A method for estimating the cumulative number of times required for a photon-counting lidar to identify a signal.

Abstract

The method for estimating the cumulative number of detections required for a photon-counting lidar signal identification belongs to the field of lidar technology. It includes calculating the signal detection probability for each detection. P s and noise detection probability P n Photon-counting lidar can be used to repeatedly detect targets within a short period, assuming minimal changes in the target and environment, i.e., constant noise rate and average signal photon count. Utilizing the different detection probabilities of signal and noise, a binomial distribution is used to derive a model and establish the minimum number of cumulative detections required to identify the signal. N The estimation method is as follows: For data with very low signal-to-noise ratio, the minimum number of cumulative counts required to identify the signal is quickly solved by approximating the binomial distribution using a normal distribution, and then corrected to obtain the minimum number of cumulative counts. This method can provide guidance for the system design of lidar. The estimation method for the minimum number of cumulative counts required to identify the signal under certain detection specifications can minimize the number of cumulative counts in the photon counting lidar system while meeting the identification specifications.

Description

Technical Field

[0001] This invention relates to a method for estimating the cumulative number of times required for a photon-counting lidar to identify a signal, belonging to the field of lidar technology. Background Technology

[0002] Photon-counting lidar is a novel detection system that applies single-photon detectors and probability distribution detection theory to traditional lidar technology. It features high accuracy and sensitivity, capable of detecting energy at the photon level reflected from a target. Extremely high sensitivity generally requires extremely high gain. While a single-photon detector can amplify a weak signal at the photon level into an electrical signal detectable by subsequent circuitry, its output cannot maintain a linear relationship with the number of incident photons. That is, a single-photon detector operating in photon-counting mode can only determine the presence or absence of a signal, not its strength. Lidar operating in photon-counting mode typically outputs only two states: "0" or "1". An output of "1" indicates that a photon has been detected, and the corresponding moment is recorded as a photon event. The result of a single detection is a discrete sequence of photon events. Furthermore, because the energy of a signal at the photon level exhibits strong randomness, even with identical input optical signals, the photon events detected each time will differ, making it impossible to independently determine whether they are signals or noise. Multiple detections are required to perform signal processing such as denoising based on the probability distribution of photon events.

[0003] This characteristic of photon-counting lidar means that when detecting a target, multiple detections are typically required to provide effective target information, which greatly limits its spatial resolution in mapping and imaging applications. Currently, the number of accumulations required for photon-counting lidar to distinguish a signal from noise is usually based on user experience. For example, the photon-counting lidar on the ICESat-2 (Ice, Cloud, and Land Elevation Satellite-2) satellite typically accumulates 200 pulses to distinguish signal photons reflected from the ground surface. In laser scanning imaging and non-line-of-sight imaging, to improve the signal-to-noise ratio of the imaging results, 5000 or even more detections are usually accumulated as the detection information for one pixel. However, these accumulation counts are mostly based on data processing results or experience, and no publicly published data provides an estimation method for the minimum number of accumulations required for a photon-counting lidar system to extract a signal under different signal and noise conditions. Summary of the Invention

[0004] The purpose of this invention is to provide a method for estimating the cumulative number of detections required for a photon-counting lidar to identify a signal based on the detection probability of signal and noise. Using the hardware parameters and target parameters of the photon-counting lidar as input, and under a specified detection success rate constraint, the method estimates the cumulative number of detections required to identify the signal from background noise based on the signal and noise levels. Specifically, it includes the following two technical solutions.

[0005] The method for estimating the cumulative number of times required for photon counting lidar to identify a signal based on binomial distribution includes the following steps:

[0006] Step 1: Estimate or measure the background noise detection probability and the average number of signal photons per detection using the lidar equation, estimate the laser echo pulse width, determine the echo signal width, and then estimate the signal detection probability P for each detection. s The noise detection probability P of a noise region with the same length as the signal pulse width n ;

[0007] Step 2: Repeated detection by a photon-counting lidar within a short period assumes minimal changes in the target and environment, i.e., the noise rate and average signal photon count remain constant. Based on this assumption, when repeatedly detecting a target, within the time slice where a signal exists, the probability distributions of signal and noise photon events (PEs) follow parameters (N, P). s ) and (N,P n The binomial distribution of is given, where N is the cumulative number of occurrences, is the quantity to be estimated, and P is the number of occurrences. s and P n These are the single detection probabilities of signal and noise within a specific time interval (laser echo pulse width), calculated from step 1.

