Transport of Event Equation: Phase Retrieval from Defocus Events

TL;DR

The paper addresses fast, robust phase retrieval under coherent illumination by replacing defocus-image capture with an event-based vision sensor (EVS) that records defocus-induced intensity changes as events. It derives the Transport of Event Equation (TEE), a Poisson-like PDE that links defocus events to the phase via the axial log-intensity derivative, enabling direct phase reconstruction without measuring absolute intensity. Through simulations and optical experiments, the authors show that TEE outperforms the traditional Transport of Intensity Equation (TIE) in low-light conditions thanks to the EVS’s wide dynamic range and exposure-free operation, achieving stable, real-time phase recovery. The approach holds promise for real-time adaptive optics and display systems by leveraging EVS hardware to efficiently extract axial derivative information and unwrap phase without explicit defocus-image acquisition.

Abstract

To time-efficiently and stably acquire the intensity information for phase retrieval under a coherent illumination, we leverage an event-based vision sensor (EVS) that can detect changes in logarithmic intensity at the pixel level with a wide dynamic range. In our optical system, we translate the EVS along the optical axis, where the EVS records the intensity changes induced by defocus as events. To recover phase distributions, we formulate a partial differential equation, referred to as the transport of event equation, which presents a linear relationship between the defocus events and the phase distribution. We demonstrate through experiments that the EVS is more advantageous than the conventional image sensor for rapidly and stably detecting the intensity information, defocus events, which enables accurate phase retrieval, particularly under low-lighting conditions.

Figures (10)

• Figure 1: Our optical system. $F_1$ and $F_2$ denote focal lengths of lenses 1 and 2, respectively. By telecentric characteristic Kingslake:10, $F_2/F_1$-scale object is projected onto focus plane. Event vision sensor (EVS) is movable along optical axis $z$. We recover phase distribution on focus plane from events triggered by defocus.
• Figure 2: Hardware implementation of our optical system. EVS is movable along optical axis (red arrow).
• Figure 3: RMSE values against intensity levels $I$.
• Figure 4: Retrieved phases w/ RMSE (simulation, $I=2$). Black frames in (b) and (c) highlight regions with small and large errors, respectively.
• Figure 5: Reconstruction error of weights (simulation, $I=2$). Non-transparent points correspond to non-zero weights in Table \ref{['tab:zernike']}.