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Lidar waveform based analysis of depth images constructed using sparse single-photon data

Yoann Altmann, Ximing Ren, Aongus McCarthy, Gerald S. Buller, Steve McLaughlin

arXiv:1507.02511v1stat.AP

TL;DR

Low-photon-count Lidar makes range and intensity estimation difficult. The paper develops a hierarchical Bayesian waveform model with spatially correlated priors and adaptive MCMC inference, then validates it on real data. The approach is reported to benefit low-count reconstruction, including empty pixels, while its intensity estimates are not directly interpretable as target reflectivity.

  • Problem

    Low photon counts make classical Lidar methods unsatisfactory for estimating target range and intensity.

  • Method

    The paper combines a hierarchical Bayesian waveform model with spatial MRF priors and adaptive MCMC estimation whose MRF parameters are tuned by maximum marginal likelihood.

  • Results

    Real Lidar data showed benefits of the proposed model and method compared with the classical method, including handling empty pixels.

  • Takeaways & Limitations

    The approach supports joint low-count estimation of depth, intensity, and background without setting MRF regularization parameters by cross-validation.

  • Takeaways & Limitations

    The estimated intensities cannot be directly related to target reflectivity because of the system's impulse response.

Abstract

from arXiv · show

This paper presents a new Bayesian model and algorithm used for depth and intensity profiling using full waveforms from the time-correlated single photon counting (TCSPC) measurement in the limit of very low photon counts. The model proposed represents each Lidar waveform as a combination of a known impulse response, weighted by the target intensity, and an unknown constant background, corrupted by Poisson noise. Prior knowledge about the problem is embedded in a hierarchical model that describes the dependence structure between the model parameters and their constraints. In particular, a gamma Markov random field (MRF) is used to model the joint distribution of the target intensity, and a second MRF is used to model the distribution of the target depth, which are both expected to exhibit significant spatial correlations. An adaptive Markov chain Monte Carlo algorithm is then proposed to compute the Bayesian estimates of interest and perform Bayesian inference. This algorithm is equipped with a stochastic optimization adaptation mechanism that automatically adjusts the parameters of the MRFs by maximum marginal likelihood estimation. Finally, the benefits of the proposed methodology are demonstrated through a serie of experiments using real data.

I. INTRODUCTION

The paper addresses low-photon-count Lidar imaging, where classical methods can estimate range and intensity poorly. It introduces a spatially structured Bayesian model and adaptive MCMC procedure for jointly estimating target properties and background from full photon-timing waveforms.

  • Motivation: Low photon flux makes classical range and intensity estimation unsatisfactory, especially with short acquisition times or low laser power.
  • Acquisition setting: The scanning design uses a fixed acquisition time per pixel, allowing some pixels to contain multiple detected photons while others are empty.
  • Proposed method: The method jointly estimates target distance, intensity, and average background from photon timing histograms in a single procedure.
  • Bayesian model: Hierarchical Bayesian priors and Markov random fields encode parameter constraints and spatial correlations in target distances and intensities.
  • Inference algorithm: An adaptive stochastic-gradient MCMC algorithm estimates the unknown quantities while automatically tuning MRF parameters by maximum marginal likelihood instead of cross-validation.
  • Validation: Real-data experiments demonstrate benefits for reconstructing a 3D object at very low photon counts, including handling empty pixels.

II. PROBLEM FORMULATION

The problem is to infer surface range, target intensity, and background levels from photon counts recorded across spatial pixels and temporal range bins. The formulation uses a Poisson waveform model with a known impulse response and positivity constraints, then introduces Bayesian priors for the unknown parameters.

