Fundamental Recovery Bounds for SPAD Signals under Stationary Flux

Technical Summary: Fundamental Recovery Bounds for SPAD Signals under Stationary Flux

Problem Statement
Single-photon avalanche diode (SPAD) arrays offer extreme sensitivity and picosecond temporal resolution, enabling applications in fluorescence lifetime imaging, LiDAR, and non-line-of-sight imaging. However, SPAD data is inherently stochastic and discrete, representing detection events rather than continuous intensity samples. Recovering a clean image or flux map from this data constitutes a nonlinear inverse problem.

Current passive SPAD sensors operate in three distinct modes, each yielding different raw signal statistics:

1. Free-running Timestamps (M1): Records the arrival time of every detection event within an exposure window.
2. Binary Bins (M2): Divides exposure into time bins, reporting only a single bit indicating if at least one event occurred (Quanta Image Sensor mode).
3. Timestamped Bins (M3): A bandwidth-limited mode where each bin reports at most one event, including its arrival time within that bin.

Prior analyses and reconstruction algorithms (specifically diffusion-based methods) have treated these modes in isolation. There was no unified framework to compare the fundamental recoverability limits across these modes or to apply a single reconstruction algorithm that adapts to the specific sensor statistics of each mode.

Methodology
The authors propose a unified approach centered on the likelihood score function (the gradient of the log-likelihood of the raw data with respect to the incident flux).

1. Derivation of Score Functions and Bounds:

   • The paper derives the likelihood score functions for all three passive modes (M1, M2, M3) from first principles, accounting for the sensor's dead time ($\tau_{dead}$) and sensing window ($\tau_{sense}$).
   • Using these score functions, the authors derive the Cramér–Rao Lower Bound (CRLB) for unbiased estimators of the photon flux. This establishes the fundamental limit of recovery for each mode.
   • The framework is extended to the Bayesian Cramér–Rao Lower Bound (BCRLB) by incorporating a learned prior. This utilizes a score-based diffusion model (SBD) to approximate the score function of the image prior ($\nabla_x \log p(x)$), allowing for the calculation of practical recovery bounds that account for spatial correlations in natural images.

2. Unified Reconstruction Algorithm:

   • The authors adapt Diffusion Posterior Sampling (DPS) to SPAD data. While previous work applied DPS to binary SPAD data, this work extends it to full timestamped cases.
   • The reconstruction algorithm (Algorithm 1) iteratively refines an image estimate by combining a learned prior score (from a trained U-Net) with the specific likelihood score function corresponding to the sensor's operation mode (M1, M2, or M3).
   • A domain adaptation step is introduced to map the non-negative flux domain to the $[-1, 1]$ range required by standard diffusion models.

Key Contributions

• Unified Statistical Analysis: The paper provides the first unified derivation of likelihood score functions and fundamental recovery bounds (CRLB) for all three major passive SPAD operation modes.
• Identification of a High-Flux Gap: The analysis reveals a qualitative difference in performance at high photon flux. The relative error bound for Binary Bins (M2) diverges exponentially as flux increases (due to saturation), whereas both Timestamped modes (M1 and M3) degrade only linearly. This demonstrates that registering event times is not merely an incremental gain but a fundamental distinction between graceful and rapid saturation.
• Extension of Diffusion Models: The work extends diffusion posterior sampling from binary data to timestamped SPAD data by deriving and integrating the correct mode-matched score functions.
• Bayesian Bound Computation: The authors demonstrate a method to compute empirical BCRLBs using a trained score network, providing a theoretical lower bound on the Mean Squared Error (MSE) for reconstruction tasks involving natural image priors.

Results

• Theoretical Bounds: Numerical plots of the CRLB confirm that at low flux, all modes perform similarly. However, as flux increases, the binary mode (M2) suffers a catastrophic loss in estimation accuracy, while timestamped modes (M1, M3) maintain linear error growth. M1 (Free-running) consistently offers the tightest bound, followed closely by M3 (Timestamped Bins).
• Simulation Performance: In simulations using the FFHQ dataset, the proposed SBD method using the correct mode-matched score function significantly outperforms Maximum Likelihood (ML) estimation and mismatched SBD approaches.
  • For medium event rates, SBD with Free-running Timestamps (M1) achieved the highest PSNR (23 dB) and SSIM (0.67).
  • SBD with Timestamped Bins (M3) and Binary Bins (M2) performed similarly to each other (22 dB PSNR) but were inferior to the continuous timestamp mode.
  • Crucially, using the wrong score function (e.g., applying the binary score to timestamped data) resulted in degraded reconstruction quality.
• Real Data Demonstration: The method was validated on real SPAD data of a rotating fan (M3 mode) and a tunnel scene. Reconstructions using the correct M3 score function produced high-fidelity images, whereas using the binary (M2) or free-running (M1) scores (which ignore specific binning constraints) resulted in artifacts or failure.

Significance and Claims
The paper claims to establish a common foundation for single-photon sensing recovery. By tying both fundamental limits and practical diffusion-based algorithms to the likelihood score function, the work answers two critical questions:

1. What is recoverable? The unified CRLB analysis quantifies the fundamental information loss in different sensor modes, highlighting the critical importance of timestamping for high-flux scenarios.
2. How to approach the bound? The proposed algorithm demonstrates that matching the reconstruction score function to the specific sensor operation mode is essential for achieving high-fidelity results.

The authors emphasize that their scope is restricted to passive sensing with stationary flux, excluding active configurations like pulsed-laser TCSPC. They modestly note that while their unified framework improves reconstruction, the performance is ultimately bounded by the physical limitations of the sensor (dead time) and the quality of the learned image prior. The work does not propose new hardware but rather provides the theoretical and algorithmic tools to maximize the utility of existing SPAD arrays across different operating modes.