[0008] Step 3: Increase the cumulative count N until the overlap area of ​​the probability distributions of the number of photon events of the signal and noise is equal to the probability that the number of noise photon events is greater than the number of signal photon events is just less than the set index D. The cumulative count N obtained at this time is the minimum cumulative count to distinguish between signal and noise.

[0009] Step 3 is described in detail below, and the relationship between the cumulative number of times N and the index D is as follows:

[0010]

[0011] In the formula, D is the set index for distinguishing between signal and noise, and P s Let N be the probability of signal detection, n1 be the cumulative number of times, n1 be the number of photons corresponding to the 100D quantile of the signal distribution, and P be the probability of signal detection. nn2 is the noise detection probability, and n2 is the number of photons corresponding to the 100(1-D) quantiles of the noise distribution. In the specific implementation, the cumulative number N is gradually increased until equation (1) is satisfied. The obtained N is the minimum cumulative number of times to distinguish between signal and noise.

[0012] After obtaining the minimum cumulative count in step 3, the photon count threshold for distinguishing between signal and noise is obtained:

[0013]

[0014] In the formula, thre is the threshold, n1 is the number of photons corresponding to the 100D quantile of the signal distribution, and n2 is the number of photons corresponding to the 100(1-D) quantile of the noise distribution.

[0015] For data with a very low signal-to-noise ratio, when the number of cumulative iterations N required to extract the signal is very large, the computation time of step 3 using the binomial distribution method is relatively long. In this case, a normal distribution can be used to approximate the binomial distribution, and the minimum number of cumulative iterations N0 can be directly estimated analytically and corrected based on empirical formulas to obtain the minimum number of cumulative iterations required to identify the signal. Therefore, another technical solution of the present invention is as follows.

[0016] The method for estimating the cumulative number of times required to identify a signal using a photon-counting lidar based on a normal distribution includes steps 1-2 and step 3' as described in the first scheme 1.

[0017] When approximating a binomial distribution using a normal distribution, the parameter transformation relationship is as follows:

[0018] μ=NP,σ 2 =NP(1-P) (3)

[0019] In the formula: N is the cumulative number of occurrences, P is the single occurrence probability of the binomial distribution, and the detection probability P for the signal is P0. s The detection probability P for noise is P n , μ is the mean of the normal distribution, and σ is the variance of the normal distribution;

[0020] Z n =μ+C1σ is the 100(1-D) quantile of the normal distribution of noise:

[0021]

[0022] In the formula, C1 is the corresponding quantile Z. n A certain constant can be obtained by looking up the quantile table of the normal distribution;

[0023] Z s =μ-C2σ is the 100D quantile of the normal distribution of the signal:

[0024]

[0025] In the formula, C2 is the corresponding quantile Z. s A certain constant can be obtained by looking up the quantile table of the normal distribution, when:

[0026] Z s ≥Z n (6) That is:

[0027]

[0028] According to equation (7), the following equation is obtained:

[0029]

[0030] The N that satisfies equation (8) is the cumulative number of times the distinguishing index D is satisfied after the normal approximation. The cumulative number of times the identification index is satisfied is obtained by modifying N using the empirical formula of the normal distribution approximating the binomial distribution.

[0031] Beneficial effects

[0032] This invention provides a method for estimating the number of cumulative counts required for a photon-counting lidar to distinguish between signal and noise, based on a binomial distribution. This method can calculate the minimum number of cumulative counts required to meet the identification criteria, thereby improving the detection efficiency of the photon-counting lidar. In low signal-to-noise ratio (SNR) scenarios, this invention provides a method for estimating the number of cumulative counts required to distinguish between signal and noise based on an approximate binomial distribution using a normal distribution. Using this approximation formula, the number of cumulative counts required to meet the identification criteria can be calculated quickly. For spaceborne photon-counting laser altimeters, the method provided by this invention can estimate the minimum number of cumulative counts required to extract the signal, improving the along-track resolution of the altimeter's height measurement data products. For ground-based photon-counting lidar systems, this invention can estimate the minimum number of cumulative counts required to meet the identification criteria for photon-based lidar systems, reducing the radar system's detection time and improving the detection efficiency of the photon-counting lidar system. Attached Figure Description

[0033] Figure 1 This is a flowchart of the present invention.

