  Many imaging science tasks can be modeled as a discrete linear inverse
problem. Solving linear inverse problems is often challenging, with
ill-conditioned operators and potentially non-unique solutions. Embedding prior
knowledge, such as smoothness, into the solution can overcome these challenges.
In this work, we encode prior knowledge using a non-negative patch dictionary,
which effectively learns a basis from a training set of natural images. In this
dictionary basis, we desire solutions that are non-negative and sparse (i.e.,
contain many zero entries). With these constraints, standard methods for
solving discrete linear inverse problems are not directly applicable. One such
approach is the modified residual norm steepest descent (MRNSD), which produces
non-negative solutions but does not induce sparsity. In this paper, we provide
two methods based on MRNSD that promote sparsity. In our first method, we add
an $\ell_1$-regularization term with a new, optimal step size. In our second
method, we propose a new non-negative, sparsity-promoting mapping of the
solution. We compare the performance of our proposed methods on a number of
numerical experiments, including deblurring, image completion, computer
tomography, and superresolution. Our results show that these methods
effectively solve discrete linear inverse problems with non-negativity and
sparsity constraints.
