The human auditory system has the ability, known as auditory induction, to estimate the missing parts of a continuous auditory stream briefly covered by noise and perceptually resynthesize them. Humans are thus able to simultaneously analyze an auditory scene and reconstruct the underlying signal. In this article, we formulate this ability as a non-negative matrix factorization (NMF) problem with unobserved data, and show how to solve it using an auxiliary function method. We explain how this method can also be generally related to the EM algorithm, enabling the use of prior distributions on the parameters. We show how sparseness is a key to global feature extraction, and that our method is ideally able to extract patterns which never occur completely. We finally illustrate on an example how our method is able to simultaneously analyze a scene and interpolate the gaps into it.