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Fast and effective image compression for multi-dimensional images has become increasingly important for efficient storage and transfer of massive amounts of high-resolution images and videos. Desirable properties in compression methods include (1) high reconstruction quality at a wide range of compression rates while preserving key local details, (2) computational scalability, (3) applicability to a variety of different image/video types and of different dimensions, and (4) ease of tuning. We present such a method for multi-dimensional image compression called Compression via Adaptive Recursive Partitioning (CARP). CARP uses an optimal permutation of the image pixels inferred from a Bayesian probabilistic model on recursive partitions of the image to reduce its effective dimensionality, achieving a parsimonious representation that preserves information. CARP uses a multi-layer Bayesian hierarchical model to achieve self-tuning and regularization to avoid overfitting---resulting in one single parameter to be specified by the user to achieve the desired compression rate. Extensive numerical experiments using a variety of datasets including 2D ImageNet, 3D medical image, and real-life YouTube and surveillance videos show that CARP dominates the state-of-the-art compression approaches---including JPEG, JPEG2000, MPEG4, and a neural network-based method---for all of these different image types and often on nearly all of the individual images.