[0034] Figure 2 This is a schematic diagram illustrating how the distribution of noise and signal photon events gradually becomes distinguishable as the cumulative number of events increases. The figure shows the signal detection probability P. s The noise detection probability P is 0.10. nThe value is 0.03. The light-colored distribution represents the probability distribution of noise photon events, and the dark-colored distribution represents the probability distribution of signal photon events. (a) is the probability distribution of signal and noise photon events within several time slices of equal length to the signal width after 50 cumulative occurrences; (b) is the probability distribution of signal and noise photon events within several time slices of equal length to the signal width after 150 cumulative occurrences; (c) is the probability distribution of signal and noise photon events within several time slices of equal length to the signal width after 200 cumulative occurrences; and (d) is the probability distribution of signal and noise photon events within several time slices of equal length to the signal width after 300 cumulative occurrences.

[0035] Figure 3 This invention uses a binomial distribution to extract the cumulative number of times required to identify the signal when the index D is 0.05.

[0036] Figure 4 This invention uses a normal distribution to approximate a binomial distribution to obtain the cumulative number of times that effectively distinguishes signals from noise, as well as the relative error diagram introduced.

[0037] Figure 5 This is a graph showing the experimental verification results of the present invention. The distribution histograms are the results after accumulating more than 50,000 times. The length of each time slice is 1 ns. The dashed box represents the signal region, and the solid box represents the noise region used during verification.

[0038] Figure 6 It corresponds Figure 5 Each case is divided into time slices with a signal width of 4 ns. The photon event image is obtained by accumulating the minimum number of accumulations N. The dashed line represents the theoretically calculated threshold. (a) Accumulated 65 times, the calculated threshold is 3; (b) Accumulated 83 times, the calculated threshold is 3; (c) Accumulated 50 times, the calculated threshold is 2; (d) Accumulated 159 times, the calculated threshold is 9. It can be seen that except for the signal region where the number of photon events exceeds the threshold, there are no cases in other regions where the number of photon events exceeds the threshold. Detailed Implementation

[0039] This invention mainly utilizes lidar equations to estimate the detection probability of signals and noise, and provides a method for estimating the number of detections required for a photon-counting lidar to identify a signal based on the detection probability of signals and noise. This method provides guidance in the design phase of photon-counting lidar.

[0040] The derivation of signal detection probability and noise detection probability is as follows: The average number of signal photons detected by the photon counting radar each time is N. s N s Influenced by many factors such as laser peak power, target distance, atmospheric transmittance, and receiver system parameters, it can be estimated using the lidar equation:

[0041]

[0042] In the formula, P t (t) represents the peak power of the emitted laser; h is Planck's constant; ν is the laser pulse frequency; T a η is the single-pass atmospheric transmittance. t For the efficiency of the laser emission system; θ t Γ is the laser emission angle; R is the distance the laser travels to the target; t η is the area of ​​the target reflective surface. r For receiving system efficiency; A r Let the area be the area of ​​the receiving telescope. For non-extended targets, when the target size is smaller than the laser spot size, the target's reflective cross-section can be expressed as:

[0043] Γ t =ρS t cosθ (II) In the formula, ρ is the target reflectivity; S t Let be the target area; θ be the angle between the laser incident direction and the normal to the target surface. For extended targets, the target size is much larger than the laser spot size. Assuming the target is a Lambertian solid, the target's reflective cross-section is:

[0044]

[0045] In the formula, ρ is the target reflectivity; θ t Let be the laser emission angle; R be the distance the laser radar reaches the target; and θ be the angle between the laser incident direction and the normal to the target surface. If the radar echo signal follows a Gaussian distribution, the average number of signal photons received in the i-th time slice is:

[0046]

[0047] In the formula, τ is the length of each time slice, and T target Let σ be the signal location and σ be the standard deviation. In the case of direct detection, the number of photons in the echo signal can be approximated by a Poisson distribution. The Poisson distribution gives the probabilities of detecting and not detecting photon events within a single time slice as follows:

[0048]

[0049]

