  Many problems of low-level computer vision and image processing, such as
denoising, deconvolution, tomographic reconstruction or super-resolution, can
be addressed by maximizing the posterior distribution of a sparse linear model
(SLM). We show how higher-order Bayesian decision-making problems, such as
optimizing image acquisition in magnetic resonance scanners, can be addressed
by querying the SLM posterior covariance, unrelated to the density's mode. We
propose a scalable algorithmic framework, with which SLM posteriors over full,
high-resolution images can be approximated for the first time, solving a
variational optimization problem which is convex iff posterior mode finding is
convex. These methods successfully drive the optimization of sampling
trajectories for real-world magnetic resonance imaging through Bayesian
experimental design, which has not been attempted before. Our methodology
provides new insight into similarities and differences between sparse
reconstruction and approximate Bayesian inference, and has important
implications for compressive sensing of real-world images.
