Maximum-likelihood estimation of a laser system pointing parameters by use of return photon counts Deva K. Borah, David Voelz, and Santasri Basu
A maximum-likelihood estimator used to determine boresight and jitter performance of a laser pointing system has been derived. The estimator is based on a Gaussian jitter model and uses a Gaussian far-field irradiance profile. The estimates are obtained using a set of return shots from the intended target. An experimental setup with a He–Ne laser and steering mirrors is used to study the performance of the proposed method. Both Monte Carlo simulations and experimental results demonstrate excellent performance of the estimator. Our study shows that boresight estimation is more challenging than jitter estimation when both quantities are estimated. Furthermore, their estimation performance improves with an increase in the number of shots. The experimental results are found to agree well with the simulation results. © 2006 Optical Society of America OCIS codes: 120.5630, 140.0140, 030.6600.
1. Introduction
Laser beam pointing is critical for a variety of applications that include active tracking, designation, and free-space communications. Two fundamental pointing errors that arise in most laser control systems are boresight (a static offset in pointing) and jitter (a temporally random pointing component). A method has been described by Lukesh et al. for estimating boresight error and beam jitter from the statistics of the laser signal reflected from a target.1 The approach does not require a special scanning mode to determine boresight, such as used with conical scanning methods,2,3 and therefore can be applied during normal system operations to estimate both boresight and jitter. The approach was initially developed for situations in which the beam is larger than the target and it utilizes knowledge of the beam profile and the target shape and reflectance profile. Beam jitter is a necessary element of the method as the changing beam position provides spatial information that is encoded in the statistics of the detected signal. The approach is suitable for either reflected signal applications or one-way propagation applications, such as
The authors are with the Klipsch School of Electrical and Computer Engineering, New Mexico State University, Las Cruces, New Mexico 88003. D. K. Borah’s e-mail address is
[email protected]. Received 2 May 2005; accepted 3 November 2005; posted 14 November 2005 (Doc. ID 61845). 0003-6935/06/112504-06$15.00/0 © 2006 Optical Society of America 2504
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free-space communications, where the illuminator is directed toward a sensor. We also note that related ideas have been applied in other specific pointing applications.4 Application, development, and extensions of the Lukesh et al. concept have been reported in a series of publications. For example, the approach was applied to the estimation of satellite laser optical cross sections5 and the estimation of target shape.6 The approach was also considered for beam control applications where a software implementation tool was developed as a real-time histogram interpretation of a numerical observations (Rhino) package7 and results of a laboratory demonstration were presented.8 Two pointing estimation techniques were discussed in these papers. The first is the key ratio method, which is a calculation involving the mean and variance of the signal and the assumption of a Gaussian beam profile and Gaussian-distributed jitter.1 The key ratio provides only a jitter estimate and assumes no boresight error. The second technique involves a chi-squared test, where the measured signal distribution is compared with a catalog of distributions that are created through simulation.1,7 The simulated distributions use models of the target and beam profiles and include a range of possible jitter and boresight values. Both techniques have been examined in simulation and兾or laboratory experiments and have shown good success in predicting jitter or boresight under certain conditions. Both the key ratio and the chi-squared methods are generally empirical and not strictly based on optimal
where K is a gain constant, ⍀ is the standard deviation of the far-field irradiance pattern along any direction, x关n兴 and y关n兴 are the angular coordinates for the nth observation, and A is the boresight. The boresight represents the mean offset in pointing the peak value of the laser beam intensity from the desired target as shown in Fig. 1. Note that the Gaussian beam pattern is circularly symmetric, and therefore the effect of the boresight error can be taken care of by perturbing any of the two coordinates. The exact beam pointing, however, is a random variable due to several factors such as tracking difficulties, mechanical vibrations in the system components, and atmospheric turbulence. This randomness is called jitter, and the variables x[n] and y[n] due to jitter have a two-dimensional probability density function (pdf) given by
p共x关n兴, y关n兴兲 ⫽
1
冉
2 exp
2j
⫺共x2关n兴 ⫹ y2关n兴兲 2j2
冊
,
(2)
where j is the jitter standard deviation along any direction. In our subsequent discussions, we refer to j simply as jitter. The target is assumed to be a point target in our model. However, the experimental results will later show that our results apply equally well to small size targets. 3. Maximum-Likelihood Estimator
Fig. 1. Laser pointing system.
