The CLEAN algorithm is one of the best known signal processing algorithms in (radio-)astronomy. It is essentially a deconvolution procedure and is used e.g. to reconstruct a sparse (star) brightness distribution from noisy ('dirty' - hence 'cleaning') observed data. In this paper we make a connection for the first time between CLEAN and l₁-denoising. We show CLEAN does a crude version of l₁-denoising and develop a modified algorithm with much improved behaviour.