Reversible Jump Markov Chain Monte Carlo (RJMCMC) is an extension of MCMCs which allows parameters to be sampled across different sample spaces. Introduced in Green (1995), three examples of applications of the RJMCMC method are provided, one of which consists of denoising a synthesized blurred image via Voronoi tesselation.
Voronoi tesselation (or Voronoi diagram) is a partition of a plane in which k generating points define the tesselation. In a 2D setting, all points (u, v) in R^2 are associated to one of the k generating points, whichever is closest in L2 distance. The voronoi_finite_polygons_2d in src/helper.py is the Python code that defines the tesselation, inspired from scipy.spatial.Voronoi.
As example of the posterior "denoised" image is provided in the first figure above titled "Deblurred image". The inference procedure is as followed: k points are sampled randomly in a 50 x 50 grid such that a Voronoi tesselation is defined (see image above). A height h_i for tile/region i in {1, ..., k} is given to each tesselation region by sampling from a Normal conditional posterior distribution centered at the mean of pixel-valued grayscale intensities of neighboring tiles. RJMCMC was used to determine the number of tiles between each Gibbs sample iteration whereby an extra tessalation point can be added or removed. As a result, k can take three possible values in any subsequent iteration: k-1, k or k+1.
This project was done around April 2018 as a final project for a graduate course in Bayesian inference with Professor David Stephens. My final report is available in this repository under report.pdf.
A major limitation is that our "Deblurred image" doesn't seem as polished as the results in the original paper. This is most likely due to an error in the Gibbs sampler (see notebooks/voronoi-tesselation-v2.ipynb) where I seem to run into issues where k falls below 3.