[0050] In the formula, f n K is the noise rate. pe Let p be the number of photon events detected in this time slice. Considering the dead zone or equivalent dead zone problem of single-photon detectors, after a photon is detected at a certain moment, the detector cannot respond to the photon event for a period of time afterwards. The detection probability p in the i-th time slice is...i for:

[0051]

[0052] In the formula, n td This represents the number of time slices contained within the dead time. The probability of detecting a photon event within the signal region can be obtained as follows:

[0053]

[0054] In the formula, T target Let σ be the signal mean location and σ be the standard deviation. The probability of noise detected within a single time slice can be given by the following formula:

[0055]

[0056] Taylor expansion of equation (ix):

[0057]

[0058] Tail item All are f n The higher-order terms of τ, when f n When τ is very small, its tail term can be ignored, and the probability of detecting noise in a single time slice can be estimated by the following formula:

[0059]

[0060] Therefore, the noise detection probability of a noise region of the same length as the signal region is:

[0061] P n =6σf n σ(12)

[0062] A single photon counting probe yields a series of time-dependent photon events, making it difficult to distinguish between signal and noise. After multiple cumulative probes, the difference in photon counts between signal and noise regions can be used to differentiate between them. This invention provides a method for estimating the number of probes required to identify a signal.

[0063] In a photon-counting lidar system, repeated detections over a short period can be assumed to indicate minimal changes in the target and environment, meaning the noise rate and average signal photon count remain constant. Under these conditions, the probability distribution of photon events (PEs) within a time slice in the signal region during multiple repeated detections follows a binomial distribution with parameters N and P, where N is the cumulative number of detections and P is the probability of a single target detection. Figure 2 As shown, when the signal detection probability P s The value is 0.1, the average signal photon number is approximately 0.025 / ns, and the noise detection probability P nThe graph shows the probability distribution of signal and noise photon events within the signal pulse width time interval under different cumulative counts when the noise photon count is 0.03 and the average noise photon count is approximately 0.0075 / ns. The lighter bars on the left represent the probability distribution of noise photon events, while the darker bars on the right represent the probability distribution of signal photon events. Figure 2 (a) is the probability distribution of photon events for signal and noise when the cumulative number of detections N is 50, at which point it is difficult to distinguish between signal and noise; Figure 2 (b) is the probability distribution of photon events of signal and noise when the cumulative number N increases to 150, at which point noise and signal can be gradually distinguished. Figure 2 (c) is the probability distribution of photon events for signal and noise when the cumulative number of times N increases to 200, at which point it is easy to distinguish between signal and noise; Figure 2 (d) is the probability distribution of photon events of signal and noise when the cumulative number of times N increases to 300, at which point the signal and noise are easier to distinguish.

[0064] The process of this invention is as follows Figure 1 As shown. The overlap between the noise binomial distribution and the signal binomial distribution is defined as less than D as an effective distinguishing criterion, as shown in the following formula:

[0065]

[0066] In the formula, P s Let N be the probability of signal detection, n1 be the cumulative number of times, n1 be the number of photons corresponding to the 100D quantile of the signal distribution, and P be the probability of signal detection. n Let n2 be the noise detection probability, and n2 be the number of photons corresponding to the 100(1-D) quantiles of the noise distribution. In the specific implementation, the cumulative count N needs to be gradually increased until equation (xiii) is satisfied, and the minimum cumulative count that can effectively distinguish between signal and noise is obtained. The threshold for distinguishing between signal and noise is:

[0067]

[0068] In the formula, thre is the threshold, n1 is the number of photons corresponding to the 100D quantile of the signal distribution, and n2 is the number of photons corresponding to the 100(1-D) quantile of the noise distribution. Figure 3 The diagram shows the minimum cumulative number of times that effectively distinguishes signal noise, satisfying equation (xiii), when the dead time is 20ns, the extraction index D is 0.05, and the extraction index D is 0.05.

