Computation of the 3D upgoing/downgoing separated wavefield at any desired position within a marine streamer spread is enabled by multicomponent streamers that can measure the crossline and vertical components of water-particle motion in addition to the pressure. We introduce the concept of simultaneous interpolation and deghosting and describe a new technique, generalized matching pursuit (GMP), to achieve this. This method is based on the matching-pursuit technique and iteratively reconstructs the signal as a combination of optimal basis functions. In the GMP method, the basis functions describing the unknown 3D upgoing wavefield are filtered by appropriate forward ghost operators before being matched to the multicomponent measurements. As a data-dependent method, GMP can operate on data samples that are highly aliased in the crossline direction without relying on assumptions about seismic events such as linearity. The technique is naturally suitable for data with only a small number of samples that may be irregularly spaced. We demonstrate the efficacy and robustness of the GMP method on several synthetic data sets of increasing complexity and in the presence of noise.