reconstruct: Image reconstruction for the Potts model with six classes

Description

This function adresses the reconstruction of an image distributed from a Potts model based on a noisy version of this image. The purpose of image segmentation (Chapter 8) is to cluster pixels into homogeneous classes without supervision or preliminary definition of those classes, based only on the spatial coherence of the structure. The underlying algorithm is an hybrid Gibbs sampler.

Usage

reconstruct(niter, y)

Value

beta: MCMC chain for the parameter \(\beta\) of the Potts model
mu: MCMC chain for the mean parameter of the blurring model
sigma: MCMC chain for the variance parameter of the blurring model
xcum: frequencies of simulated colours at every pixel of the image

Arguments

niter: number of Gibbs iterations
y: blurred image defined as a matrix

Details

Using a Potts model on the true image, and uniform priors on the genuine parameters of the model, the hybrid Gibbs sampler generates the image pixels and the other parameters one at a time, the hybrid stage being due to the Potts model parameter, since it implies using a numerical integration via integrate. The code includes (or rather excludes!) the numerical integration via the vector dali, which contains the values of the integration over a 21 point grid, since this numerical integration is extremely time-consuming.

Examples

Run this code

if (FALSE) data(Menteith)
lm3=as.matrix(Menteith)
#warning, this step is a bit lengthy
titus=reconstruct(20,lm3)
#allocation function
affect=function(u) order(u)[6]
#
aff=apply(titus$xcum,1,affect)
aff=t(matrix(aff,100,100))
par(mfrow=c(2,1))
image(1:100,1:100,lm3,col=gray(256:1/256),xlab="",ylab="")
image(1:100,1:100,aff,col=gray(6:1/6),xlab="",ylab="")