  • II. PROBLEM FORMULATION: The observations are Nrow×Ncol Lidar waveforms, with each pixel represented by photon counts across T temporal bins corresponding to range.
  • II. PROBLEM FORMULATION: Each pixel is associated with a surface position ti,j representing the range of an object surface from the sensor.
  • II. PROBLEM FORMULATION: Each temporal-bin photon count yi,j,t follows a Poisson distribution under the observation model.
  • II. PROBLEM FORMULATION: The waveform model combines a known photon impulse response with target intensity ri,j and a background level bi,j constant across bins within each pixel.
  • II. PROBLEM FORMULATION: The estimation task is to recover target positions, intensities, and background levels from the observed Nrow×Ncol×T data array.
  • III. BAYESIAN MODEL: The unknown parameter vector comprises the intensity matrix R, target-position matrix T, and background matrix B, all subject to positivity constraints.
  • A. Likelihood: The likelihood assumes independent Poisson noise realizations across bins and wavelengths.

B. Prior for the target positions

The model uses Markov random fields to encode spatial structure in target positions and intensities. Depths are modeled with a discrete, piecewise-constant prior, while intensities use a hidden gamma-MRF allowing spatial smoothness without restricting values to a finite set.

  • Prior for target positions: A discrete Markov random field models target positions conditional on neighboring pixels.The prior assigns finite-valued depths and uses a neighborhood-based correlation structure.
  • Prior for target positions: Total-variation regularization promotes piecewise-constant depth representations, consistent with finite depth values.The paper notes that this representation is generally sufficient for the depth resolution of recent Lidar imaging systems.
  • Prior for target positions: The eight-pixel, second-order neighborhood is selected because it produces smoother depth images in practice.Alternative neighborhood structures and cost functions are possible.
  • Prior for target intensity: A hidden gamma-MRF models intensity coefficients whose neighboring values are expected to vary smoothly.Auxiliary gamma variables create the bipartite dependence structure underlying the intensity prior.
  • Prior for target intensity: Unlike TV-based intensity regularization, the gamma-MRF produces spatially smooth intensities without constraining them to finitely many values.The global parameter α0 controls the prior deviation of intensity coefficients and is adjusted with the depth parameter c by maximum marginal likelihood.

D. Prior for the background levels

The background levels receive gamma priors with fixed, weakly informative hyperparameters, while the complete posterior combines the likelihood, background prior, depth MRF, and intensity GMRF.

  • Prior for the background levels: Gamma priors are assigned to background levels using positive hyperparameters η and ν.The hyperparameters are chosen to represent limited prior knowledge about background levels.
  • Prior for the background levels: The experiments use the weakly informative setting (η, ν) = (1, 10) for all reported results.The authors state that this choice has limited impact on estimation performance and can be adapted when observation information is available.
  • Joint posterior: The joint posterior combines the waveform likelihood with priors for background, target positions, intensity, and auxiliary variables.The posterior is conditioned on the spatial regularization parameters c and α0, which are determined during inference.
  • Bayesian estimators: Bayesian estimators summarize posterior knowledge through an MMSE intensity estimate and a MAP estimate of target positions.The intensity estimator marginalizes uncertainty in the other unknown parameters.
  • Adaptive regularization: Empirical Bayes replaces the MRF regularization parameters with maximum marginal likelihood estimates computed from the observed data.This permits automatic image-specific adjustment and is reported to improve estimation relative to one fixed parameter value across datasets.
  • Adaptive regularization: The adaptive strategy has lower computational cost than including the MRF parameters in the model and marginalizing them during inference.The cited comparison describes the cost as several times lower.

B. Bayesian algorithm

The Bayesian estimators require high-dimensional posterior integration and intractable marginal-likelihood optimization. The paper addresses both through a stochastic-gradient MCMC algorithm that jointly adapts regularization parameters and generates posterior samples.

  • Computational challenge: Computing the estimators requires posterior expectations over a very high-dimensional space.The difficulty comes from evaluating the full posterior and integrating over nuisance parameters.
  • Computational challenge: Estimating c and α0 is also intractable because the marginal likelihood and its gradient cannot be evaluated.The marginalization over unknown parameters and the model structure make direct optimization unavailable.
  • Bayesian algorithm: A stochastic-gradient MCMC algorithm jointly estimates the MRF parameters and generates posterior samples for R, T, B, and Γ.The algorithm combines stochastic optimization with simulation rather than solving the two tasks separately.
  • Bayesian algorithm: The estimators are approximated by Monte Carlo integration after discarding the initial burn-in iterations.The algorithm uses posterior samples generated by the MCMC chain.
  • Bayesian algorithm: Target positions are sampled sequentially from conditional distributions, with a Gibbs sampler enabling parallel updates through a coloring scheme.For a second-order neighborhood, the implementation requires nine coloring steps.

