  Conventional algorithms for sparse signal recovery and sparse representation
rely on $l_1$-norm regularized variational methods. However, when applied to
the reconstruction of $\textit{sparse images}$, i.e., images where only a few
pixels are non-zero, simple $l_1$-norm-based methods ignore potential
correlations in the support between adjacent pixels. In a number of
applications, one is interested in images that are not only sparse, but also
have a support with smooth (or contiguous) boundaries. Existing algorithms that
take into account such a support structure mostly rely on non-convex methods
and---as a consequence---do not scale well to high-dimensional problems and/or
do not converge to global optima. In this paper, we explore the use of new
block $l_1$-norm regularizers, which enforce image sparsity while
simultaneously promoting smooth support structure. By exploiting the convexity
of our regularizers, we develop new computationally-efficient recovery
algorithms that guarantee global optimality. We demonstrate the efficacy of our
regularizers on a variety of imaging tasks including compressive image
recovery, image restoration, and robust PCA.