[0069] The above calculation method is used in P s and P nWhen the numbers are very close, the calculated cumulative count N will be very large, requiring a long computation time. When the cumulative count N exceeds a certain number, a normal distribution can be used to approximate the binomial distribution. The minimum required cumulative count can be quickly calculated analytically, and then corrected to obtain the minimum required cumulative count. When using a normal distribution to approximate the binomial distribution, the parameter transformation relationship is as follows:

[0070] μ=NP,σ 2 =NP(1-P) (XV) In the formula: N is the number of repetitions of the binomial distribution, P is the probability of a single occurrence of the binomial distribution, μ is the mean of the normal distribution, and σ is the variance of the normal distribution. Z n =μ+C1σ is the 100(1-D) quantile of the normal distribution of noise:

[0071]

[0072] In the formula, C1 is the corresponding quantile Z. n A certain constant, The operator represents the floor function. Z s =μ-C2σ is the 100D quantile of the normal distribution of the signal, and the signal threshold Z that meets the condition is... s The requirements are:

[0073]

[0074] In the formula, C2 is the corresponding quantile Z. s A certain constant. When it satisfies:

[0075] Z s ≥Z n (18) That is:

[0076]

[0077] At this point, a threshold can be obtained for the cumulative number N:

[0078]

[0079] In the formula, N is the minimum cumulative number of times that the discrimination index D is satisfied after normal approximation. The minimum cumulative number of times that the discrimination index is satisfied can be obtained by modifying N using the empirical formula of normal distribution approximating binomial distribution.

[0080] If the identification index D is selected as 0.05, the overlap between the noise and signal probability distributions is less than 5%. Therefore, the constants corresponding to equation (20) should be the constants corresponding to the 95th quartile of the normal noise distribution and the 5th quartile of the normal signal distribution, i.e., C1 = C2 = 1.6449. The minimum cumulative number N0 calculated after approximating the normal distribution and the introduced relative error are as follows: Figure 4As shown, when N0 is greater than 300, the relative error caused by the normal approximation is less than 10%, and at this time the minimum number of cumulative iterations required to extract the signal is N = 1.1N0.

[0081] Instance verification

[0082] The experimental data is generated by a laser emitted from a 532nm wavelength laser. The laser pulse passes through multiple attenuators before being incident on the target surface and reflected. The reflected light signal is received by a single-photon detector and converted into an electrical signal. Finally, the arrival time of the photon event is recorded by a time-of-flight instrument.

[0083] Figure 5 It is a cumulative histogram of over 50,000 detection data under four different signal and noise conditions. The signal detection probability P is calculated after statistically analyzing all data for each condition. S and noise detection probability P n Then, the minimum cumulative number of detections N for the signal under this index is given by equation (13). 1000 starting points are randomly selected from the data for testing. After selecting the starting points, the detection data is continuously accumulated for the minimum cumulative number of N times. For each accumulated data, the photon count in the signal region and the noise region are calculated respectively. A successful detection is defined as the photon count in the signal region being greater than a threshold, and a false detection is defined as the photon count in the noise region being greater than a threshold. After obtaining the number of successful detections and the number of false detections, they are divided by the total number of tests, 1000, to calculate the successful detection rate and the false detection rate, where the threshold is given by equation (14). Figure 5 As shown, (a) is the verification result under a noise rate of 1.56MHz, where the minimum cumulative number of attempts N required to identify the signal is 50, the successful detection rate is 96.1%, and the false detection rate is 5%; (b) is the verification result under a noise rate of 1.87MHz, where the minimum cumulative number of attempts N required to identify the signal is 83, the successful detection rate is 94.9%, and the false detection rate is 2.9%; (c) is the verification result under a noise rate of 3.01MHz, where the minimum cumulative number of attempts N required to identify the signal is 65, the successful detection rate is 95.6%, and the false detection rate is 5.6%; (d) is the verification result under a noise rate of 7.12MHz, where the minimum cumulative number of attempts N required to identify the signal is 159, the successful detection rate is 95.5%, and the false detection rate is 5.1%. For the minimum cumulative number of attempts calculated with an identification index D of 0.05, the lowest successful detection rate in the above verifications is 94.9%, and the highest false detection rate is 5.6%, which basically conforms to the successful detection rate of 95% and the false detection rate of 5% given by the identification index.

[0084] Figure 6 It corresponds Figure 5In each case, the time slice is divided with a signal width of 4ns as the time slice width. The minimum cumulative number N is calculated using equation (13) to obtain the photon event image. The dashed line represents the threshold theoretically calculated by equation (14). (a) 65 cumulative times, the calculated threshold is 3; (b) 83 cumulative times, the calculated threshold is 3; (c) 50 cumulative times, the calculated threshold is 2; (d) 159 cumulative times, the calculated threshold is 9. In the figure, except for the signal region where the number of photon events is greater than the threshold, the number of photon events in other regions is not greater than the threshold. The above results show that the target position on the time axis can be correctly located according to the threshold method, indicating that the cumulative number calculated according to this patent can be used to guide the signal measurement of photon counting lidar.

