Perfect Reconstruction Filter Banks Research Articles

Classification of Nonexpansive Symmetric Extension Transforms for Multirate Filter Banks

This paper describes and classifies a family of invertible discrete-time signal transforms, referred to as symmetric extension transforms(SET's), for finite-length signals. SET's are algorithms for applying perfect reconstruction multirate filter banks to symmetric extensions of finite-length signals, thereby avoiding the boundary artifacts introduced by simple periodic extension. A key point is when such symmetric decompositions can be formed with no increase in data storage requirements (“nonexpansive decompositions”). Transforms based on three types of symmetric extension and four classes of linear phase filters are analyzed in terms of their memory requirements for general M-channel perfect reconstruction filter banks. The classification is shown to be complete in the sense that it contains all possible nonexpansive SET's. Completeness is then used to deduce design constraints on the construction of nonexpansive M-channel SET's, including new obstructions to the existence of certain classes of filter banks. This paper also forms the principal technical reference on the SET algorithms incorporated in the Federal Bureau of Investigation's digital fingerprint image coding standard.

Design of 2-D perfect reconstruction filter banks using transformations of variables: IIR case

We have previously (1993) presented a design technique for 2-channel multidimensional filter banks. The technique is based on the transformation of variables and provides the flexibility of controlling the frequency characteristics of the resulting subband filters with ease. In this paper we extend the design technique to yield IIR filters. We emphasize the 2-D diamond subband case (with quincunx sampling) but the technique can be used for the parallelogram and diagonally quadrant subband cases as well. The IIR nature of the filters occurs from the use of IIR transformation functions instead of FIR. Two cases are considered. The first is with zero-phase IIR transformations which yield linear phase subband filters but require noncausal filtering. The second is with nonlinear phase IIR transformations which yield nonlinear phase subband filters that can be implemented in a causal manner.

Design of linear‐phase M‐channel perfect reconstruction finite impulse response filter banks using lagrange‐newton method

AbstractFilter banks are separated roughly into paraunitary and nonparaunitary ones. Paraunitary filter banks are excellent and have several good characteristics, but it is difficult to design such a system. This research focuses on nonparaunitary filter banks and presents a design method for the linear‐phase (LP) M‐channel perfect reconstruction (PR) finite impulse response (FIR) filter banks. In this method, M‐2 analysis filters are arbitrarily designed and then the remaining ones are optimized using the Lagrange‐Newton method. Using the proposed method, PR filter banks with arbitrary order can be designed easily. Furthermore, the design method that considers the pairwise‐shift symmetry is shown and such a system is easy to design.

Adaptive filtering using filter banks

This paper derives a theory for adaptive filters which operate on filter bank outputs, here called filter bank adaptive filters (FBAFs). It is shown how the FBAFs are a generalization of transform domain adaptive filters and adaptive filters based on structural subband decompositions. The minimum mean-square error performance and convergence properties of FBAFs are determined as a function of filter bank used. A parametrization for a class of FIR perfect reconstruction filter banks is derived which is used to design FBAF's having optimal error performance given prior knowledge of the application. Simulations are performed to illustrate the derived theory and demonstrate the improved error performance of the FBAFs relative to the LMS algorithm, when prior knowledge is incorporated.

Minimal and complete factorization for a class of paraunitary filter banks

M channel maximally decimated filter banks have been used in the past to decompose signals into subbands. The theory of perfect-reconstruction filter banks has also been studied extensively. A class of filter banks that has received particular attention are the so-calledparaunitary filter banks. These filter banks are attractive because there exist robust structures for implementing these systems, based on factorizations of their polyphase matrices. In this paper, we will impose two other conditions on the paraunitary filter bank, the linear phase property of the filters, and their pairwise mirror-image symmetry in the frequency domain. These two properties are useful in several applications, including image processing. In this paper we will propose a factorization for filter banks having all the three properties simultaneously, namely, paraunitariness, linear phase, and pairwise mirror-image symmetry. We will then prove that our factorization isminimal andcomplete. This means that the number of delays used is equal to the McMillan degree, and that all filter banks satisfying the three properties simultaneously can be factored in terms of our factorization. Our factorization therefore gives us a robust structure for implementing these systems. All the three properties are guaranteed to be satisfied inspite of multiplier quantization.We are not aware of any other factorizations that are proved minimal and complete for any three properties simultaneously.

