High quality noise-free images constitute an integral part of all image processing applications. Image acquisition and transmission stages often result in corruption of images with impulse noise, which is of two main types: salt-and-pepper noise and random-valued noise. Among the two types, random-valued noise is the most difficult to be removed due to its randomness and this problem is addressed in the paper. The proposed method for random-valued impulse noise removal takes advantage of two important properties of images — local sparsity and non-local self-similarity. The technique for random-valued impulse noise removal has two stages. The first stage called Impulse Detection stage identifies the outlier candidates affected by impulse noise. The second stage called Inpainting-based Reconstruction stage reconstructs the image from the unaffected partial random samples. A robust split Bregman iterative algorithm is used to solve the optimization problem. Experimental results support the effectiveness of the algorithm.