Design of Two-channel 2-d Nonseparable Multi-plet Perfect Reconstruction Filter Banks

and the frequency characteristics of the prototype FB. Also, the lengths and the passband delays of the analysis/synthesis filters This paper proposes a new design method for a class of can be made closer to each other, which simplifies practical two-channel 2D non-separable perfect reconstruction (PR) filter implementation and reduces the number of additional delay banks (FBs) using the multi-plet FBs. ID multi-plet FBs are PR elements required for processing two subband signals. Finally, FBs that can be obtained by frequency transformation of a similar to the ID to 2D mapping used in [1] and [11], the ID prototype PR FB in the conventional lifting structure so that a multi-plet FB can be readily transformed to construct 2D PR better frequency characteristics can be obtained and varied FB with quincunx, hourglass and parallelogram spectral online to process different signals. By employing the ID to 2D supports. Note that these multi-plet 2D nonseparable PR FBs transformation of Phoong et al, new 2D PR multi-plet FBs with can also be implemented as a lifting structure with the same quincunx, hourglass, and parallelogram spectral support are number of lifting steps as its prototype. Furthermore, they can obtained. These nonseparable multi-plet FBs can be cascaded to be cascaded appropriately in a tree-structure to obtain new PR realize new PR directional FB for image processing and motion directional FBs, using the approach previously proposed in analysis. The design procedure is very general and it can be [12]. Such directional FBs found important applications in applied to both linear-phase and low-delay 2D FBs. Design segmentation, directional decomposition of images, motion examples are given to demonstrate the usefulness of the analysis of videos, etc [13]. Design examples are given to proposed method. illustrate the potential flexibility of the proposed approaches in processing ID and 2D signals. The paper is organized as follows: the ID multi-plet FBs and the concept of frequency transformation are introduced in Perfect reconstruction (PR) filter banks (FBs) have section II. Their generalizations to 2D nonseparable PR FBs important applications in signal analysis, coding and the design with quincunx, hourglass and parallelogram spectral supports of wavelet bases. An efficient structure of two-channel are discussed in Section III. Several design examples are given biorthogonal FIR/IIR FBs, which structurally satisfies the PR in section IV to illustrate the effectiveness of the proposed condition, is the structural PR FB proposed by Phoong et al. approach, and finally, conclusion is drawn in section V. [1]. Moreover, the FBs of this structure can be transformed, via a simple ID to 2D transformation, to obtain 2D PR FIR II. MULTI-PLET TWO-CHANNEL STRUCTURAL PR FBs nonseparable filter banks with quincunx spectral support. One limitation of this structure is that the magnitudes of the lowpass The general structure of the multi-plet two-channel FBs is and highpass analysis filters at c = gz/2 in the linear-phase case shown in Fig. 1. It is parameterized by L subfilters Qm(z), L are respectively restricted to 0.5 and 1, or vice versa. In another delay parameters Nm , L lifting coefficients Pm , and two structural PR FBs called triplet FBs [2] [4], a generalization scaling constants C0 and C1 for m = 0,1,...,L 1. It can be seen of the structure in [1], more degree of freedom is available and from Fig. 1 that z-transforms of the analysis and synthesis it is possible to achieve a more symmetric frequency response. filters in the lifting structure can be written as follows: More recently, Chan et al. extended the structural PR FBs [1] and the triplet FBs [2] [4], which involve two and three lifting Ho (z) = COH(L) (z), H1 (z) = CiH(L) (z), steps [5], respectively, and studied a new class of two-channel Fo (z) = HI (-z) and F, (z) = -Ho (-z) (1) structural PR FBs with multiple lifting steps called the multiplet FBs [6]. In particular, they showed that the concept of where H(0) (z) z2Noi +p Q (Z2) frequency transformation of digital filters studied in [7] can be H(1) (z) = Z-2Nl + P1 Q1 (z2 )H(°) (z), and applied directly to the lifting structure to obtain another PR FB with the same number of lifting steps having the similar H(m) (z) = z2Nm H(m2) (z) + Pm Q (Z2 )H(m4) (Z) frequency characteristics but an arbitrary sharp transition for m = 2,3,.,L 1. We shall consider a special case of lifting bandwidth. This can also be viewed as an extension of the with identical subfilters: work in [8]. QL_1 (Z ) =QL-2 (Z2) Q1 (Z2 Q (Z2 Q(Z2 (2) In this paper, we shall extend the ID to 2D transformation Qhe (zl ) prQt2 z2 are t Q ivzn Q Qz of Phoong et al. [1] to the multi-plet FBs. The design of such The delay parameters Nm are then given by: PR 2D nonseparable FBs can be divided into three steps. NL-i= =N2 = N1 =G and No= (G-1)/2, (3) Firstly, a low order prototype PR FB with a rather wide where C is the passband group delay of Q(Z2). As a result, the transition bandwidth is first designed to meet certain group delays of the analysis filter pair, Ho(z) and Hi(z), are specifications on passband and stopband ripples and it is then respectively given by: factorized into lifting structure [9]. Alternatively, the well CO =(L -1) G and C==L G C. (4) known lifting structured FB, say (9,7) filter pair [10], can be For the simplest case where identical subfilter having the form used. Secondly, by properly designing the subfilter of of( -l1 h Bi efre oa rttp itn transformation, the frequency response of the prototype FB can ',r vr o be warped by means of frequency transformation to meet the structured FB. Furthermore, it can be transformed to a new FB desired transition bandwidth, while preserving the PR condition using the following substitution of variable: 0-7803-9390-2/06/$20.00 ©C2006 IEEE 2033 ISCAS 2006 x = RQ (x) = zGQ(z2), (5) transformation is also valid when Q(z) is a low-delay FIR/IR function. For more details, interested readers are referred to [6]. where RQ (x) is the zero-phase response of the subfilter Q(z2) for some positive integer G. Since the transformed FB is III. DESIGN OF 2D MULTI-PLET FILTER BANK obtained by replacing x in each lifting step of Fig. 1 by RQ(X), it can also be implemented by the same number of Let x(n) be a N-dimensional discrete-time signal with lifting steps as the prototype. n = (nO, nfN-l )T and n, E X, the set of integers. The output To analyze the effect of the transformation, let us consider y(n) of a decimator with an integer sub-sampling matrix M the zero-phase responses of the prototype FB, subfilter and can be written as y(n) = x(Mn) The decimation factor is equal transformed FB in Fig. 2. From (5), the digital frequencies to det(M) Different choices of M give rise to different before and after transformation are related by: x = cos(6) = RQ (x) = R. (cos(c)), spectral support of x(n) while achieving aliasing-free where c and 6 are respectively the digital radian frequencies decimation. The points in x(n) that are retained in of the prototype and transformed FBs. If RQ (X) is y(n) = x(Mn) lie on the lattice {LAT(M) t = Mn, neN}. appropriately designed to have a sharper characteristic than Fig. 3c shows the lattice generated by the 2D sampling matrix x=cos c around c = z /2, then the transformed FB will have MQ =[MQO MQ ] associated with the quincunx spectral a much narrower transition band. Furthermore, if we want to support in Fig. 3a where MQ_O = [I I]T and MQ-1 = [I I]T preserve the passband and stopband ripples of the prototype FB in~~~~~~~~ th.rnfre B ~(? hudmprsetvl h It can also be seen