Image Deconvolution with Markov-Chain Monte Carlo

I've decided to morph this blog to be more about short updates pertaining to current ideas I'm thinking about, rather than the long-winded philosophical rants I've posted so far. Hopefully this might keep me more engaged as a blogger and maybe even help me keep better track of the things I'm working on. Starting up this blog again, by the way, is a shameless procrastination technique, since my dissertation is due in about 1 month, and all my writing energy should really be focused on that...

After attending an SKA Imaging Workshop in Socorro, NM a couple of weeks ago, I've developed an interest in Bayesian statistics and Markov-Chain Monte Carlo (MCMC) techniques as they pertain to interferometric imaging. Having never taken a stats course, I'm scrambling a little to absorb the vocabulary I need to understand papers written on the subject. Fortunately, in this era of wikipedia, getting up to speed isn't that hard. After reading wiki articles on MCMC, Markov Chains, and the Metropolis-Hastings algorithm, I dived into EVLA Memo 102, which talks about a first shot at using MCMC for image deconvolution.

The Maximum Entropy Method (MEM) is a classic deconvolution technique (one I've already reimplemented for AIPY), but I'd like to go a bit further down this road. According to the standard implementation (which I gleaned from reading Cornwell & Evans (1984) and Sault (1990)) this algorithm uses a steepest descent minimization technique based on the assumption of a nearly diagonal pixel covariance matrix (i.e. the convolution kernel is approximately a single pixel). While this is an effective computation-saving assumption, I found that for the data I was working with, this assumption lead to the fit diverging when I started imaging at finer resolutions.

I think MCMC, by not taking the steepest decent, might be able to employ the diagonality assumption more robustly. I also think it's high time that deconvolution algorithms make better use of priors. The spatial uniformity prior in MEM makes it powerful for deconvolving extended emission, while the brightest-pixel selection technique in CLEAN makes it effective for deconvolving point sources. There's no reason we can't build a prior explicitly for a deconvolution algorithm that tells it to prefer single strong point sources over many weaker point sources, but also tells it that when all else is equal, entropy should be maximized.