A closed-form expression for thin lens image irradiance

Computer vision tasks often have the goal of inferring 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. 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 irradiance equation. Through both numerical simulations and comparison with the intensity values of actual images, our equation is shown to provide better accuracy than Horn's equation. Improvement is most notable for 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.