Unsupervised Image Segmentation with Pairwise Markov Chains Based on Nonparametric Estimation of Copula Using Orthogonal Polynomials

Abstract

Copula were introduced in Markov models in the early 2000s to better model the relationship between the observation data involved in these models. However, their estimation is difficult. This paper presents a new approach in the estimation of copula in Markov models. The proposed approach is based on a nonparametric method of estimating the density of the copula. The decomposition of an orthonormal basis of the unit interval support polynomials is used to estimate this density. The family of polynomial used is built from the family of Legendre polynomials. Our approach has the major advantage of reducing the problem of unsupervised image segmentation by Pairwise Markov chains to the problem of the estimated marginal distributions unlike the conventional approach which requires both an estimation of the density the marginal distributions and the estimation of the density distribution of the copula. Moreover, the problems of boundary effect encountered in the density estimation of copulas are solved by the use of orthonormal basis functions in the unit interval. The new model is validated from experiments performed on synthetic images and real images including optical and radar satellite images not necessarily affected by the Gaussian noise. The results are encouraging and show the proposed model as an interesting alternative to Pairwise Markov chains commonly used in literature.