2-D fast Kalman algorithms for adaptive parameter estimation of nonhomogeneous Gaussian Markov random field model

In this paper, a two-dimensional (2-D) nonhomogeneous Gaussian Markov Random Field (GMRF) model is presented and the problem of adaptive parameter estimation for this model is addressed. Two 2-D fast Kalman algorithms are proposed as extensions of the 1-D fast Kalman algorithm, which utilize the shift-invariant and near-to-Toeplitz properties of the coefficient matrix of the normal equation resulting from the least squares (LS) criterion. In the first algorithm the space-varying model parameters are updated by sliding a data window with a constant size. By first shifting the data window from left to right and then from top to bottom, the spatial adaptive algorithm covers a whole image. In the second algorithm the model parameters are updated by absorbing new pixel data or deleting old pixel data. The computational complexities of the proposed two algorithms are O(Lm²)+O(L ²m) MADPR (Multiplications And Divisions Per Recursion) acid O(m^3/2) MADPR respectively, compared with O(L²m ²)+O(m³) and O(m³) needed in the corresponding direct least squares method, m and L being respectively the total number of model parameters to be estimated and the size of data window. For computer simulation two sample images which obey two sets of known parameters are first synthesized, and are then merged, resulting in a non-homogeneous image. It is shown that the 2-D fast Kalman algorithms developed in the paper reduce the computational complexity significantly and can track the model parameters very well. The estimated model parameters are as same as those obtained by using direct LS method. The algorithms derived in this paper can be used in many applications where an image is considered as a nonstationary one