To derive the ML estimator, we first rewrite Eq. (1) as theoretical estimation approaches. In this paper we derive a maximum-likelihood (ML) estimator for determining boresight and jitter using a laser source with a Gaussian far-field irradiance profile. The jitter is assumed to be random with a Gaussian distribution. Our estimation is based on the availability of N return shots from the target. Although the target is modeled as a point target in our theoretical derivation, the experimental results show that the method works well even with extended targets. Our study shows that boresight estimation is more challenging than jitter estimation when both quantities require estimation. The estimation performance is found to improve with the number of return shots. However, the improvement is rather slow when both jitter and boresight are estimated. 2. System Model
We consider N shots directed from a laser source at a target in the far field as shown in Fig. 1. The far-field irradiance pattern is assumed to be Gaussian so that the nth observation r[n] can be written as
冉
r关n兴 ⫽ K exp
⫺共共x关n兴 ⫹ A兲2 ⫹ y2关n兴兲
n ⫽ 1, 2, . . . , N,
2
2⍀
冊
冉 冊
z关n兴 ⫽ 2⍀2 log
(1)
(3)
where z关n兴 ⫽ 共x关n兴 ⫹ A兲2 ⫹ y2关n兴. Next, defining z˜ 关n兴 ⫽ z 关n兴兾j 2, we observe that z˜ 关n兴 has a noncentral chi-squared pdf9 with two degrees of freedom and noncentrality parameter ⫽ A2兾j 2. Hence the pdf of z˜ 关n兴 is given by p共z˜ 关n兴兲 ⫽
冉
冊
1 1 exp ⫺ 共z˜ 关n兴 ⫹ 兲 I0共冑z˜ 关n兴兲u共z˜ 关n兴兲, 2 2 (4)
where I共·兲 is the modified Bessel function of the first kind and order , and u(·) is the unit step function. The pdf of z[n] is then obtained as
p共z关n兴兲 ⫽
1 2j
2
冉
exp ⫺
冉
⫻ I0
,
K , r关n兴
A j2
1 2j2
冊
共z关n兴 ⫹ A2兲
冊
冑z关n兴 u共z关n兴兲.
(5)
We define the observation vector z ⫽ 关z关1兴, z关2兴, . . ., z关N兴兴. Assuming independent observations, the joint 10 April 2006 兾 Vol. 45, No. 11 兾 APPLIED OPTICS
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pdf for z is given by p共z兲 ⫽
1
共2j 兲 ⫻
N
冉
exp ⫺
2 N
冉
兿 I0 n⫽1
A j2
冉NA ⫹ 兺 z关n兴冊冊 2 1
N
2
2
冊
n⫽1
j
冑z关n兴 u共z关n兴兲.
(6)
The pdf of r ⫽ 关r关1兴, r关2兴, . . . , r关N兴兴 for a given set of K, A, and j is then obtained from Eq. (6) as p共r; K, A, j兲 ⫽
冉 冊 兿冉 冊 ⍀ j
2N N
i⫽1
冉
1 r关i兴 1
冉NA ⫹ 2N⍀ log K ⫺ 2⍀ 兺 log r关n兴冊冊
⫻ exp ⫺
2
2
2j 2
2
n⫽1
⫻
Fig. 2. Experimental setup.
N
冉
N
兿 I0
n⫽1
A j
冑 2 2 2⍀ log共K兾r关n兴兲
冊
⫻ u共2⍀2 log共K兾r关n兴兲兲.
