Analysis and Improvement of Rank-Ordered Mean Algorithm in Single-Photon LiDAR

TL;DR

The paper analyzes the rank-ordered mean (ROM) filter used for depth estimation in single-photon LiDAR, deriving a phase-transition threshold that predicts ROM success based on per-pixel signal-to-background conditions. It then introduces the Neighborhood Consensus Filter, which leverages temporal clustering of signal timestamps across an enlarged spatial neighborhood, along with robust outlier rejection and timestamp-consecutive-difference analysis, to dramatically improve photon efficiency and noise tolerance. Theoretical results are validated via simulations, and the new method achieves orders-of-magnitude better depth RMSE than ROM, with successful imaging at very low signal-to-background ratios (e.g., $\text{SBR} \ge 0.06$). The approach is amenable to parallelization and offers a practical path to high-fidelity 3D imaging in challenging low-light and high-noise environments.

Abstract

Depth estimation using a single-photon LiDAR is often solved by a matched filter. It is, however, error-prone in the presence of background noise. A commonly used technique to reject background noise is the rank-ordered mean (ROM) filter previously reported by Shin \textit{et al.} (2015). ROM rejects noisy photon arrival timestamps by selecting only a small range of them around the median statistics within its local neighborhood. Despite the promising performance of ROM, its theoretical performance limit is unknown. In this paper, we theoretically characterize the ROM performance by showing that ROM fails when the reflectivity drops below a threshold predetermined by the depth and signal-to-background ratio, and its accuracy undergoes a phase transition at the cutoff. Based on our theory, we propose an improved signal extraction technique by selecting tight timestamp clusters. Experimental results show that the proposed algorithm improves depth estimation performance over ROM by 3 orders of magnitude at the same signal intensities, and achieves high image fidelity at noise levels as high as 17 times that of signal.

Figures (7)

• Figure 1: Comparison between our proposed method in \ref{['section:difference']} and the ROM filter b1 under different signal-to-background ratios (SBR).
• Figure 2: (a): How ROM works: $8$ neighboring pixels (dark green) of target pixel $(i, j)$ (center, orange) form $\mathcal{P}_{i, j}$ and carry independent timestamps that are combined to form $\mathcal{T}_{i, j}$. Timestamps belonging to $(i, j)$ are left out. (b): Timestamp censoring of ROM, where only timestamps within $\Delta T_{i, j}^{\text{sig}}$ centered at the ROM estimate $t_{i, j}^{\text{ROM}}$ are taken into account for depth estimation (Figure 2(b) is not drawn in scale).
• Figure 3: Behavior of ROM estimate in two typical scenarios. Figure (A1) and (B1) displays predictor $\pi_{i, j}$ for each pixel for the toy scene with SBR = 0.1 and 1.0 respectively. Histograms (A2) and (B2) shows timestamps collected for the same pixel under each case. (A2) denotes a failure of ROM as increased background count pulls ROM estimator towards the halfway time $t_{\frac{1}{2}} = cT_r/4$, while (B2) denotes a success.
• Figure 4: Simulated processing results for Art and Bowling scenes b6 at SBR = 0.2 and 2.0 signal PPP.
• Figure 5: Absolute error of ROM estimates against predictor $\pi_{i, j}$. Timestamps were generated for the toy scene with SBR = 1.0 and 2.0 signal PPP.

Theorems & Definitions (2)

• proof
• proof