Resolution Limit of Single-Photon LiDAR

TL;DR

The paper derives a closed-form resolution limit for single-photon LiDAR sensor arrays by formulating photon arrivals as an inhomogeneous Poisson process and performing a bias-variance decomposition of the depth MLE. Through a sequence of novel approximations—linearizing τ(x), replacing the boxcar with a Gaussian, and using space-time convolutions—the authors obtain a concise MSE expression: $ \text{MSE} = \frac{c^2}{12N^2} + \frac{N}{\alpha_0}igl(c^2 \sigma_x^2 + \sigma_t^2\bigr)$. The theory predicts that increasing flux, smoothing the scene, and shrinking the temporal pulse reduce error, but increasing pixel count $N$ initially reduces bias yet raises per-pixel noise, producing a U-shaped MSE vs. $N$ curve; an optimal $N$ may exceed the sensor’s native resolution. The authors validate the bound with 1D and 2D simulations and a real SPAD dataset, showing close agreement and providing actionable insights for sensor design and post-processing choices. Overall, the work offers a first closed-form fundamental limit on depth estimation accuracy for SPAD-based LiDAR and highlights how both scene structure and hardware constraints shape achievable resolution.

Abstract

Single-photon Light Detection and Ranging (LiDAR) systems are often equipped with an array of detectors for improved spatial resolution and sensing speed. However, given a fixed amount of flux produced by the laser transmitter across the scene, the per-pixel Signal-to-Noise Ratio (SNR) will decrease when more pixels are packed in a unit space. This presents a fundamental trade-off between the spatial resolution of the sensor array and the SNR received at each pixel. Theoretical characterization of this fundamental limit is explored. By deriving the photon arrival statistics and introducing a series of new approximation techniques, the Mean Squared Error (MSE) of the maximum-likelihood estimator of the time delay is derived. The theoretical predictions align well with simulations and real data.

Figures (26)

• Figure 1: As we pack more pixels in a unit space, we gain the spatial resolution with a reduction in the SNR. The goal of this paper is to understand the trade-off between the two factors.
• Figure 2: Matched filter: Given a known pulse shape, we shift the pulse until it matches with the measured samples.
• Figure 3: The space-time signal $\lambda(x,t)$ in the unit length $0 \le x \le 1$ and time span $[0,T]$, and its corresponding "effective" returned pulse $\overline{\lambda}(x,t)$ where each individual returned pulse is $\lambda_n(t)$.
• Figure 4: Our core proof involves an approximation of the boxcar kernel by a Gaussian. Doing so will allow us to replace the integration with a convolution.
• Figure 5: 1D simulation. Comparing simulation and the theoretically predicted MSE. Note the excellent match between the theory and the simulation.

Theorems & Definitions (23)

• Example 1
• Theorem 1: Bar-David_1969 Joint distribution of $M$ time stamps
• Corollary 1: Probability of $M$ occurrence
• Example 2
• Corollary 2: Probability over the sample space
• Theorem 2
• Example 3
• Lemma 1
• Theorem 3
• Theorem 4: Overall MSE