Real-Time Markov Modeling for Single-Photon LiDAR: $1000 \times$ Acceleration and Convergence Analysis

Abstract

Asynchronous single-photon LiDAR (SP-LiDAR) is an important imaging modality for high-quality 3D applications and navigation, but the modeling of the timestamp distributions of a SP-LiDAR in the presence of dead time remains a very challenging open problem. Prior works have shown that timestamps form a discrete-time Markov chain, whose stationary distribution can be computed as the leading left eigenvector of a large transition matrix. However, constructing this matrix is known to be computationally expensive because of the coupling between states and the dead time. This paper presents the first non-sequential Markov modeling for the timestamp distribution. The key innovation is an equivalent formulation that reparameterizes the integral bounds and separates the effect of dead time as a deterministic row permutation of a base matrix. This decoupling enables efficient vectorized matrix construction, yielding up to $1000 \times$ acceleration over existing methods. The new model produces a nearly exact stationary distribution when compared with the gold standard Monte Carlo simulations, yet using a fraction of the time. In addition, a new theoretical analysis reveals the impact of the magnitude and phase of the second-largest eigenvalue, which are overlooked in the literature but are critical to the convergence.