  Reconstruction of multidimensional signals from the samples of their partial
derivatives is known to be a standard problem in inverse theory. Such and
similar problems routinely arise in numerous areas of applied sciences,
including optical imaging, laser interferometry, computer vision, remote
sensing and control. Though being ill-posed in nature, the above problem can be
solved in a unique and stable manner, provided proper regularization and
relevant boundary conditions. In this paper, however, a more challenging setup
is addressed, in which one has to recover an image of interest from its noisy
and blurry version, while the only information available about the imaging
system at hand is the amplitude of the generalized pupil function (GPF) along
with partial observations of the gradient of GPF's phase. In this case, the
phase-related information is collected using a simplified version of the
Shack-Hartmann interferometer, followed by recovering the entire phase by means
of derivative compressed sensing. Subsequently, the estimated phase can be
combined with the amplitude of the GPF to produce an estimate of the point
spread function (PSF), whose knowledge is essential for subsequent image
deconvolution. In summary, the principal contribution of this work is twofold.
First, we demonstrate how to simplify the construction of the Shack-Hartmann
interferometer so as to make it less expensive and hence more accessible.
Second, it is shown by means of numerical experiments that the above
simplification and its associated solution scheme produce image reconstructions
of the quality comparable to those obtained using dense sampling of the GPF
phase.
