  We propose a novel approach and algorithm based on two preliminary tests of
the optical system elements to enhance the super-resolved complex-valued
imaging. The approach is developed for inverse phase imaging in a single-shot
lensless optical setup. Imaging is based on wavefront modulation by a single
binary phase mask. The preliminary tests compensate errors in the optical
system and correct a carrying wavefront, reducing the gap between real-life
experiments and computational modeling, which improve imaging significantly
both qualitatively and quantitatively. These two tests are performed for
observation of the laser beam and phase mask along, and might be considered as
a preliminary system calibration. The corrected carrying wavefront is embedded
into the proposed iterative Single-shot Super-Resolution Phase Retrieval
(SSR-PR) algorithm. Improved initial diffraction pattern upsampling, and a
combination of sparse and deep learning based filters achieves the
super-resolved reconstructions. Simulations and physical experiments
demonstrate the high-quality super-resolution phase imaging. In the
simulations, we showed that the SSR-PR algorithm corrects the errors of the
proposed optical system and reconstructs phase details 4x smaller than the
sensor pixel size. In physical experiment 2um thick lines of USAF phase-target
were resolved, which is almost 2x smaller than the sensor pixel size and
corresponds to the smallest resolvable group of used test target. For phase
bio-imaging, we provide Buccal Epithelial Cells reconstructed in computational
super-resolution and the quality was of the same level as a digital holographic
system with 40x magnification objective. Furthermore, the single-shot advantage
provides the possibility to record dynamic scenes, where the framerate is
limited only by the used camera. We provide amplitude-phase video clip of a
moving alive single-celled eukaryote.