1 N A⫺ 兺 N n⫽1 (7)
Since the observations r关n兴 ⱕ K, we ignore the unit step function from subsequent discussions. Taking the logarithm on both sides, we obtain log p共r; K, A, j兲 ⫽ 2N log ⍀ ⫺
N
兺 log r关n兴 ⫺ 2N n⫽1
冉A N ⫹ 2⍀ ⫻ 兺 log共K兾r关n兴兲冊 ⫻ log j ⫺
1
2
2
2j 2
冉 冉
I1 I0
A j 2 A j 2
冑2⍀2 log共K兾r关n兴兲 冑2⍀2 log共K兾r关n兴兲
冊 冊
⫻ 冑2⍀2 log共K兾r关n兴兲 ⫽ 0,
2⍀2 N A ⫺ 2j ⫹ 兺 log共K兾r关n兴兲 N n⫽1 A 冑2⍀2 log共K兾r关n兴兲 I1 N 2A j 2 ⫺ 兺 A N n⫽1 冑2⍀2 log共K兾r关n兴兲 I0 j 2 2
(9)
2
冉 冉
N
冊 冊
⫻ 冑2⍀2 log共K兾r关n兴兲 ⫽ 0.
(10)
n⫽1
N
冉冉
⫹ 兺 log I0 n⫽1
A j2
冑2⍀2 log共K兾r关n兴兲
冊冊
From Eqs. (9) and (10), we obtain, .
(8)
To obtain the ML estimate for the unknown boresight and jitter parameters, we have to maximize the likelihood [Eq. (8)] over j and A. The gain K can be estimated by sampling the intensity of the directed beam. Alternatively, since Eq. (1) implies that r关n兴 ⱕ K, we can approximately use the maximum value of r[n] as the gain estimate. This estimate becomes more accurate as the boresight reduces and兾or the number of observations increases. Finding the best parameters from Eq. (8) is a nonlinear optimization problem and there are standard techniques to obtain the optimal solution set.10 However, since the possible parameter space is typically not too large, we propose to do a grid search over the unkown parameters by directly computing Eq. (8) for different values of A and j. Alternatively, the two-dimensional grid search can also be reduced to a one-dimensional search as follows. Differentiating Eq. (8) with respect to A and j, respectively, we get 2506
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A2 ⫹ 2j 2 ⫽
2⍀2 N 兺 log共K兾r关n兴兲. N n⫽1
(11)
Thus we need to do only a one-dimensional search on either A or j since the other variable can be obtained from Eq. (11). The log pdf expression of Eq. (8) is then maximized to obtain the ML estimate. Finally, for the special case when the boresight is zero, an explicit expression for the jitter can be obtained. Putting A ⫽ 0 in Eq. (10), we obtain the estimate ˆ j as ˆ j ⫽
冑
冉 冊
⍀2 N K log . 兺 N n⫽1 r关n兴
(12)
4. Experimental Setup
We have performed a laboratory experiment to investigate the pointing performance of the proposed estimator in a practical laser system. A diagram of the laboratory system is shown in Fig. 2. A 1.5 mW He–Ne laser beam is input to a 10⫻ beam expander, which
produces an output beam with a full width at halfmaximum (FWHM) of ⬃0.64 cm (the relationship between the FWHM and the beam-width parameter ⍀ is given by a FWHM of 冑8 ln 2⍀). The expanded beam is reflected off a manually adjusted steering mirror and sent to a voltage-driven fast steering mirror (FSM). The manual mirror is used to adjust the initial position of the beam. The FSM is controlled with a PC and is used to introduce beam jitter. Gaussian random number generators (each for x and y coordinates) programmed in LabVIEW create a sequence of random positions, which are converted to voltages through a digital-to-analog converter board and sent to command the FSM. This approach allows a controlled and repeatable implementation of beam jitter. A constant voltage offset can also be sent to the FSM to simulate the effect of boresight misalignment. The light reflected off the FSM is incident on a reflective tape target fixed to a piece of glass. A square target of approximately 2 mm ⫻ 2 mm is used for the results in this paper. The large target size relative to the beam 共FWHM of 6.4 mm兲 is necessary to provide high signal levels, but, as demonstrated in the following section, the fact that it is not a point target has not affected the results significantly. A 2 in. 共5 cm兲 diameter lens collects the flux reflected from the target and focuses the light on the photodiode detector. A small percentage of the input laser beam is also picked off with a glass plate and directed to a second photodetector. This arrangement is necessary to monitor power changes in the laser. The signals from both detectors are digitized and recorded in the computer through an analog-to-digital converter board. The LabVIEW program generates the x- and y-angle inputs for the FSM every 300 ms where the delay is necessary to ensure that all movement has stopped before the photodetector readings are captured. Therefore a data collection run of 1000 points requires ⬃5 min. 5. Performance Results
Fig. 3. Estimates on the jitter– boresight plane for ten random cases each with N ⫽ 20 observations.