Pade table, continued fraction expansion, and perfect reconstruction filter banks

We investigate the relationships among the Pade table, continued fraction expansions and perfect reconstruction (PR) filter banks. We show how the Pade table can be utilized to develop a new lattice structure for general two-channel bi-orthogonal perfect reconstruction (PR) filter banks. This is achieved through characterization of all two-channel biorthogonal PR filter banks. The parameterization found using this method is unique for each filter bank. Similar to any other lattice structure, the PR property is achieved structurally and the quantization of the parameters of the lattice does not effect this property. Furthermore, we demonstrate that for a given filter, the set of all complementary filters can be uniquely specified by two parameters, namely, the end-to-end delay of the system and a scalar quantity. Finally, we investigate the convergence of the successive filters found through the proposed lattice structure and develop a sufficient condition for this convergence.

BOT's based on nonuniform filter banks

Parallels between orthogonal transforms and filter banks have been drawn before. Block orthogonal transform (BOT) is a special case of orthogonal transform where a nonoverlapping window is used. We relate BOTs to filter banks. Specifically, we show that any BOT can be shown as a perfect reconstruction filter bank, and any tree-structured perfect reconstruction filter bank or any orthonormal filter bank for which no filter length exceeds its decimation factor can be shown as a BOT. We then show that all conventional BOTs map to uniform filter banks. A construction method to design a BOT from any nonuniform filter bank is presented, and finding an optimal tree structure (in the sense of transform coding gain) for a given source is also discussed. The results show that the optimal, nonuniform BOT outperforms uniform BOTs having either the same number of bands or the same size in most cases.

A binary wavelet decomposition of binary images

We construct a theory of binary wavelet decompositions of finite binary images. The new binary wavelet transform uses simple module-2 operations. It shares many of the important characteristics of the real wavelet transform. In particular, it yields an output similar to the thresholded output of a real wavelet transform operating on the underlying binary image. We begin by introducing a new binary field transform to use as an alternative to the discrete Fourier transform over GF(2). The corresponding concept of sequence spectra over GF(2) is defined. Using this transform, a theory of binary wavelets is developed in terms of two-band perfect reconstruction filter banks in GF(2). By generalizing the corresponding real field constraints of bandwidth, vanishing moments, and spectral content in the filters, we construct a perfect reconstruction wavelet decomposition. We also demonstrate the potential use of the binary wavelet decomposition in lossless image coding.

New framework for modulated perfect reconstruction filter banks

A new formulation for the analysis and design of modulated filter banks is introduced. The formulation provides a broad range of design flexibility within a compact framework and allows for the design of a variety of computationally efficient modulated filter banks with different numbers of bands and virtually arbitrary lengths. A unique feature of the formulation is that it provides explicit control of the input-to-output system delay in conjunction with perfect reconstruction. Design examples are given to illustrate the methodology.

Some Results in the Theory Modulated Filter Banks and Modulated Wavelet Tight Frames

Perfect reconstruction (PR) FIR filter banks, obtained by modulation of a linear-phase, lowpass, prototype filter and of length 2Mm are well known. Recently, PR modulated filter banks (MFBs) with the analysis and synthesis banks obtained from different prototypes have been reported. This paper describes a general form of modulation that includes modulations used in the literature. This modulation depends on an integer parameter, the modulation phase. The PR property is characterized for MFBs with finite and infinite impulse response filters. The MFB PR problem reduces to roughly M/2 two-channel PR problems. A natural dichotomy in the PR conditions leads us to the concepts of Type 1 and Type 2 MFBs. Unitary MFBs are characterized by the M/2 two-channel PR filter banks also being unitary (for FIR filters of length N = 2Mm, these results are given in (Malvar, Electr Lett. 26, June 1990, 906–907; Koilpillai and Vaidyanathan, IEEE Trans. SP40, No. 4, Apr. 1992, 770–783)). We also give a necessary and sufficient condition for a large class (including FIR) unitary MFB prototypes to have symmetric (even or odd) prototype filters, and exhibit unitary MFBs without symmetric prototypes. A parameterization of all FIR unitary MFBs is also given. An efficient design procedure for FIR unitary MFBs is developed. It turns out that MFBs can be implemented efficiently using Type III and Type IV DCTs. Compactly supported modulated wavelet tight frames are shown to exist and completely parameterized. K-regular modulated WTFs are designed numerically and analytically by solving a set of non-linear equations over the parameters. Design of optimal modulated WTFs for the representation of any given signal is described with examples, and this is used to design smooth modulated WTFs.

