Accelerated convergence of an iterative implementation of a two-dimensional IIR filter

An investigation is conducted of the acceleration of the iterative implementation of the two-dimensional infinite impulse response (IIR) filter proposed by D.E. Dudgeon (IEEE Trans. Acoust. Speech Signal Proc., vol.ASSP-28, no.6, p.666-71, 1980). The conventional procedure results in a sequence of estimations that converge linearly to the desired filter output. Applying an accelerated convergence algorithm for linearly distorted signal reconstruction problems, the authors use a product expansion of the denominator of the transfer function of an IIR filter to derive a class of iterative algorithms that have a pth-order rate of convergence. Specifically, k iterations of the proposed pth-order algorithm are equal to the p^k iterations of the linear algorithm. Therefore, the number of iterations required to obtain a given approximation is reduced from p^k to k. The authors show that the spatial truncation error does not affect the computation of the desired result in the region of interest, provided that the boundary values outside the area of interest are known. Finally, the authors present two examples in order to compare the convergence rates for the linear and the accelerated algorithms