IEEE TRANSACTIONS ON IMAGE PROCESSING 1 Special Paraunitary Matrices, Cayley Transform, and Multidimensional Orthogonal Filter Banks Jianping Zhou, Minh N. Do, Member, IEEE, and Jelena Kovaceviˇ c,´ Fellow, IEEE Abstract— We characterize and design multidimensional or- I. INTRODUCTION thogonal filter banks using special paraunitary matrices and the Cayley transform. Orthogonal filter banks are represented Multidimensional (MD) filter banks have gained particular by paraunitary matrices in the polyphase domain. We define attention in the last decade [1]–[9]. Nonseparable filter banks special paraunitary matrices as paraunitary matrices with unit can capture geometric structures in MD data and offer more determinant. We show that every paraunitary matrix can be freedom and better frequency selectivity than traditional sepa- characterized by a special paraunitary matrix and a phase factor. Therefore the design of paraunitary matrices (and thus of or- rable filter banks constructed from one-dimensional (1D) filter thogonal filter banks) becomes the design of special paraunitary banks. Nonseparable filter banks also provide flexible direc- matrices, which requires a smaller set of nonlinear equations. tional decomposition of multidimensional data [10]. Therefore, Moreover, we provide a complete characterization of special nonseparable filter banks are more suited to image and video paraunitary matrices in the Cayley domain, which converts applications. nonlinear constraints into linear constraints. Our method greatly simplifies the design of multidimensional orthogonal filter banks Orthogonal filter banks are special critically sampled perfect and leads to complete characterizations of such filter banks. reconstruction filter banks where the synthesis filters are time- reversals of the analysis filters. Orthogonal filter banks can Index Terms— Cayley Transform, Filter Banks, Multidimen- sional Filter Banks, Nonseparable Filter Design, Orthogonal be used to construct orthonormal wavelet bases [11], [12]. Filter Banks, Paraunitary, Polyphase, Special Paraunitary. Because of orthogonality, orthogonal filter banks offer certain conveniences; for example, the best M-term approximation is simply done by keeping those M coefficients with largest magnitude. Designing nonseparable MD orthogonal filter banks is a challenging task. Traditional design methods for 1D orthog- onal filter banks cannot be extended to higher dimensions directly due to the lack of an MD factorization theorem. In the infinite impulse response (IIR) case, Fettweis et al. applied wave digital filters and designed a class of orthogonal filter banks [13]. In the finite impulse response (FIR) case, there are only a few design examples (for example, [3]). In the polyphase domain, the polyphase synthesis matrix of an orthogonal filter bank is a paraunitary matrix, U(z), that satisfies UT (z−1)U(z) = I, for real coefficients. (1) A paraunitary matrix is an extension of a unitary matrix when the matrix entries are Laurent polynomials. Paraunitary matrices are unitary on the unit circle. For simplicity, we con- sider only filter banks with real coefficients. The paraunitary condition (1) requires solving a set of nonlinear equations — a difficult problem. Vaidyanathan and Hoang provided a Jianping Zhou is with the Department of Electrical and Computer En- gineering and the Coordinated Science Laboratory, University of Illinois at complete characterization of paraunitary FIR matrices for 1D Urbana-Champaign (email: [email protected]). orthogonal filter banks via a lattice factorization [8](pp. 302– Minh N. Do is with the Department of Electrical and Computer Engi- 322). However, in multiple dimensions the lattice structure is neering, the Coordinated Science Laboratory, and the Beckman Institute, University of Illinois at Urbana-Champaign (email: [email protected]). not a complete characterization. Jelena Kovaceviˇ c´ is with the Department of Biomedical Engineering and Recently we proposed a complete characterization of MD the Department of Electrical and Computer Engineering, Carnegie Mellon orthogonal filter banks using the Cayley transform and de- University (email: [email protected]). This work was supported in part by the National Science Foundation under signed some orthogonal filter banks for both IIR and FIR cases Grant CCR-0237633 (CAREER). [14]. The Cayley transform maps a paraunitary matrix to a 2 IEEE TRANSACTIONS ON IMAGE PROCESSING para-skew-Hermitian matrix H(z) that satisfies IIR Infinite Impulse Response. MD Multi-Dimensional. H(z−1) = HT (z), for real coefficients. (2) − PSH Para-Skew-Hermitian. Conversely, the inverse Cayley transform maps a para-skew- SPSH Special Para-Skew-Hermitian. Hermitian matrix to a paraunitary matrix. Therefore, the Cay- SPU Special Paraunitary. ley transform establishes a one-to-one mapping between pa- raunitary matrices and para-skew-Hermitian matrices. A para- II. MULTIDIMENSIONAL ORTHOGONAL FILTER BANKS skew-Hermitian matrix is an extension of a skew-Hermitian