2) Sampling the intensity coefficients:

The algorithm exploits conditional structure to sample intensity and auxiliary variables efficiently, while stochastic MCMC updates approximate otherwise unavailable regularization-parameter gradients. Despite bias from inexact simulation, the approach is reported to be less expensive and to produce accurate depth and intensity results.

  • Sampling the intensity coefficients: The intensity coefficients and auxiliary variables can be updated simultaneously because their posterior conditional structure permits parallel updates.The same analysis applies to background sampling under the corresponding conditional distributions.
  • Sampling the intensity coefficients: Intensity coefficients can be sampled by selecting a finite gamma-mixture component and then drawing from its corresponding gamma distribution.This conditional representation follows from the polynomial likelihood structure.
  • Updating MRF parameters: The stochastic gradient replaces the unavailable likelihood gradient with estimates computed from MCMC-generated auxiliary-variable samples.Two MCMC kernels generate auxiliary samples for intensity-related and depth-related variables.
  • Updating MRF parameters: Projection onto bounded intervals enforces positivity of c and α0 and stabilizes the stochastic optimization.The implementation uses upper bounds of 20 for both parameters.
  • Updating MRF parameters: Exact simulation from the conditional distributions is computationally infeasible, so the algorithm uses MCMC kernels and obtains a biased gradient estimator.The estimator converges toward a neighborhood of the maximum marginal likelihood parameters rather than exactly to them.
  • Updating MRF parameters: The biased estimator is reported to cost significantly less than approximate Bayesian computation while yielding very accurate depth and intensity results.This comparison motivates using the approximate stochastic procedure despite its bias.

V. SIMULATION RESULTS

The experiments use real outdoor TCSPC Lidar data across acquisition times from 60µs to 30ms, with photon counts varying from sparse to abundant. The setup includes impulse-response calibration and measurements of a polystyrene head at 40m.

  • Experimental setup: The outdoor measurements used an 841nm fibre-coupled source, a silicon SPAD detector, and 95ps full-width-at-half-maximum system jitter.Measurements were performed on the Heriot-Watt University campus in November 2014.
  • Experimental setup: The evaluation compares depth reconstruction across acquisition times of 30ms, 6ms, 3ms, 600µs, 300µs, and 60µs.Timed-event data enable shorter effective acquisition histograms after measurement.
  • Calibration: The calibrated instrumental impulse response was estimated from a Spectralon panel at 40m using a 60s acquisition and background removal.The resulting response is shown in Fig. 5.
  • Photon-count regime: Almost 49% of pixels contained no detected photons at 60µs, and this proportion decreased with longer acquisition times.These measurements provide the sparse-photon conditions used to evaluate the algorithms.

B. Estimation performance

The proposed Bayesian method is compared with standard cross-correlation and maximum-likelihood procedures for depth and intensity estimation across decreasing acquisition times. It remains more robust when photons are sparse or absent, although MCMC makes it substantially more computationally demanding.

  • Depth estimation: The proposed method and the standard method produce similar depth maps at long acquisition times.The comparison is based on Fig. 6.
  • Depth estimation: As acquisition time decreases, cross-correlation increasingly fails to identify target positions, especially in pixels without detected photons.The proposed method appears more robust to absent signal.
  • Intensity estimation: The two methods produce similar intensity maps at the longest acquisition times, while the proposed method is more robust to missing photons.At 60µs, spatial correlation produces a smoother intensity image around the head.
  • Spatial regularization: At 300µs, the proposed model handles low photon returns by using spatial correlation in depths and intensities.The comparison includes both depth and intensity estimates.
  • Quantitative evaluation: The proposed method is more robust as acquisition time decreases and provides more consistent depth and intensity estimates.Its cumulative error distributions also process pixels with no detected photons.
  • Computational cost: The Bayesian method is significantly more computationally demanding because of its MCMC procedure.This cost is balanced against improved depth and intensity estimation when detected-photon counts are low.

