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 par ameter estimation for this model is addressed. Two 2-D fast Kalman alg orithms are proposed as extensions of the 1-D fast Kalman algorithm, w hich utilize the shift-invariant and near-to-Toeplitz properties of th e coefficient matrix of the normal equation resulting from the least s quares (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 th e second algorithm the model parameters are updated by absorbing new p ixel data or deleting old pixel data. The computational complexities o f the proposed two algorithms are O(Lm2) + O(L2m) MADPR (Multiplicatio ns And Divisions Per Recursion) and O(m3/2) MADPR respectively, compar ed with O(L2m2)+O(m3) and O(m3) needed in the corresponding direct lea st squares method, m and L being respectively the total number of mode l 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-homogen eous 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 algorithm s derived in this paper can be used in many applications where an imag e is considered as a nonstationary one.