  We propose an image reconstruction method for an X-ray telescope system with
an angular resolution booster proposed by Maeda et al.(2018). The system
consists of double multi-grid masks in front of an X-ray mirror and an
off-focused two-dimensional imager. Because the obtained image is off-focused,
additional image reconstruction process is assumed to be included. Our image
reconstruction method is an extension of the traditional Richardson-Lucy
algorithm with two regularization terms, one for sparseness and the other for
smoothness. Such a combination is desirable for astronomical imaging because
astronomical objects have variety in shape from point sources, diffuse sources
to mixtures of them. The performance of the system is demonstrated with
simulated data for point sources and diffused X-ray sources such as Cas A and
Crab Nebula. The image resolution is improved from a few arcmin of focused
image without the booster to a few arcsec with the booster. Through the
demonstration, the angular resolution booster with the image reconstruction
method is shown to be feasible.
