Paiiepistimiopolis, TYPA Buildings, 157 71 ATHENS, GR.EECE. A B S T R A C T . This paper is concerned with the development of fast batch and adaptive algorithms for Two Dimensional (2-D) least squares (LS) FIR filtering. The proposed algorithms accommodate filter masks with arbitrary boundaries shape, and utilize scalar operations, only. 1. P r o b l e m forinulation We shall deal with the general 2-D FIR filter Such filters are used in a wide range of applications.Typical examples include digital restoration of coloured images, classification and segmentation, pattern recognition and object detection, and 2-D spectral estimation, [11-[51. Extension of the results to multidimensional signals can be carried out with analogous arguments. We shall allow for general support regions, (filter masks), R. Thus, R can be strongly causal, causal, semicausal and/or noncausal. More precisely, R consists of a union of intervals, C l e a r l y , P = m a x { k : ( k , l ) E R } , c r = m i n { k : ( k , l ) E R},Examples of causal and noncausal regions are illustrated in Fig. 1, (a-b). The engagement of varying length intervals Sl;, ~k , enables the incorporation of masks with very general boundaries, for instance, smooth curves like circles or parabolas, see Fig. 1, (c).Let z(m, 72) denote the desired response 2-D signal.We seek to determine the FIR structure that minimizes some function of the error, e(m,n) = z(m,n)y(m,vx). Two criteria will be considered. The mean squared error and the total squared error In the former case, both input and desired response 2-D signals are random fields, possessing certain stationarity properties that will be explained below. In the second case, samples of z ( m , n ) and s(nz,n) are assuiiied available on the observation region [ M I N I ] x [MzNz]. Efficient algorithms for batch and adaptive LS filtering can be derived if a shift invariance property prevails. The method of [6], uses extended filter masks, R, R E. R, that enables the formulation of fast adaptivestructures. This overparametrization approach, however, increases computational complexity and memory needs.In this paper, we introduce suitable permutations that unravel shift invariance properties and enable the development of fast batch and adaptive LS algorithms for 2-D FIR filters with general masks. Fast algorithms for batch LS filteringTo come up with a format amenable to efficient coinputation we manipulate eq. (1) as follows,-3009 -CH2977-719110000-3009 $1.00 0 1991 IEEE
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