Diffusion in SPAD Signals

TL;DR

The paper tackles reconstructing high-fidelity images from SPAD timing data under photon-starved conditions, where the detection process is nonlinear and stochastic. It adopts diffusion-based posterior sampling with a learned score network to impose a strong prior while accommodating a non-linear forward model that includes SPAD dead time. A domain-adaptation step maps diffusion-domain outputs to physically meaningful photon flux, enabling reconstruction via Diffusion Posterior Sampling (DPS). Through forward-model simulations across diverse flux levels and two reconstruction setups, the approach demonstrates improved image quality over conventional methods, validating diffusion priors for photon-starved imaging with timing data.

Abstract

We derive the likelihood of a raw signal in a single photon avalanche diode (SPAD), given a fixed photon flux. The raw signal comprises timing of detection events, which are nonlinearly related to the flux. Moreover, they are naturally stochastic. We then derive a score function of the signal. This is a key for solving inverse problems based on SPAD signals. We focus on deriving solutions involving a diffusion model, to express image priors. We demonstrate the effect of low or high photon counts, and the consequence of exploiting timing of detection events.

Figures (9)

• Figure 1: Visualizations of the Poisson and Erlang distributions in photon counting. [Top] The Poisson PMF for varying expected counts $\lambda T$, showing the probability of observing exactly $N$ events. [Middle] The Erlang PDF for varying $N$, representing the distribution of detection time for the $N$-th event. [Bottom] The duality relationship showing that the Poisson probability (shaded region) is the difference between two Erlang CDFs.
• Figure 2: The shifted exponential distribution representing the inter-detection time $\Delta t$ in the presence of dead time. The standard exponential distribution (gray dashed line, $\tau_d=0$) is shifted by the dead time period $\tau_d$, creating a region $[0, \tau_d)$ where the probability of detection is zero.
• Figure 3: Case I: The $N$-th event occurs early enough ($t_N \leq T-\tau_d$) that the dead time ends before the exposure time $T$, leaving a window where further detection is possible.
• Figure 4: Case II: The $N$-th event occurs late ($t_N > T-\tau_d$), meaning the dead time window extends beyond $T$, preventing any further detection.
• Figure 5: Simulation of photon detection events for different lux values. For visualization, a simple reconstruction method is used.