Exact interpolation and iterative subdivision schemes

We examine the circumstances under which a discrete-time signal can be exactly interpolated given only every Mth sample. After pointing out the connection between designing an M-fold interpolator and the construction of an M-channel perfect reconstruction filter bank, we derive necessary and sufficient conditions on the signal under which exact interpolation is possible. Bandlimited signals are one obvious example, but numerous others exist. We examine these and show how the interpolators may be constructed. A main application is to iterative interpolation schemes, used for the efficient generation of smooth curves. We show that conventional bandlimited interpolators are not suitable in this context. A better criterion is to use interpolators that are exact for polynomial functions. We demonstrate that these interpolators converge when iterated, and show how these may be designed for any polynomial degree N and any interpolation factor M. This makes it possible to design interpolators for iterative schemes to make best use of the resolution available in a given display medium.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Synthesis of extremal wavelet-generating filters using Gaussian quadrature

Orthogonal wavelets can be generated from finite impulse response quadrature mirror filters; these filters are also used in perfect reconstruction filter banks. This paper addresses the problem of efficiently synthesizing such filters. A class of extremal filters is defined by the property that their magnitude spectrum maximizes an integral criterion. It is found that these filters are characterized by their zeros on the unit circle, which frequently can be obtained from a set of orthogonal polynomials. A family of filters is constructed that minimize the subband aliasing energy and can generate wavelets with an arbitrary number of vanishing moments. The algorithm for generating these filters makes use of the Levinson recursions, Gaussian quadrature and a fast version of Euclid's algorithm. Similar to other methods for constructing quadrature mirror filters, the spectral factorization of a polynomial is the computationally expensive part of this algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Role of anticausal inverses in multirate filter-banks .II. The FIR case, factorizations, and biorthogonal lapped transforms

For pt. I see ibid., vol.43, no.5, p.1090, 1990. In part I we studied the system-theoretic properties of discrete time transfer matrices in the context of inversion, and classified them according to the types of inverses they had. In particular, we outlined the role of causal FIR matrices with anticausal FIR inverses (abbreviated cafacafi) in the characterization of FIR perfect reconstruction (PR) filter banks. Essentially all FIR PR filter banks can be characterized by causal FIR polyphase matrices having anticausal FIR inverses. In this paper, we introduce the most general degree-one cafacafi building block, and consider the problem of factorizing cafacafi systems into these building blocks. Factorizability conditions are developed. A special class of cafacafi systems called the biorthogonal lapped transform (BOLT) is developed, and shown to be factorizable. This is a generalization of the well-known lapped orthogonal transform (LOT). Examples of unfactorizable cafacafi systems are also demonstrated. Finally it is shown that any causal FIR matrix with FIR inverse can be written as a product of a factorizable cafacafi system and a unimodular matrix.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Perfect reconstruction with critically sampled filter banks and linear boundary conditions

This work is concerned with the (image) boundary conditions involved in processing a finite discrete-time signal with a critically sampled perfect reconstruction filter bank. It is desirable that the boundary conditions reduce edge effects and define a transformation into a space having the same dimensionality as the original signal. The complication that arises is in the computation of the inverse transform. Although it is straightforward to reconstruct the signal values that were not influenced by the boundary conditions, recovering those values on the boundaries is nontrivial. The solution of this problem is discussed for general linear boundary conditions. No symmetry assumptions are made on the boundary conditions or on the impulse responses of the analysis filters. A low-rank linear transform is derived that expresses the boundary values in terms of the transform coefficients, which in turn provides a method for inverting the subband decomposition. The application of the results in the case of two-channel orthonormal wavelet filters is discussed, and the effects of the filter support on the conditioning of the inverse problem are investigated.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A comparison of design methods for 2-D FIR orthogonal perfect reconstruction filter banks

