This paper is concerned with the design of a non-negative matrix factorization algorithm for image analysis. This can be used in the context of blind source separation, where each observed image is a linear combination of a few basis functions, and that both the coefficients for the linear combination and the bases are unknown. In addition, the observed images are commonly corrupted by noise. While algorithms have been developed when the noise obeys Gaussian or Poisson statistics, here we take it to be Laplacian, which is more representative for other leptokurtic distributions. It is applicable for cases such as transform coefficient distributions and when there are insufficient noise sources for the central limit theorem to apply. We formulate the problem as an L1 minimization and solve it via linear programming.