Noise and defocus trade-off in depth of field extension with a co-designed binary phase mask

The use of a phase mask in the pupil of an optical system, whose parameters are optimized in conjunction with digital processing, makes it possible to increase a system's depth of field (DoF). However, this co-design strategy is usually conducted with a fixed aperture and at a fixed noise level. Yet the aperture directly drives the trade-off between noise and defocus blur in the captured images that both have to be corrected by the restoration processing. Hence, its optimization within the extended DoF task offers a new range of solutions. In this paper, we co-design a camera using a binary phase mask optimized jointly with either two types of processing: a finite filter or a neural network with the same receptive field. We first consider the traditional case of co-design with a fixed aperture size, but in a challenging use-case of DoF extension and conduct a robustness study as a function of the signal-to-noise ratio for fixed processing. We then include the aperture in the parameters to be co-optimized and improve restoration performance, showing that taking this parameter into account in the optimization leads to a new, to the best of our knowledge, trade-off between blur and noise. Moreover, we show that the optimal setting differs according to the type of processing considered. These results are validated experimentally.