 shows that even at 500 m distance to the space target three-dimensional imaging, the simulation results obtained by the parameters in Table 4, when the laser incidence angle and the space target surface normal vector angle difference is large, the corresponding parts will still present serious noise, which is also consistent with the current status that the space target 3D imaging working distance is still within 200 m. The direct reason for the noise is the poor reflection ability of the space target with respect to the laser in some attitudes and the low echo power, and the indirect reason is that the space-based platform limits the laser power and lens aperture. The noise in the point cloud does not show obvious statistical characteristics in the individual imaging results and is difficult to filter out directly in the point cloud.

5. Conclusions

We first analyze the main devices associated with the laser echo in the LM-APD array 3D imaging link and point out that the space target material and the laser incidence angle have a great influence on the waveform and energy of the laser echo. The influence of the relative motion of the space target and the imaging system on the imaging results during the pulse time is analyzed, and it is pointed out that the relative motion during the pulse time can be neglected. The performance and main influencing factors of the 3D imaging of the space target LM-APD array are analyzed from various aspects, such as spatial resolution, distance resolution, range error, and false alarm rate. Finally, considering the 3D imaging characteristics of the LM-APD array and the motion characteristics of the space target, we propose a simulation method for the 3D imaging of the space target LM-APD array to realize the simulation of the laser echo waveform, which can simulate the laser echo waveform according to the distance and attitude of the target. The laser echo waveform simulation can form a surface array space target point cloud according to the changes of laser echo power and waveform caused by the target distance and attitude. The results of different imaging conditions are analyzed and compared to validate the rationality of the simulation method. The advantages of our method are that it can accurately describe the imaging results of the LM-APD array 3D imaging system for space targets with different shapes, materials, and motion states. The shortcomings are that it is based on the assumption that the shape and material of space targets are known, and it ignores the influence of factors, such as satellite surface folds on the echoes. However, there are great similarities in terms of the shape and material of various space targets, so the characteristics of the LM-APD array 3D imaging system for space targets should be similar. Our method can be applied to the selection of parameters in the design process of a space-based LM-APD array 3D imaging system and to provide point cloud data under various imaging conditions for the study of space-based LM-APD array 3D imaging point cloud data processing. The next step of our research is to compare with real space-based LM-APD array 3D imaging data and optimize the model. Moreover, combining intelligent imaging quality evaluation methods [25,26] to achieve optimal selection of system design parameters under different conditions is also an important future research direction.