  We propose a novel framework for the systematic design of lensless imaging
systems based on the hyperuniform random field solutions of nonlinear
reaction-diffusion equations from pattern formation theory. Specifically, we
introduce a new class of imaging point-spread-functions (PSFs) with enhanced
isotropic behavior and controllable sparsity. We investigate the PSFs and the
modulated transfer functions (MTFs) for a number of nonlinear models and
demonstrate that two-phase isotropic random fields with hyperuniform disorder
are ideally suited to construct imaging PSFs with improved performances
compared to PSFs based on the Perlin noise. Additionally, we introduce a phase
retrieval algorithm based on the non-paraxial Rayleigh-Sommerfeld diffraction
theory and introduce diffractive phase plates with PSFs designed from
hyperuniform random fields, called hyperuniform phase plates (HPPs). Finally,
using high-fidelity object reconstruction, we demonstrate improved image
quality using engineered HPPs across the visible range. The proposed framework
is suitable for high-performance lensless imaging systems for on-chip
microscopy and spectroscopy applications.
