Parametrization of multi-dimensional Markov chains for rock type modeling
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A parametrization of a multidimensional Markov chain model (MDMC) is studied with the goal of capturing texture in training images. The conditional distribution function of each row in the image, given the previous rows, is described as a one-dimensional Markov random field (MRF) that depends only on information in the immediately preceding rows. Each of these conditional distribution functions is then an element of a Markov chain that is used to describe the entire image. The parametrization is based on the cliques in the MRF, using different parameters for different clique types with different colors, and for how many rows backward we can trace the same clique type with the same color. One of the advantages with the MDMC model is that we are able to calculate the normalizing constant very efficiently thanks to the forward-backward algorithm. When the normalizing constant can be calculated we are able to use a numerical optimization routine from R to estimate model parameters through maximum likelihood, and we can use the backward iterations of the forward-backward algorithm to draw realizations from the model. The method is tested on three different training images, and the results show that the method is able to capture some of the texture in all images, but that there is room for improvements. It is reasonable to believe that we can get better results if we change the parametrization. We also see that the result changes if we use the columns, instead of the rows, as the one-dimensional MRF. The method was only tested on images with two colors, and we suspect that it will not work for images with more colors, unless there are no correlation between the colors, due to the choice of parametrization.