  Effective learning of asymmetric and local features in images and other data
observed on multi-dimensional grids is a challenging objective critical for a
wide range of image processing applications involving biomedical and natural
images. It requires methods that are sensitive to local details while fast
enough to handle massive numbers of images of ever increasing sizes. We
introduce a probabilistic model-based framework that achieves these objectives
by incorporating adaptivity into discrete wavelet transforms (DWT) through
Bayesian hierarchical modeling, thereby allowing wavelet bases to adapt to the
geometric structure of the data while maintaining the high computational
scalability of wavelet methods---linear in the sample size (e.g., the
resolution of an image). We derive a recursive representation of the Bayesian
posterior model which leads to an exact message passing algorithm to complete
learning and inference. While our framework is applicable to a range of
problems including multi-dimensional signal processing, compression, and
structural learning, we illustrate its work and evaluate its performance in the
context of image reconstruction using real images from the ImageNet database,
two widely used benchmark datasets, and a dataset from retinal optical
coherence tomography and compare its performance to state-of-the-art methods
based on basis transforms and deep learning.