[0085] The specific embodiments described herein are merely illustrative examples of the invention. Those skilled in the art to which this invention pertains may make various modifications or additions to the described specific implementation processes or use similar methods to replace them, without departing from the spirit of the invention or exceeding the scope defined by the appended claims.

Claims

1. A method for estimating the cumulative number of times required to identify a signal using a photon-counting lidar based on a binomial distribution, characterized by: Includes the following steps: Step 1: Utilize the lidar equation to estimate or measure the background noise detection probability and the average number of signal photons per detection. Estimate the laser echo pulse width, determine the echo signal width, and then estimate the signal detection probability for each detection. P s The noise detection probability of a noise region with the same length as the signal pulse width P n ; Step 2: Repeated detection by a photon-counting lidar within a short period assumes minimal changes in the target and environment, i.e., the noise rate and average signal photon count remain constant. Based on this assumption, when repeatedly detecting the target, the probability distributions of signal and noise photon events within the time slice containing the signal follow parameters (…). N , P s )and( N , P n The binomial distribution of ) where N This is the cumulative number of times, and it is a quantity to be estimated. P s and P n These are the single detection probabilities of the signal and noise within a specific time interval, calculated from step 1; Step 3, increase the cumulative number of times N The probability distributions of photon events for both signal and noise overlap until the area of ​​overlap, i.e., the probability that the number of noise photon events is greater than the number of signal photon events, is just less than a set target. D The cumulative number of times obtained at this time N That is, the minimum number of cumulative counts required to distinguish between signal and noise.

2. The method for estimating the cumulative number of times required to identify a signal using a photon-counting lidar based on a binomial distribution as described in claim 1, characterized in that: Step 3 is as follows: cumulative number of times N With indicators D The relationship is as follows: In the formula, D The set index for distinguishing between signal and noise, P s For signal detection probability, N For cumulative times, n 1 represents signal distribution 100 D The number of photons corresponding to the quantile site P n For noise detection probability, n 2 represents the noise distribution 100(1- D The number of photons corresponding to the quantile; in the specific implementation, this is achieved by gradually increasing the cumulative number of times. N Until equation (1) is satisfied, the result is... N That is, the minimum number of cumulative counts required to distinguish between signal and noise.

3. The method for estimating the cumulative number of times required to identify a signal using a photon-counting lidar based on a binomial distribution as described in claim 2, characterized in that... After obtaining the minimum cumulative count in step 3, the photon count threshold for distinguishing between signal and noise is obtained: In the formula, thre For the threshold, n 1 represents signal distribution 100 D The number of photons corresponding to the quantile site n 2 represents the noise distribution 100(1- D The number of photons corresponding to the quantile.

4. A method for estimating the cumulative number of times required to identify a signal using a photon-counting lidar based on a normal distribution, characterized by: Including steps 1-2 and step 3' as described in claim 1, When approximating a binomial distribution using a normal distribution, the parameter transformation relationship is as follows: In the formula: N For cumulative times, P The probability of a single occurrence is given by a binomial distribution, representing the probability of signal detection. P Right now P s Detection probability of noise P Right now P n , μ The mean is a normal distribution. σ The variance is a normal distribution. Z n =μ+C 1 σ The noise follows a normal distribution 100(1- D quantiles: In the formula, C 1 is the corresponding quantile. Z n A certain constant can be obtained by looking up the quantile table of the normal distribution; Z s =μ-C 2 σ 100% of the normal distribution of the signal D quantiles: In the formula, C 2 is the corresponding quantile. Z s A certain constant can be obtained by looking up the quantile table of the normal distribution, when: Right now: According to equation (7), we obtain the following equation: Satisfying equation (8) N After approximating the normal distribution, the discrimination index is satisfied. D The cumulative number of occurrences is used to approximate the binomial distribution using the empirical formula for the normal distribution. N After making corrections, the cumulative number of times the identification criteria are met is obtained.

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