Forward and inverse diffraction in phasor fields

Non-line-of-sight (NLOS) imaging is an inverse problem that consists of reconstructing a hidden scene out of the direct line-of-sight given the time-resolved light scattered back by the hidden scene on a relay wall. Phasor fields transforms NLOS imaging into virtual LOS imaging by treating the relay wall as a secondary camera, which allows reconstruction of the hidden scene using a forward diffraction operator based on the Rayleigh-Sommerfeld diffraction (RSD) integral. In this work, we leverage the unitary property of the forward diffraction operator and the dual space it introduces, concepts already studied in inverse diffraction, to explain how phasor fields can be understood as an inverse diffraction method for solving the hidden object reconstruction, even though initially it might appear it is using a forward diffraction operator. We present two analogies, alternative to the classical virtual camera metaphor in phasor fields, to NLOS imaging, relating the relay wall either as a phase conjugator and a hologram recorder. Based on this, we express NLOS imaging as an inverse diffraction problem, which is ill-posed under general conditions, in a formulation named inverse phasor fields, that we solve numerically. This enables us to analyze which conditions make the NLOS problem formulated as inverse diffraction well-posed, and propose a new quality metric based on the matrix rank of the forward diffraction operator, which we relate to the Rayleigh criterion for lateral resolution of an imaging system already used in phasor fields.