Closed-form solution for thin lens image irradiance under arbitrary solid angle.

Optical imaging systems are found everywhere in modern society. They are integral to computer vision, where the goal is often to infer geometric and radiometric information about a 3D environment given limited sensing resources. It is helpful to develop relationships between these real-world properties and the actual measurements that are taken, such as 2D images. To this end, we propose a new relationship between object radiance and image irradiance based on power conservation and a thin lens imaging model. The relationship has a closed-form solution for in-focus points and can be solved via numerical integration for points that are not focused. It can be thought of as a generalization of Horn's commonly accepted irradiance equation. Through both ray tracing simulations and comparison to the intensity values of actual images, we believe our equation provides better accuracy than Horn's equation. An improvement is most notable for large lenses and near-focused images where the pinhole imaging model implicit in Horn's derivation breaks down. Outside of this regime, our model validates the use of Horn's approximation through a more thorough theoretical foundation.