detailed understanding of the chi-squared method if needed. The jitter and boresight values are expressed in the normalized form as j兾⍀ and A兾⍀. Figure 3 shows ten estimates of boresight and jitter using ten independent sets of N ⫽ 20 observations. A similar set of estimates is shown for N ⫽ 100 in Fig. 4. The true values of the jitter and boresight parameters are 0.4 and 0.7, respectively. Both simulation and experimental results show performance improvement for N ⫽ 100 over N ⫽ 20 as the estimates become closer to the true values in Fig. 4. Figures 3 and 4 also show that the estimates are more spread along the boresight axis, implying that boresight is relatively more difficult to estimate than the jitter. It is to be noted that, although Fig. 4 shows a large fraction of the estimates above the true boresight value, generating a larger number of random estimates can result in a more uniform scattering of estimates about the true value.
We present both Monte Carlo simulation results and experimental results to demonstrate the performance of the proposed ML estimator. The jitter estimation results are compared with the key ratio method,1 which is an empirical method providing an estimate of the jitter standard deviation ˆ j as
冑
ˆ j ⫽ ⍀2共2 ⫹ 冑2 ⫹ 1兲, where  ⫽ r兾r is the key ratio, and r and r are the standard deviation and the mean, respectively, of the return photons. We choose the key ratio method over the chi-squared method1 for several reasons. First, for low boresight, the key ratio method gives good estimation results. Second, the performance of the chi-squared method depends on the choice of the histogram bin size, and therefore a comparison will involve many possible cases. Finally, readers can always refer to the work by Lukesh et al.1 for a more
Fig. 4. Estimates on the jitter– boresight plane for ten random cases each with N ⫽ 100 observations. 10 April 2006 兾 Vol. 45, No. 11 兾 APPLIED OPTICS
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Fig. 5. Jitter estimation performance of ML and key ratio methods with increasing number of observations in the presence of zero boresight error. The true jitter value is 1.5.
To statistically characterize the performance of the ML estimator, we use the mean-squared-error (MSE) metric defined as MSE ⫽
1 M 兺 ⱍq ⫺ qˆiⱍ2, M i⫽1
where q is the true parameter (jitter or boresight), and qˆi is its estimate from the ith data set of N observations. Thus the set of N observations is repeated M times to calculate the MSE. In our simulation results, we use M ⫽ 15,000, whereas the experimental results use only M ⫽ 50. The smaller averaging value of M in the experimental results is due to the practical difficulty in obtaining a large experimental data set. Thus the limited experimental data factor alone can cause some mismatch between the theoretical and the experimental results.
Fig. 6. Estimation performance of the ML method with increasing number of observations in the presence of boresight error. The true jitter value is 0.8 and the boresight is 1.0. 2508
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Fig. 7. Jitter estimation performance of ML and key ratio methods in the presence of zero boresight. The estimates are obtained using N ⫽ 50 samples.