that those points on LAT(M) are in fact in the transformed F , RQ X) s o l map respectively the geradbytelnrco iainsfM adM values of x = cos w in the passband and stopband of the ga0 adM prototype FB to the new passband and stopband of the Fig. 4 shows the general structure of a two-channel transformed FB. Therefore, the transformed FB can achieve a critically decimated N-D FB with sub-sampling matrix M and much narrower and prescribed transition bandwidth by properly Idet(M)l = 2. The analysis and synthesis filters are expressed designing the subfilter using conventional filter design respectively in their type-I and type-II polyphase technique, while preserving the frequency characteristics of the representations, where E(z) and R(z) are their polyphase prototype FB. matrices. k and k1 belong to the set of integer vectors, Using these results, the relations of the prototype FB, m k subfilter and transformed FB can be summarized as follows: N(M), which lie inside the fundamental parallelepiped of M, (i) Zero-phase response of the transformed FB: FPD(M), where FPD(M) is the region spanned by M x, I1-.po< Ro(X) < I+po, 0 < . c with xi c[0,1), i=0,...,N-I The gray parallelogram area in Lowpass: l0<. Ro( ). , <TFoG <_<g Fig. 3c spanned by the two vectors (MQ,:oMQ1) is the ip gPl < Rx < 1 + I,1 ;Tc.c< ) < T FPD(M) associated with MQ. If the product R(zM)E(zM) Highpas P L, < R, (5?) . 7 61 0< . < is equal to a constant multiplies of signal delay, then the system c ~~~~isPR. where wc specifies the cutoff frequency; 8i and 8si are Similar to [1], a simple method for achieving this condition respectively the passband and stopband ripples of the can be done by transformation of ID PR FB. More precisely, transformed analysis filters R, (5), for i = 0,1 Q(z2 ) is transformed to Q(zMO )Q(zM1) and the 1-D delay z2 (ii) Zero-phase response of the prototype FB: is replaced with zMo zM1 in Fig. 1. Then, the 2D analysis filters I gpo <,ko (x) < I + gpo, 0 < ~~< c-o can be written as follows: Lowpass: (x) 0,< 6 <6 Ho(z) = COH(L-2)(z) and HI(z) = C H(Ll1)(z) (9) 1s < (x) .6.7<) where He )(z) -=zk(ZMZM)-No +P . Q(ZMO)Q(ZMl) Highpass: l1 <, (x) < I + , < < H(1) (z) (zMozMy)-N1 +p, Q(zMO)Q(zM1)H(O)(z)), and Ls < R1(x) <81 0<sl<c H(m) (z) (zMO zM1 )-N H(m-2) (z) + pmQ(ZM 0)Q(zM1 )H(ml) (z), where ik is the cutoff frequency of the prototype FB. for m = 2,3,...,L 1. Hence, the problem remaining is to (iii) Zero-phase response of the subfilter: determine the sampling matrix M and the integer vectors ko ros(,~~C) < R ^ < 1, 0 < co < co and k1 according to the desired spectral support. Suppose {1.Rg(5?).cos(6<) 2Z-(0v<W.2T c ^ (8) x(n) is decimated by M , its aliasing-free spectral support I < RQ < cs(,~~C ), ~r coc < co < ~r should be equal to the spectral support Q of the analysis If the prototype FB is monotonic decreasing at the stopband, -T 2 say a maximally-flat function in 9/7 filter pair [10], then by lowpass filter, which iS given by {Q :w= zMT x,x [-1,1)2} choosing an appropriate subfilter, an arbitrary small stopband [14]. For notational convenience, let 7zM-T= [aw ,w] In attenuation can be achieved after transformation. Alternatively, general, if [ws0, w1l] can be determined from the desired spectral a set of prototype PR FBs with different passband/ stopband ripples can be designed offline. The subfilter can then be support, then M can be computed from M= [wO WtiV]T /1~. designed so that the transformed FB will be able to achieve a Consider the quincunx sampling as an example, the spectral narrower transition bandwidth, while preserving the ripples of support is a parallelogram defined by the two vectors, the prototype FB. Optimized prototype FBs usually lead to a = ~2~2T adw=[-/2 11T a hw nFg better frequency selectivity. It should be noted that this " 3a. Using this result, it can easily be verified that the sampling