RECURSIVE-IN-ORDER LEAST-SQUARES PARAMETER-ESTIMATION ALGORITHM FOR 2-D NONCAUSAL GAUSSIAN MARKOV RANDOM-FIELD MODEL

We present in this paper a recursive-in-order least-squares (LS) algor ithm to compute efficiently the parameters of a 2-D Gaussian Markov ra ndom field (GMRF) model. The algorithm is based on the fact that the l east-squares estimation of the parameters of a 2-D noncausal GMRF mode l is consistent and the coefficient matrix in the normal equation has near-to-block-Toeplitz structure. Hence, it has a Levinson-like form f or the updating of model parameters by introducing auxiliary variables . Moreover, this paper proposes the concept of recursive path for 2-D recursive-in-order algorithms, and points out that there exists a trad eoff between fast computation of the parameters and accurate choice of model support; a compromise recursive path is then suggested where th e orders change alternately in two directions. The computational compl exity of the developed algorithm is analyzed, and the results show tha t the algorithm is more efficient when either the image size or the mo del support is larger. It is found that the total number of multiplica tions (mps) involved in the new algorithm is only about 14% of that in the conventional LS method when the image size is 512 x 512 and the n eighbor set of the model is a 17 x 17 window. Computer simulation resu lts using the recursive-in-order algorithm developed in this paper and the conventional LS method are given to verify the correctness of the new algorithm.