- Research Article
- Open Access
Full Waveform Analysis for Long-Range 3D Imaging Laser Radar
© Andrew M. Wallace et al. 2010
- Received: 27 December 2009
- Accepted: 21 June 2010
- Published: 18 August 2010
The new generation of 3D imaging systems based on laser radar (ladar) offers significant advantages in defense and security applications. In particular, it is possible to retrieve 3D shape information directly from the scene and separate a target from background or foreground clutter by extracting a narrow depth range from the field of view by range gating, either in the sensor or by postprocessing. We discuss and demonstrate the applicability of full-waveform ladar to produce multilayer 3D imagery, in which each pixel produces a complex temporal response that describes the scene structure. Such complexity caused by multiple and distributed reflection arises in many relevant scenarios, for example in viewing partially occluded targets, through semitransparent materials (e.g., windows) and through distributed reflective media such as foliage. We demonstrate our methodology on 3D image data acquired by a scanning time-of-flight system, developed in our own laboratories, which uses the time-correlated single-photon counting technique.
- Markov Chain Monte Carlo
- Single Mode Optical Fiber
- Full Waveform
- Reversible Jump Markov Chain Monte Carlo
- Markov Chain Monte Carlo Analysis
In general, laser range finding can be achieved on the basis of triangulation or time-of-flight, of which the latter method is more suited to long-range measurement. In the context of time-of-flight, the principal methodologies include measurement of phase-shift in an amplitude-modulated signal, measurement of frequency shift in a frequency modulated signal, or measurement of transmit-receive pulse separation in a pulsed system . To build a 3D image, either the laser beam must be scanned across the scene, or a static laser beam diverges to encompass the target, and a focal plane array of independent pixels records the received radiation.
Full waveform ladar [2, 3] requires the analysis of multiple returns that occur within a single measurement or pixel. One of the major applications for full waveform topographic ladar analysis is in the survey of forest canopies to monitor environmental changes [4, 5], but this analysis also has important applications in defense and security . One key application is the detection and classification of targets on the ground under tree cover using airborne imagery, which is related to environmental mapping and is the focus of the Jigsaw  and Swedish Defence Research  systems. However, full waveform analysis is also required in many other situations where single pixel returns are composed of multiple reflections within the laser footprint. For example, this occurs at an occluding boundary, that is, one object behind another, where objects are partially obscured, for example, behind foliage, camouflage, or blinds, when imaging through semitransparent surfaces, or where a single surface may be distributed in depth or moving during exposure. If selected infrared wavelengths are used, then these can penetrate better through the atmosphere or glass , and if multiple wavelengths are used, then this be can more informative in surface classification [9, 10].
In many defense and security applications, it is also desirable that the active laser pulse is eye-safe and "covert", that it be of short duration and low energy. To that end, we have developed a 3D imaging ladar system based on a low-power pulsed laser source and a time-correlated single photon counting detector, for which the detailed optical design is described in .
The twin demands of low power and multiwaveform analysis place significant demands on the signal processing methodology. Typical techniques within the frequentist framework are to calculate the maximum likelihood estimates (MLE) of parameters for every possible number of signal returns, and then use information theoretic criteria, such as akaike (AIC), bayesian information criterion (BIC) and minimum description length (MDL) , to determine the signal number. One popular tool for finding MLE is expectation-maximization (EM) . Compared with centroid method and matched filter, this algorithm is computationally more expensive, but it may give estimates of higher accuracy. However, EM holds a potential risk in that it might converge to a local maximum likelihood  or diverge to an infinite value . Additionally, it is sensitive to initial values and not efficient for data set containing numerous observed events, in our case the timing information for the received photons. Moreover, even though AIC, BIC, and MDL introduce penalty terms to avoid overfitting the data, that is adding more returns to increase the likelihood, they still have the tendency to produce more complicated models which correspond to more signal returns .
In , a hybrid approach is proposed, which first applies a deterministic nonparametric bump-hunting process for initial estimates of signal returns, and second Poisson-MLE to refine the estimates. Although it is effective in many cases, it fails to resolve two closely separated peaks and is not able to produce satisfactory results when the background noise level is comparable or higher than the signal amplitudes.
In order to detect multiple, small returns embedded in background, noise, and clutter, we have been developing concurrently ladar signal analysis methods within the Bayesian framework based on reversible-jump Markov chain Monte Carlo (RJMCMC) techniques for both single pixel and image data . In this paper, we report the development and application of these methods to process images from the new covert, depth imaging sensor, and compare our results with conventional cross-correlation and peak detection applied to the same data.
