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When a band-pass filter is applied to a natural image, the distribution of the output has a consistent, distinctive form across many different images, with the distribution sharply peaked at zero and relatively heavy-tailed. This prior has been exploited for several image processing tasks. We show how this prior on the appearance of natural images can also be used to estimate full-resolution images from incomplete data. The unobserved image pixels are modeled with a factor graph. The constraints in the factor graph are based on the characteristic distribution of image derivatives. We introduce an efficient representation for finding candidate values for patches of the image being estimated, avoiding combinatorial explosion. The usefulness of this approach is demonstrated by applying it to two applications: extracting a high-resolution image from a low-resolution version and estimating a full-color image from an image with one color sample per pixel. We show how the super resolution system produces noticeably sharper images, with few significant artifacts. The demosaicing system produces full-color images with fewer color-fringing artifacts than images from other methods.