[Paper Review] Lidar waveform based analysis of depth images constructed using sparse single-photon data

Yoann Altmann, Ximing Ren+3 more|arXiv (Cornell University)|Jul 9, 2015

Advanced Optical Sensing TechnologiesPhysics and Astronomy28 references139 citations

TL;DR

This paper proposes a hierarchical Bayesian model with adaptive Markov chain Monte Carlo (MCMC) inference for depth and intensity profiling from sparse single-photon Lidar waveforms under low photon counts. By modeling spatial correlations via gamma Markov random fields (MRFs) for target intensity and depth, and using stochastic optimization to adapt MRF parameters via maximum marginal likelihood, the method enables robust estimation—even in empty pixels—outperforming classical methods in depth and intensity MSE under low photon flux.

ABSTRACT

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.

Motivation & Objective

• Address the challenge of accurate depth and intensity estimation in time-of-flight Lidar when photon counts are extremely low.
• Overcome limitations of classical methods that fail under sparse photon detection, especially in empty pixels or high background noise.
• Incorporate spatial correlation between neighboring pixels using hierarchical priors to improve estimation stability and accuracy.
• Eliminate the need for cross-validation in regularization parameter selection by using adaptive MCMC with maximum marginal likelihood estimation.
• Enable joint estimation of target depth, intensity, and background levels within a single coherent Bayesian framework.

Proposed method

• Model each Lidar waveform as a Poisson-distributed sum of a known impulse response (weighted by target intensity) and a constant background level.
• Use gamma Markov random fields (MRFs) to encode spatial correlation in target intensity and depth, enforcing smoothness across neighboring pixels.
• Formulate a hierarchical Bayesian model with conjugate priors to embed prior knowledge and positivity constraints on intensity and background.
• Develop an adaptive stochastic gradient MCMC (SGMCMC) algorithm that jointly samples from the posterior of target parameters and tunes MRF hyperparameters via maximum marginal likelihood.
• Use polynomial expansion of the likelihood to derive a mixture-of-gamma conditional distribution for intensity, enabling efficient MCMC sampling.
• Implement automatic parameter adaptation using stochastic optimization to avoid manual tuning of regularization strength.

Experimental results

Research questions

• RQ1 Can spatial correlation between neighboring pixels be effectively leveraged to improve depth and intensity estimation in ultra-low photon count Lidar?
• RQ2 How does the proposed adaptive MCMC algorithm compare to classical methods in terms of mean squared error (MSE) for depth and intensity under sparse photon detection?
• RQ3 To what extent can the method reliably estimate parameters in pixels with zero detected photons?
• RQ4 Does the automatic parameter adaptation via maximum marginal likelihood outperform cross-validation-based regularization in low-photon regimes?
• RQ5 How do background levels and signal-to-noise ratio affect the performance of the proposed Bayesian model in real-world data?

Key findings

• The proposed method achieves significantly lower depth and intensity MSE than the standard method, especially as acquisition time decreases and photon counts drop.
• Cumulative distribution functions (CDFs) of MSE show that the proposed method processes more pixels consistently, including those with zero detected photons, due to spatial regularization.
• The method successfully estimates parameters in empty pixels by leveraging spatial correlation, with CDFs upper-bounded by the proportion of pixels that can be processed.
• The adaptive MCMC algorithm automatically tunes MRF parameters via maximum marginal likelihood, eliminating the need for cross-validation and improving robustness.
• Performance gains are most pronounced under low photon flux, where classical methods fail due to noise and empty pixels, while the Bayesian model maintains stability.
• Despite higher computational cost due to MCMC sampling, the method provides superior estimation accuracy when photon counts are low, justifying the trade-off.