Estimating Sparse Signals with Smooth Support via Convex Programming and Block Sparsity

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.