Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

Two-Dimensional FIR Filters 22-2522.6 Two-Dimensional Filter Banks2-D subband decomposition of signals using filter banks (that implement a 2-D wavelet transform) findapplications in a wide range of tasks including image and video coding, restoration, denoising, and signalanalysis. For example, in recently finalized JPEG-2000 image coding standard an image is first processedby a 2-D filter bank. Data compression is then carried out in the subband domain. In this section webriefly discuss the case of four-channel separable filter banks. Chapter 24 provides a detailed descriptionof 1-D filter banks.In most cases, 2-D filter banks are constructed in a separable form with the use of the filters of 1-Dfilter banks, i.e., as a product of two 1-D filters [49,52]. We confine our attention to a 2-D four-channelfilter bank obtained from a 1-D two-channel filter bank. Let h 0 and h 1 denote the analysis filters of a 1-Dtwo-channel filter bank. The four analysis filters of the separable 2-D filter bank are given byh i,j (n 1 , n 2 ) ¼ h i (n 1 )h j (n 2 ), i, j ¼ 0, 1: (22:53)The filters h 0 and h 1 can be either FIR or IIR. If they are FIR (IIR), then the 2-D filters, h i,j , are also FIR(IIR). Frequency responses of these four filters, H i,j (v 1 , v 2 ), i,j ¼ 0, 1, are given byH i, j (v 1 , v 2 ) ¼ H i (v 1 )H j (v 2 ), i, j ¼ 0, 1, (22:54)ω 2where H 0 (v 1 ) and H 1 (v 2 ) are the frequency responses of the 1-D low-pass (approximating an ideal cutofffrequency at p=2) and high-pass filters of a 1-D subband filter bank, respectively [51]. Any 1-D filter bankdescribed in Chapter 24 can be used in Equation 22.53 to design 2-D filter banks. Feature-rich structuresfor 1-D filter banks are described in Ref. [6].The 2-D signal is decomposed by partitioning its frequency domain support into four rectangularregions of equal areas. The ideal passband regions of the filters, H i,j (v 1 , v 2 ), are shown in Figure 22.15. Forexample, the ideal passband of H 0,0 (v 1 , v 2 ) is the square region [ p=2, p=2] 3 [ p=2, p=2]. The 2-D filterbank is shown in Figure 22.16.Corresponding 2-D synthesis filters are alsoconstructed in a separable manner from thesynthesis filters of the 1-D filter bank. If the1-D filter bank has the perfect reconstructionπ(PR) property, then the 2-D filter bank also hasH 01H 00H 11H 10the PR property. Subband decomposition filterbanks (or filter banks implementing the 2-Dwavelet transform) consist of analysis and synthesisfilters, upsamplers, and downsamplers as–ππdiscussed in Chapter 24. In the separable 2-Dω 1 filter bank, downsampling is carried out bothhorizontally and vertically as follows:x 0 (n 1 , n 2 ) ¼ x a (2n 1 ,2n 2 ): (22:55)–πFIGURE 22.15 Ideal passband regions of the separablefilters of a rectangular filter bank.Here we consider the input 2-D signal x a to bean image. The downsampled image x 0 is aquarter-size version of x a . Only one sample outof four is retained in the downsampling operationdescribed in Equation 22.55. The upsamplingoperation is the dual of the downsampling