Expectation propagation for scalable inverse problems in imaging
Abstract
The solutions of ill-posed inverse problems in imaging are usually non-unique,
making it important to quantify the uncertainties associated with the estimates.
Bayesian inference provides a powerful theoretical framework to derive various summaries of the posterior distribution of the unknown parameter of interest, such as
the posterior mean and covariance. However, using Bayesian inference to find such
solutions requires to compute integrals over the high-dimensional unknown image
vectors. While Markov chain Monte Carlo sampling based methods are classically
and widely used to draw samples from the posterior distributions, sampling methods
for accurate evaluation of higher-order moments are still computationally expensive
and not yet fully scalable for fast inference.
Expectation Propagation provides a fast alternative to sampling methods and
has recently become popular for approximate Bayesian inference. In this thesis,
a set of new Expectation Propagation algorithms are proposed to achieve scalable
posterior approximation for high-dimensional imaging inverse problems. The main
contribution of this thesis is to construct new Expectation Propagation algorithms
to provide both point estimate and uncertainty quantification for imaging inverse
problems. By designing the factorization over the posterior distribution and tailoring
the covariance matrix structure of the approximating distributions, the resulting Expectation Propagation algorithms are scalable to address high-dimensional imaging
problems. The main novelty considers three aspects: (1) block diagonal covariance
matrix structure is, to the best of this author’s knowledge, proposed for the first
time in applying Expectation Propagation for Bayesian models with patch-based
image prior, (2) factorization over convex and non-convex gradient-based priors is
designed to allow for highly parallel computation in using Expectation Propagation
to solve high-dimensional imaging inverse problems, and (3) the proposed Expectation Propagation algorithms are embedded within larger inference schemes where the
prior regularization parameters are unknown. Without significantly increasing the
computational footprint, the resulting Expectation Propagation based algorithms
allow for greater scalability with unsupervised hyperparameter tuning.