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Resolution Limit of Single-Photon LiDAR
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Derives a closed-form MSE for maximum-likelihood depth estimation from an array of single-photon LiDAR pixels, exposing the bias-variance trade-off between pixel count and per-pixel SNR.
Problem
Single-photon LiDAR arrays face a trade-off: packing more pixels per unit space improves spatial resolution but lowers per-pixel photon flux and SNR. The paper asks for a closed-form characterization of depth-estimation MSE as a function of pixel count.
Approach
Extends Bar-David's single-pixel photon arrival theory to a pixel array by approximating the boxcar pixel integration with a Gaussian kernel and replacing projection with convolution, yielding a one-line closed-form MSE with bias term c^2/(12N^2) and variance term N/α0 (c^2σx^2+σt^2).
Key results
- Closed-form MSE (Theorem 4) matches Monte Carlo simulation in 1D, with optimal N around 64 for the 1D example.
- The closed-form 2D prediction matches simulated MSE and the measured MSE on real 192x128 SPAD data.
- Simplified variance expression omitting the c^2 σx^2 term fails to match simulation, showing the full approximation is needed.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase total scene flux | Single-photon LiDAR array · 1D resolution-noise analysis | MSE | Helped |
| Use a smoother scene with smaller depth slope | Single-photon LiDAR array · 1D resolution-noise analysis | MSE | Helped |
| Use a narrower pulse | Single-photon LiDAR array · 1D resolution-noise analysis | MSE | Helped |
| Increase the number of pixels per unit space | Single-photon LiDAR array · 1D resolution-noise analysis | MSE | Mixed |
| Omit the c²σ²x term from the variance model (ablation) | Theorem 4 MSE model · 1D simulated experiment | Agreement with simulated variance | Hurt |
| Approximate the pixel boxcar kernel with a Gaussian and use convolution | Proposed MSE analysis · 1D simulated experiment | MSE prediction agreement | Helped |
| Apply the theoretical MSE model to a 2D scene | Proposed MSE analysis · 2D simulated experiment | MSE prediction agreement | Helped |
| Reduce spatial resolution by using fewer, larger pixels | SPAD sensor array · Real 2D SPAD data | Variance of estimated time of arrival | Helped |
Surprising findings
- On real SPAD data the MSE curve shows no optimal valley because the optimal N can exceed the sensor's physical resolution when the pulse is short.
Limitations
- Assumes Gaussian pulse, constant reflectivity, no dark current, no dead-time or pile-up, and no background flux in the main theorem.
- Results are for maximum-likelihood estimators; advanced post-processing may outperform the predicted bound.
- Relies on approximations (linearized depth, boxcar-to-Gaussian) whose accuracy is validated empirically rather than bounded.
Experimental setup
Datasets: real SPAD data from Henderson et al. 192x128 sensor
Benchmarks: simulated 1D MSE simulated 2D MSE real 2D SPAD measurements
Metrics: mean squared error variance
Builds on: Bar-David photon arrival model
Closest papers by meaning
- Image Reconstruction from Readout-Multiplexed Single-Photon Detector Arrays (CVPR 2025 · similarity 0.82)
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- Opportunistic Single-Photon Time of Flight (CVPR 2025 · similarity 0.77)
- Snapshot Lidar: Fourier Embedding of Amplitude and Phase for Single-Image Depth Reconstruction (CVPR 2024 · similarity 0.76)
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- Separating the Wheat from the Chaff: Spatio-Temporal Transformer with View-interweaved Attention for Photon-Efficient Depth Sensing (AAAI 2025 · similarity 0.76)
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