The performance of the estimator with respect to the number of observations (N) is studied by simulations in Figs. 5 and 6. Whereas Fig. 5 shows results for no boresight error with a jitter value of 1.5, the results in Fig. 6 demonstrate the case for a jitter value of 0.8 in the presence of a boresight value of 1.0. Although both Figs. 5 and 6 show an improvement in estimation performance, the improvement is slower in the presence of boresight error. Furthermore, boresight error performance is always found to be worse than the jitter estimation error. Figures 7, 8, and 9 show simulation and experimental results and include a comparison with the key ratio method. Note that the key ratio method can estimate only jitter. In general, as the jitter value increases, the MSE performance of the ML estimator is found to deteriorate. Apparently, positional accuracy declines as samples of the Gaussian irradiance
Fig. 8. Estimation performance of ML and key ratio methods for a boresight of 1.2. The estimates are obtained using N ⫽ 50 samples.
results are used to study the performance of the estimator. It is observed that the estimator outperforms the key ratio method in the presence of boresight error. The performance of both boresight and jitter estimation is found to improve with an increase in the number of observations. Future work will involve the study and development of algorithms under a broader range of conditions and in the presence of noise. The authors acknowledge valuable discussions with Gordon Lukesh and Susan Chandler. This work was supported by the U.S. Air Force Office of Scientific Research grant FA9550-05-C-0010. The New Mexico State University supports Nukove Scientific Consulting as a subcontractor on this project. References Fig. 9. Estimation performance of ML and key ratio methods for a jitter of 0.4. The estimates are obtained using N ⫽ 50 samples.
surface spread over a wider area. The boresight estimation is seen to suffer more than the jitter estimation for larger jitter values. Both the ML and the key ratio approaches show excellent jitter estimation performance for the zero boresight case (Fig. 7), although the ML performance is slightly superior. In the presence of nonzero boresight (Fig. 8), the ML method clearly outperforms the key ratio method, which is to be expected since the key ratio was developed for zero boresight situations, although the ML error also becomes significant for large jitter. In the case of varying boresight as shown in Fig. 9, the ML jitter estimation is found to be relatively insensitive, whereas the boresight estimation slightly improves with increasing boresight. The performance of the key ratio method is found to decline with increasing boresight, which is again expected since the method assumes zero boresight error. 6. Conclusion
A maximum-likelihood estimator for estimating pointing performance of a laser pointing system in terms of boresight and jitter has been derived. Both Monte Carlo simulation and experimental validation
1. G. Lukesh, S. Chandler, and D. G. Voelz, “Estimation of laser system pointing performance by use of statistics of return photons,” Appl. Opt. 39, 1359 –1371 (2000). 2. A. Erteza, “Boresighting a Gaussian beam on a specular target point: a method using conical scan,” Appl. Opt. 15, 656 – 660 (1976). 3. P. S. Neelakantaswarmy and A. Rajaratram, “Boresight error in the conical scan method of autoboresighting a laser beam on a specular point-target,” Appl. Opt. 21, 3607–3612 (1982). 4. S. Arnon, “Use of satellite natural vibrations to improve performance of free-space satellite laser communication,” Appl. Opt. 37, 5031–5036 (1998). 5. G. W. Lukesh, S. M. Chandler, and D. G. Voelz, “Analysis of satellite laser optical cross sections from the active imaging testbed,” in Proc. SPIE 4538, 24 –33 (2002). 6. G. W. Lukesh and S. M. Chandler, “Non-imaging active system determination of target shape through a turbulent medium,” in Proc. SPIE 4167, 111–119 (2000). 7. S. M. Chandler, G. W. Lukesh, D. Voelz, S. Basu, and J. Sjogren, “Model-based beam control for illumination of remote objects. Part I: Theory and near real-time feasibility,” in Proc. SPIE 5552, 105–113 (2004). 8. S. Basu, D. Voelz, S. M. Chandler, G. W. Lukesh, and J. Sjogren, “Model-based beam control for illumination of remote objects. Part II: Laboratory testbed,” in Proc. SPIE 5552, 114 – 122 (2004). 9. L. L. Scharf, Statistical Signal Processing Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991). 10. D. G. Luenberger, Linear and Nonlinear Programming (Kluwer Academic, 2004).
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