The organisation of the paper is as follows. In Section 2, we describe briefly the 3D image sensor, and the conditions for data acquisition. In Section 3, we describe the processing methodology. In Section 4, we apply this methodology to images acquired by the sensor to detect wholly visible and partially concealed targets at a moderate range of 325 meters, using our own test facility. We also show how the RJMCMC method can improve our interpretation of the data. Finally, in Section 5, we conclude and summarise some of the key issues that must be addressed to develop these ideas further.
3.1. Bayesian Modelling of Ladar Signals
In previous work, we have shown how Bayesian analysis (using the reversible jump markov chain monte carlo (RJMCMC) computational methodology ) can be used to construct multilayered 3D images  when the laser return consists of multiple peaks due to the footprint of the beam impinging on a target with surfaces distributed in depth. In dense ladar images, one can improve the quality of the 3D data by considering spatial context through a markov random field (MRF) . We have also shown how multispectral LiDAR can be used to classify different types of surface response on the basis of different colour responses, using a maximum of six wavelengths . We have applied these techniques successfully to both Burst Illumination Laser (BIL)  and TCSPC  ladar systems. As pointed out by Mallet and Bretar  in their survey on full waveform LiDAR for remote sensing, our RJMCMC method is robust (finding a global minimum in a multimodal distribution), no initialization or gradient computations are required, and the grammar of instrumental models is extensible.
where the likelihood function, is defined by (4) and the full joint prior distribution is given by .
3.2. RJMCMC Methodology for Ladar Signal Analysis
Updating the positions .
Updating the amplitudes .
Updating the background .
Random birth or death of a peak.
Random splitting of a peak into two peaks or merging of two peaks into a single peak.
In this case, can be , , or dependent on whether it is a move of type (1), (2) or (3).
where is the probability of move type when in state , is the density function of a continuous random vector and the Jacobian term arises from deterministic transfer from variable to .
3.3. Convergence Assessment
A properly designed Markov chain Monte Carlo sampler should generate a convergent Markov sequence whose limiting distribution is the true joint posterior distribution of interest . However, in practical applications, only a finite number of samples can be produced, and it is therefore important to choose the chain length appropriately and assess the convergence of the Markov chain to the stationary distribution.
Three separate but related issues need to be considered when determining the chain length [24, 25]. First, evaluate the length of the burn-in period, which is to determine from which observation point the chain has "forgotten" its starting value and escaped from its influence. At this point, the chain has reached the stationary distribution and the previous samples should be discarded to eliminate the estimation bias introduced by the transient period. Second, determine if the chain is long enough to fully represent the underlying distribution and conclude its convergence to an asymptotic distribution. Third, evaluate if the samples are adequate to achieve a certain precision of estimation.
Over the last two decades, a number of different convergence diagnostics have been proposed, which can be classified into two categories. For theoretical approaches, the attempt is to predetermine the number of iterations required to ensure convergence by analyzing the Markov transition kernel and stationary distribution; a collection of approaches can be found in [26, 27] and references therein. Although they hold formal guarantees, these algorithms are not feasible in practice due to sophisticated mathematical calculation and loose convergence bounds. Therefore, as pointed out in , empirical methods are almost always applied, relying on the outputs of MCMC samplers and diagnostics computed from the produced sequence to check convergence. On the one hand, they provide evidence of convergence; on the other hand, all the diagnostics are unreliable since in practice the target limiting distribution always remains unknown and it is impossible to conclude with certainty that the finite MCMC samples are sufficient to cover the whole support of the underlying stationary distribution. From this point of view, we should be cautious about the diagnostic results.
In the literature, empirical methods seek to conclude the convergence through bias and/or variance evaluation. The Gelman and Rubin diagnostic methodology presented in [23, 28] compares the samples drawn from several independent sequences with different starting points and quantitatively evaluates mixing by analyzing the within-sequence and between-sequence variance. The estimation bias arising from the produced samples is uncovered by multiple separate chains rather than a single chain, and therefore it has comparatively higher diagnostic reliability in terms of detecting if the underlying stationary distribution has been fully explored and the chains have converged to the same limiting distribution. This is particularly significant when applied to multimodal posterior distributions.
As , the total variance estimation should decrease while the within-sequence variance should increase, and finally the PSRF should theoretically decline to 1. If is large, it indicates the posterior distribution should be further explored. Once the PSRF is close to 1, we assume the Markov chain has converged to the target distribution.