This paper examines the design of nonseparable 2-D FIR perfect reconstruction filter banks. We restrict our attention to orthogonal filter banks, i.e., analysis and synthesis filter banks with paraunitary polyphase matrices. The design problem reduces to the selection of a 2-D FIR paraunitary polyphase matrix, such that the filtering characteristics of the analysis or synthesis filters approximate as closely as possible the ideal bandpass characteristics of their desired frequency bands. Three design methods are compared. The first method, which was developed by Viscito and Allebach (1991), designs the impulse response of the polyphase matrix in the space-domain. A second technique was proposed in a paper by Karlsson Vetterli (1990) and relies on the subclass of 2-D paraunitary transfer matrices which can be represented as a product of paraunitary matrices of McMillan degree one in each variable. Finally, we develop a third approach based on a parametrization of 2-D FIR paraunitary matrices in terms of orthonormal minimal 2-D state-space realizations. The design examples that we present indicate that the third approach may have significant advantages, at least for the design of orthogonal filter banks with small spatial support.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Two-dimensional filter bank design for optimal reconstruction using limited subband information

In this correspondence, we propose design techniques for analysis and synthesis filters of 2-D perfect reconstruction filter banks (PRFB's) that perform optimal reconstruction when a reduced number of subband signals is used. Based on the minimization of the squared error between the original signal and some low-resolution representation of it, the 2-D filters are optimally adjusted to the statistics of the input images so that most of the signal's energy is concentrated in the first few subband components. This property makes the optimal PRFB's efficient for image compression and pattern representations at lower resolutions for classification purposes. By extending recently introduced ideas from frequency domain principal component analysis to two dimensions, we present results for general 2-D discrete nonstationary and stationary second-order processes, showing that the optimal filters are nonseparable. Particular attention is paid to separable random fields, proving that only the first and last filters of the optimal PRFB are separable in this case. Simulation results that illustrate the theoretical achievements are presented.

Time-varying analysis-synthesis systems based on filter banks and post filtering

Perfect reconstruction (PR) time-varying analysis-synthesis filter banks are those in which the filters are allowed to change from one set of PR filter banks to another as the input signal is being processed. Such systems have the property that, in the absence of coding, they faithfully reconstruct every sample of the input. Various methods have been reported for the time-varying filter bank design; all of them, however, utilize structures for conventional PR filter banks. These conventional structures that have been applied in the past result in different limitations in each method. This paper introduces a new structure for exactly reconstructing time-varying analysis-synthesis filter banks. This structure consists of the conventional filter bank followed by a time-varying post filter. The new method requires neither the redesign of the analysis sections nor the use of any intermediate analysis filters during transition periods. It provides a simple and elegant procedure for designing time-varying filter banks without the disadvantages of the previous methods.

Design of two-channel perfect-reconstruction linear-phase filter banks for subband image coding by the Lagrange multiplier approach

In this paper, a new approach is proposed for the design of two-dimensional two-channel perfect-reconstruction FIR filter banks employing diamond-shaped linear-phase filters. When one of the analysis filter pair is properly designed, the condition of perfect-reconstruction is formulated into a spatial-domain constraint equation for designing the other. The method is based on minimizing the integrated square error for the frequency response over the passband and stopband while imposing the spatial-domain constraint. One design example is given to demonstrate the perfect-reconstruction for the proposed system. >

Principal component filter banks for optimal multiresolution analysis

An important issue in multiresolution analysis is that of optimal basis selection. An optimal P-band perfect reconstruction filter bank (PRFB) is derived in this paper, which minimizes the approximation error (in the mean-square sense) between the original signal and its low-resolution version. The resulting PRFB decomposes the input signal into uncorrelated, low-resolution principal components with decreasing variance. Optimality issues are further analyzed in the special case of stationary and cyclostationary processes. By exploiting the connection between discrete-time filter banks and continuous wavelets, an optimal multiresolution decomposition of L/sub 2/(R) is obtained. Analogous results are also derived for deterministic signals. Some illustrative examples and simulations are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Symmetric extension methods for M-channel linear-phase perfect-reconstruction filter banks

The symmetric extension method has been shown to he an efficient way for subband processing of finite-length sequences. This paper presents an extension of this method to general linear-phase perfect-reconstruction filter banks. We derive constraints on the length and symmetry polarity of the permissible filter banks and propose a new design algorithm. In the algorithm, different symmetric sequences are formulated in a unified form based on the circular-symmetry framework. The length constraints in symmetrically extending the input sequence and windowing the subband sequences are investigated. The effect of shifting the input sequence is included. When the algorithm is applied to equal-length filter banks, we explicitly show that symmetric extension methods can always be constructed to replace the circular convolution approach.