AND SPECIAL PARAUNITARY MATRICES matrix when the matrix entries are Laurent polynomials. Para-skew-Hermitian matrices are skew-Hermitian on the unit A. Multidimensional Orthogonal Filter Banks circle. In contrast to solving for the nonlinear paraunitary We start with notations. Throughout the paper, we will condition in (1), the para-skew-Hermitian condition amounts always refer to M as the number of dimensions, and N to linear constraints on the matrix entries in (2), leading to an as the number of channels. In MD, z stands for an M- T −1 easier design problem. dimensional variable z = [z1,z2,...,zM ] and z stands −1 −1 −1 T z The new contribution of this paper is the introduction of for [z1 ,z2 ,...,zM ] . Raising to an M-dimensional T k the special paraunitary (SPU) matrix that leads to a simplified integer vector power k = [k1, k2,...,kM ] yields z = M ki A and complete characterization of MD orthogonal filter banks. i=1 zi . For a matrix , we use Ai,j for its entry at (i, j). U z I A paraunitary matrix ( ) is said to be special paraunitary QWe use N to denote the N N identity matrix, and omit the if its determinant equals 1. We will show that any paraunitary subscript when it is clear from× the context. For a matrix A, the matrix can be characterized by an SPU matrix and a phase entry of its adjugate (denoted by adjA) at (i, j) is defined as i+j factor that applies to one column, as illustrated in Fig. 1. This ( 1) det Aj,i, where Aj,i is the submatrix of A obtained leads to an important signal processing result that any N- by− deleting its jth row and ith column. channel orthogonal filter bank is completely determined by its N 1 synthesis filters and a phase factor in the last synthesis ANALYSIS SYNTHESIS filter.− Although this result was shown for 1D two-channel z−l0 D D zl0 orthogonal filter banks [15] and MD two-channel orthogonal filter banks [3], to the best of our knowledge, this is the first time it is proved for general orthogonal filter banks of any z−l1 D D zl1 dimension and any number of channels. Moreover, the design H G + ˆ problem of orthogonal filter banks can be converted into that X p p X of SPU matrices, leading to solving a smaller set of nonlinear equations. In other words, the SPU condition provides the core of the orthogonal condition for a filter bank. Finally, since the l l z− N−1 D D z N−1 characterization of SPU matrices in the Cayley domain is also simpler than that of the general paraunitary matrices, SPU (a) matrices also simplify the characterization of MD orthogonal filter banks in the Cayley domain. ANALYSIS SYNTHESIS H0 D D G0 Orthogonal Paraunitary Special Paraunitary Filter Banks Matrices Phase H1 D D G1 Matrices X + Xˆ Fig. 1. Relationship among orthogonal filter banks, paraunitary matrices, and special paraunitary matrices: Orthogonal filter banks are characterized by paraunitary matrices in the polyphase domain; Paraunitary matrices are D D characterized by special paraunitary matrices and phase factors. HN−1 GN−1 The rest of the paper is organized as follows. In Section II, (b) we study the link between multidimensional orthogonal filter Fig. 2. Multidimensional filter banks and polyphase representation. (a) A banks and special paraunitary matrices. In Section III, we multidimensional N-channel filter bank: Hi and Gi are analysis and synthesis D study the Cayley transform of special paraunitary matrices. filters, respectively; is an M × M sampling matrix. (b) Polyphase representation: Hp and Gp are N × N analysis and synthesis polyphase The characterization of two-channel special paraunitary ma- l N−1 Dt matrices, respectively; { j }j=0 is the set of integer vectors of the form , trices in the Cayley domain and the design of two-channel such that t ∈ [0, 1)M . orthogonal filter banks are given in Section IV. We conclude in Section V. The glossary of abbreviations is given as follows. Consider an MD N-channel filter bank as shown in CT Cayley Transform. Fig. 2(a). For implementation purposes, we only consider filter FCT FIR-Cayley Transform. banks with rational filters. We are interested in the critically FIR Finite Impulse Response. sampled filter bank in which the sampling rate is equal to the ZHOU, DO, AND KOVACEVIˇ C:´ SPECIAL PARAUNITARY MATRICES, CAYLEY TRANSFORM, AND FILTER BANKS 3 number of channels, that is, det D = N. In the polyphase paraunitary condition (1) leads to N(N + 1)/2 equations domain, the analysis and synthesis| parts| can be represented by with N 2 unknowns. The condition (4) leads to N(N 1)/2 2 − N N polyphase matrices Hp(z) and Gp(z) respectively, as equations with N N unknowns. Moreover, it can be seen shown× in Fig. 2(b). In particular, IIR filter banks lead to IIR that the set of nonlinear− equations generated by (4) is a polyphase matrices, entries of which are rational functions, subset of that generated by (1). Therefore, solving the SPU while FIR filter banks lead to FIR polyphase matrices, entries condition instead of the paraunitary condition saves us N of which are polynomials.