In this section, we present the analysis of images acquired under bright daylight conditions of two distant outdoor scenes, comparing methods based on cross-correlation and fixed and variable dimension Markov chain Monte Carlo analysis. Our images are of a life-sized mannequin (a human figure) in full view of the sensor, and of the same mannequin partially concealed behind a fence. The data were acquired at a range of approximately 325 meters. The equivalent scene dimensions were 0.8 m width by 2.0 m height, and the scanned image resolution was 32 by 128 pixels for the whole mannequin. The pulse repetition frequency was 2 MHz, resulting in an average optical power of 40 W. The pixel dwell time was 1.0 s.
To assess the ability of RJMCMC algorithm for multiple peak detection and particularly the resolution capacity for closely separated peaks, we set up a remote target containing several distributed surfaces with known separations, which provides the ground truth and allows us to compare the performance with cross-correlation method.
4.1. Mannequin in Full View: Cross-Correlation and MCMC
In the first example, the mannequin is in full view, standing in front of a concrete pillar, as shown in Figure 5. It was anticipated that the majority of pixels would have clear and distinct, single returns from the surface of either the mannequin or the pillar. Given the divergence of the beam there may be some mixed pixels at the occluding boundary of the mannequin, and there may be pixels with no return as they miss the targets all together. In short, this is a situation in which a cross-correlation detector based on the system instrumental response should perform well and there should be questionable need for the added complexity of Markov chain Monte Carlo analysis. Further, since the expectation in processing this data set is to estimate the range of a single surface return from either the mannequin or the pillar, we apply the fixed dimension Markov chain Monte Carlo (MCMC) approach to avoid redundant computation caused by trans-dimension jumps. Accordingly, only the first three steps in Section 3.2 are used.
The unknowns ( ) subject to inference have independent priors. To completely eliminate any prior knowledge of the peak position, is drawn from a uniform distribution on . The peak amplitudes ( ) and background ( ) follow Gamma distributions and with the shape parameters set to be and , while the scale parameters are and respectively, where is the histogram of photon counts. The previously unspecified proposal distributions are set as follows: all of the parameter updates employ the Gaussian random walk whose proposal means are the current sample values. The standard deviations for amplitude ( ) and background ( ) are both 0.3. For position updates, a delayed rejection step  is carried out to allow movement between posterior estimates that correspond to more widely separated channels. When using delayed rejection, the scale in each step is characterized by and respectively.
We first generate multiple chains for each pixel and evaluate the convergence. After finding a safe convergence length, we then run single MCMC chains with on all the pixels with a bounded number of iterations (5000) including the 500 samples burn-in period. This is consistent with the initial estimate. Subsequently, to assess the convergence of the MCMC chains, we produce four independent sequences for each pixel, and monitor the Gelman and Rubin diagnostic statistic (PSRF) defined in Section 3.3 every 100 samples. The chain generation is terminated when the convergence is concluded, that is when the PSRF reduces to less than a preset threshold , at which the posterior distributions obtained from all the sample trajectories becomes approximately the same.
For the low-amplitude return in Figure 4(e), the cross-correlation approach gives several extrema as displayed in Figure 4(g). Such low amplitude may be caused primarily by lower reflectance back towards the receiver, either because of the material properties or its angle to the beam direction. In this case, it is difficult to decide with certainty where the surface return is located, although we can always define it to be the one corresponding to the maximum cross-correlation value. In comparison, the power of the MCMC methodology lies in supplying Bayesian evidence of the final answer. In other words, the histogram of indicates the posterior distribution of the estimates. As the parameter space becomes more complex, the posterior distribution is spread over a wider channel range and becomes bimodal, which in turn results in a slower convergence rate and an increased chain length in excess of 4000 samples.
Another example is shown in Figure 4(k). For this pixel, the bin index for the maximum cross-correlation does not equal the one for the posterior mode. Hence, the MCMC chain gives a different and better substantiated estimate of the true value, further demonstrating the power of the Bayesian approach.
3D images based on these two methods are provided in Figure 5, where a target range gate is set and those pixels with with target position estimates beyond this preset gate are treated as zero return. It is observed that there are a few more pixels beyond the target range with cross-correlation, which implies the maximum values do not always correspond to the correct surface position. This is consistent with the discussion of illustrative pixel data showing the strength of the MCMC method in processing low amplitude ladar signals hidden in backgrounds in that the posterior mode is more informative, robust, and reliable.
