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The problem of recovering sparse signals and sparse gradient signals from a small collection of linear measurements is one that arises naturally in many scientific fields. The recently developed Compressed Sensing Framework states that such problems can be solved by searching for the signal of minimum L 1-norm, or minimum Total Variation, that satisfies the given acquisition constraints. While L 1 optimization algorithms, based on Linear Programming techniques, are highly effective at generating excellent signal reconstructions, their complexity is still too high and renders them impractical for many real applications. In this paper, we propose a novel approach to solve the L 1 optimization problems, based on the use of suitable nonlinear filters widely applied for signal and image denoising. The corresponding algorithm has two main advantages: low computational cost and reconstruction capabilities similar to those of Linear Programming optimization methods. We illustrate the effectiveness of the proposed approach with many numerical examples and comparisons.