VI. CONCLUSION

The paper concludes that its adaptive Bayesian model and MCMC algorithm improve low-photon Lidar depth and intensity estimation on real data. The conclusions also identify scope limits involving multiple surfaces, empty pixels, reflectivity, atmospheric effects, and computational cost.

  • Contributions: The paper proposes a Bayesian model for low-photon Lidar imaging with spatially correlated target depths and intensities.An adaptive MCMC algorithm estimates unknown parameters, including spatial regularization parameters, without cross-validation.
  • Validation: Validation on real Lidar data shows benefits over the classical method when detected-photon counts are low.The reported comparison concerns the proposed model and method against the classical approach.
  • Scope limitations: The model assumes a single surface per pixel; with multiple surfaces, one peak is treated as principal and other peaks as background noise.The principal surface depends on peak amplitude and neighboring-pixel returns.
  • Scope limitations: The model assumes a target in every pixel, so empty pixels are processed using the single-layered-target model.Accounting for absent targets in some pixels remains an unresolved issue.
  • Reflectivity limitations: Estimated intensity coefficients can exceed one and therefore cannot be directly interpreted as target reflectivity values.Constraining coefficients below one may not suffice for accurate reflectivity estimation.
  • Future work: Atmospheric scintillation effects may have limited impact on depth estimation but must be modeled in future work to improve reflectivity estimation.The issue is especially relevant for long-range targets.

APPENDIX: ON THE CONDITIONAL DISTRIBUTION OF THE INTENSITY COEFFICIENTS

The appendix derives the conditional distribution of the target intensity coefficient by expressing its polynomial form through photon-count-dependent roots. The resulting conditional law is represented as a mixture of gamma distributions.

  • Conditional distribution: The conditional distribution of the intensity coefficient is expressed up to a multiplicative constant as a polynomial function of the coefficient.The polynomial structure follows from the conditional model.
  • Polynomial representation: Each positive photon-count bin contributes a root at -b_i,j/g_0(t−t_i,j), with multiplicity y_i,j,t.The polynomial order depends on the detected photon counts.
  • Polynomial representation: The polynomial coefficients are obtained from its roots and used to derive the conditional distribution.The appendix then rewrites the result as a mixture distribution.
  • Mixture representation: The conditional distribution can be expressed as a mixture of O_i,j + 1 gamma distributions.This representation supplies the form used for intensity-coefficient sampling.

ALGORITHM 1

The algorithm outputs Bayesian depth and intensity estimates from the proposed model, using MCMC samples after burn-in. The associated figures compare neighborhood structures, reconstructions, and processing measurements with standard methods.

  • Algorithm 1: The algorithm fixes inputs including the Lidar impulse response, burn-in iterations, and total iteration count before sampling model quantities.The output consists of posterior samples {R(n), T(n)} for n=1,...,NMC.
  • Algorithm 1: It samples T, R, and B sequentially, then advances the iteration counter and discards burn-in iterations.The listed steps explicitly sample T(n), R(n), and B(n), conditionally increment n, and output retained samples.
  • Spatial model: The model uses spatial neighborhood structures, including four- and eight-pixel neighborhoods and a proposed GMRF structure, to represent relationships among pixels.The figures distinguish neighborhood choices and show the proposed first-order GMRF structure.
  • Experiments: The proposed Bayesian algorithm estimates depth and intensity maps for a 40 m target across different per-pixel acquisition times.The depth and intensity figures report reconstructions produced by the proposed method, while the comparison text identifies corresponding standard-method results.
  • Experiments: At 300 µs, depth and intensity reconstructions are compared between the Bayesian and standard methods, with black pixels indicating cases where no photons were detected.For the standard cross-correlation method, such pixels are cases where the target distance cannot be identified.
  • Evaluation: The evaluation also reports distance and intensity RMSE cumulative distributions, background-level maps, and processing-time measurements.The supplied figure and table references cover RMSE CDFs, background maps, average detected photons, empty-pixel proportions, and processing time.
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