An efficient affine projection algorithm for 2-D FIR adaptive filtering and linear prediction

Abstract

In this paper, a fast affine projection algorithm (FAPA), for the two-dimensional (2-D) adaptive linear filtering and prediction, is presented. The derivation of the proposed algorithm is based on the spatial shift-invariant properties of the 2-D discrete time signals. The proposed algorithm has low computational complexity, comparable to that of the 2-D LMS algorithm. The performance of the proposed scheme is comparable to that of the higher complexity 2-D RLS algorithms. The convergence speed and the tracking ability of the proposed 2-D FAPA algorithm are illustrated by computer simulation.

Introduction

Two-Dimensional (2-D) system identification has received extensive attention in a wide range of image processing and image analysis applications. Typical examples include modeling of 2-D data, filtering and smoothing of noisy images, image enhancement and edge detection, prediction and spectrum estimation of 2-D data, image restoration, stochastic texture analysis and synthesis, image segmentation, image compression, target detection, change detection in image sequences, etc. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14].

A widely used class of 2-D models consists of linear 2-D finite impulse response (2-D-FIR) models, i.e., models that provide estimates of the underlying process using a finite support, linear combination of the 2-D data. Linear 2-D modeling is very popular due to the underlying model simplicity and the ability to obtain the optimum model parameters by solving a set of linear equations, for the case when the mean squared error (MSE) has been adopted as the measure of the estimation performance [6]. 2-D MSE parameters estimation has been considered in the past, for the case of stationary 2-D models, with known data statistics. In many applications, however, the classical 2-D MSE solution cannot be directly applied, because either the underlying data model is nonstationary, or the data statistics are not known in advance. In these cases, the optimum design is obtained by applying adaptive least-squares (LS) 2-D techniques, that operate directly on the available data set [13], [15], [16], [17].

LS 2-D filtering aims to shape an input signal $x(m,n)$ so that the corresponding output $y(m,n)$ matches a desired response signal $z(m,n)$. Following the standard notation, indices m, n denote the row and the column of the image, respectively. 2-D data are obtained by a particular scanning of a discrete 2-D rectangular region, consisting of an $M\times N$ regularly spaced lattice. Raster scanning is assumed, i.e., left to right, top to bottom. In the LS formulation we assume that data records of input and the desired response $x(m,n)$ and $z(m,n)$, over the range $(m,n)\in [1,1]\times [M,N]$, provide our knowledge basis, and that we select the filter which minimizes a cost function over the available data record. The minimization procedure is carried out recursively, i.e., the optimum LS filter is re-estimated (or adapted) each time a new pair of measurements is collected.

Two different approaches, leading to two widely used algorithmic families, have been adopted for the adaptive estimation of the optimum LS 2-D FIR filter, on the basis of the available data set. The first one is based on a stochastic approximation of the steepest descent method and is known as the 2-D LMS family [1], [15], [18], [19], [20]. The latter is based on a stochastic approximation of the Gauss–Newton method and is known as the 2-D RLS family [2], [17], [21], [22], [23], [24], [25], [26]. Although 2-D LMS-type algorithms have small computational cost, they suffer from slow convergence rate, especially when the input signal autocorrelation matrix has a large eigenvalue spread. 2-D RLS-type algorithms do not suffer from such a drawback. They have, however, increased computational complexity. In an attempt to improve the convergence characteristics of the 2-D LMS-type algorithms, a Quasi-Newton adaptive scheme, called ‘the 2-D Affine Projection Algorithm (2-D-APA)’, has been proposed [28]. The good convergence performance of the 2-D-APA adaptive algorithm, is, however, overshadowed by the high computational complexity of the method.

In this paper, fast algorithms for the efficient implementation of the 2-D-APA scheme will be presented. The derivation of the fast 2-D-APA algorithms is based upon the spatial shift invariance property that the 2-D regressor vector possesses. The proposed have reduced computational complexity, compared to that of the original 2-D-APA algorithm. The performance of the proposed fast 2-D-APA algorithms is illustrated by means of computer simulation.