4.2. Mannequin Concealed by Fence: Cross-Correlation and RJMCMC
In one sweep of the RJMCMC algorithm, the fixed-dimensional parameter updates (steps 1–3 of Section 3.2) follow the MCMC sampler settings. Jumps between parameter subspaces with different dimensions are accomplished by steps 4 and 5 in the same manner as . Although our expectation would be that the number of surface returns in any single pixel would not be greater than three in this example, we are conservative in allowing the varying dimension sampler to explore values from to .
The first row of Figure 7 illustrates a pixel in which the beam misses all three targets, so that no surface return exists. The use of the cross-correlation method is difficult when there is no surface return as shown in Figure 7(a) to 7(d). In comparison with Figure 4(g) from a small signal-to-background ratio pixel, Figure 7(c) shows the probable existence of at least one surface return. However, according to the asymptomatic posterior probability estimate of , no target return is the most probable conclusion. If we examine the second and third rows of Figure 7 then we see the situations analogous to Figure 4(e) in that there are single returns from fence and mannequin, respectively. The difference in this case is that we have applied full RJMCMC chains, so that the posterior probability estimate, , shows one return.
Of more interest are those pixels containing more than one return, shown in Figures 7(m)–7(x). The fourth row has distinct returns from the fence and mannequin, and the RJMCMC sampler has a very strong preference for two returns. The fifth row is far less distinct, but the sampler again shows a strong posterior probability estimate of two peaks, although the second one might be difficult to detect automatically on a cross-correlation detector, for example, using a fixed (or even proportional) threshold. Due to the varying surface reflectances and angles, pixels can have different photon intensities, which makes it a difficult problem to choose a reliable threshold. The corresponding parameter estimates of the two surface returns shown in Figure 7(q) correspond in depth to the known ground truth of the relative separation. Finally, the last row shows one of the pixels in which the beam partially reflects from the fence, partially transmits through a gap and hence reflects from the mannequin, but near an occlusion boundary so that part reflects from the pillar behind. The posterior estimate of favours surfaces but it is by no means as clear cut as the earlier examples, and the parameter estimates of the 3 surface positions shown in Figure 7(u) correspond to the fence, mannequin, and pillar separations at this point.
4.3. Real Data with Known Geometry: Cross-Correlation and RJMCMC
According to Figure 10, both RJMCMC and cross-correlation methods succeed in detecting distinct return signals. For the two surfaces separated at 30 mm, they merge to be a single peak in cross-correlation values. In comparison, with assistance of Merge/Split updates, RJMCMC can easily separate them. However, both methods fail to distinguish the peaks 10 mm (17 channel bins) away from one another, and instead place a combined return, which results in the increased estimated distances from the combined signal to its neighboring peaks, that is, the two peaks corresponding to the surfaces separated by 450 and 200 mm.
In this paper, we have demonstrated the application of Bayesian analysis using Markov chains to analyse full-waveform Ladar pixel and image data acquired by a new scanning sensor. The sensor uses time-correlated photon counting technology, and coupled with algorithmic development, we are able to detect multiple surface returns within the field of view of single pixels, creating multilayer images. This has application in defence and security when objects of interest may be partially concealed, or viewed through semitransparent surfaces, such as through windows.
To demonstrate the method, and compare with thresholded correlation analysis, we have used selected data from two images of a distant target, the first in full view, the second viewed through a trellis fence. In general, RJMCMC analysis is advantageous in supplying principled estimates of both the number of surface returns and the associated parameter vectors (range, amplitude, and background level). This allows us to construct multilayered 3D images. The methodology is effective in dealing with low amplitude returns, a few photons at maximum in a single bin. This adds to the covert capability of the sensor, aimed at detecting returns from uncooperative surfaces at medium range using a low-power source laser diode.
However, there are a number of outstanding problems that require future work. In the long term, we need to acquire image data at an approximate rate of one frame per second, or better, and to process the data in comparable time frames. Currently, we are investigating the use of convergence diagnostics to better control the chain length, the validity of initialising the chains by correlation data, and multicore programming in combination with vector processor and FPGA technology. In general, all of these can lead to faster, single pixel processing. Another possibility is to promote an investigation on the Dirichlet process (DP) mixture model developed in  and recently studied in , which provides natural estimates for Bayesian inference in both model number and associated parameters with efficient simulations.
The work reported in this paper was funded in part by the UK Engineering and Physical Sciences Research Council, and in part by the Electro-Magnetic Remote Sensing (EMRS) Defence Technology Centre, established by the UK Ministry of Defence and run by a consortium of SELEX Sensors and Airborne Systems (now SELEX Galileo), Thales Defence, Roke Manor Research and Filtronic.
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