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In this paper we show how to obtain full-resolution depth maps from a single image obtained from a plenoptic camera. Previous work showed that the estimation of a low-resolution depth map with a plenoptic camera differs substantially from that of a camera array and, in particular, requires appropriate depth-varying antialiasing filtering. In this paper we show a quite striking result: One can instead recover a depth map at the same full-resolution of the input data. We propose a novel algorithm which exploits a photoconsistency constraint specific to light fields captured with plenoptic cameras. Key to our approach is handling missing data in the photoconsistency constraint and the introduction of novel boundary conditions that impose texture consistency in the reconstructed full-resolution images. These ideas are combined with an efficient regularization scheme to give depth maps at a higher resolution than in any previous method. We provide results on both synthetic and real data.