Abstract
This final chapter covers the analysis of pixelized images through Markov random field models, towards pattern detection and image correction. We start with the statistical analysis of Markov random fields, which are extensions of Markov chains to the spatial domain, as they are instrumental in this chapter. This is also the perfect opportunity to cover the ABC method, as these models do not allow for a closed form likelihood. Image analysis has been a very active area for both Bayesian statistics and computational methods in the past 30 years, so we feel it well deserves a chapter of its own for its specific features.
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Notes
- 1.
Taken from Gaetan and Guyon (2010), kindly provided by the authors.
- 2.
Wikipedia: “Carex is a genus of plants in the family Cyperaceae, commonly known as sedges. Most (but not all) sedges are found in wetlands, where they are often the dominant vegetation.” Laîche is the French for sedge.
- 3.
We will indiscriminately use site and pixel in the remainder of the chapter.
- 4.
This dependence immediately forces the neighbourhood relation to be symmetric.
- 5.
It is no surprise that computational techniques such as the Gibbs sampler stemmed from this area, as the use of conditional distributions is deeply ingrained in the imaging community.
- 6.
For those that do not want nor do not need to worry, the end of this section can be skipped, it being of a more theoretical nature and not used in the rest of the chapter.
- 7.
This representation is by no means limited to MRFs: it holds for every joint distribution such that the full conditionals never cancel. It is called the Hammersley–Clifford theorem, and a two-dimensional version of it was introduced in Exercise 3.10.
- 8.
The very name “Gibbs sampling” was proposed in reference to Gibbs random fields, related to the physicist Willard Gibbs. Interestingly, both of the major MCMC algorithms are thus named after physicists and were originally developed for problems that were beyond the boundaries of (standard) statistical inference.
- 9.
In fact, there exists a critical value of β, β c = 2.269185 in the case of the four neighbor relation, such that, when β > β c , the Markov chain converges to one of two different stationary distributions, depending on the starting point. In other words, the chain is no longer irreducible. In particle physics, this phenomenon is called phase transition.
- 10.
Similar to the Ising model mentioned in Footnote 9, there also exist a phase transition phenomenon and a critical value for β in this model.
- 11.
The upper bound on β in the above prior is chosen for a very precise reason: As mentioned in the previous footnotes, when \(\beta \geq 2\), the Potts model associated with a four-neighbor relation is almost surely concentrated on single-color images. It is thus pointless to consider larger values of β.
- 12.
Note that, for Laichedata, it is possible to wait for the equality S(x) = S(y) with a sufficiently high probability. In that case, since S is a sufficient statistic, we are simulating from the exact posterior.
- 13.
The setting of Markov random fields like the Ising and the Potts models is an exception in that it allows for a sufficient statistic, while being intractable via classical approaches.
- 14.
Besides image segmentation, another typical illustration of such structures is character recognition where a machine scans handwritten documents, e.g., envelopes, and must infer a sequence of symbols (i.e., numbers or letters) from digitized pictures. Hastie et al. (2001) provide an illustration of this problem.
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Marin, JM., Robert, C.P. (2014). Image Analysis. In: Bayesian Essentials with R. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8687-9_8
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