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The patch transform represents an image as a bag of overlapping patches sampled on a regular grid. This representation allows users to manipulate images in the patch domain, which then seeds the inverse patch transform to synthesize modified images. Possible modifications include the spatial locations of patches, the size of the output image, or the pool of patches from which an image is reconstructed. When no modifications are made, the inverse patch transform reduces to solving a jigsaw puzzle. The inverse patch transform is posed as a patch assignment problem on a Markov random field (MRF), where each patch should be used only once and neighboring patches should fit to form a plausible image. We find an approximate solution to the MRF using loopy belief propagation, introducing an approximation that encourages the solution to use each patch only once. The image reconstruction algorithm scales well with the total number of patches through label pruning. In addition, structural misalignment artifacts are suppressed through a patch jittering scheme that spatially jitters the assigned patches. We demonstrate the patch transform and its effectiveness on natural images.