  The power-balanced hybrid optical imaging system is a special design of a
computational camera, introduced in this paper, with image formation by a
refractive lens and Multilevel Phase Mask (MPM) as a diffractive optical
element (DoE). This system provides a long focal depth and low chromatic
aberrations thanks to MPM, and a high energy light concentration due to the
refractive lens. This paper additionally introduces the concept of a optimal
power balance between lens and MPM for achromatic extended-depth-of-field
(EDoF) imaging. To optimize this power-balance as well as to optimize MPM using
Neural Network techniques, we build a fully-differentiable image formation
model for joint optimization of optical and imaging parameters for the designed
computational camera. Additionally, we determine a Wiener-like inverse imaging
optimal optical transfer function (OTF) to reconstruct a sharp image from the
defocused observation. We numerically and experimentally compare the designed
system with its counterparts, lensless and just-lens optical systems, for the
visible wavelength interval (400-700) nm and the EDoF range (0.5-1000) m. The
attained results demonstrate that the proposed system equipped with the optimal
OTF overcomes its lensless and just-lens counterparts (even when they are used
with optimized OTFs) in terms of reconstruction quality for off-focus
distances.
