Perfect Reconstruction Filter Banks Research Articles

Theory and factorization for a class of structurally regular biorthogonal filter banks

Regularity is a fundamental and desirable property of wavelets and perfect reconstruction filter banks (PRFBs). Among others, it dictates the smoothness of the wavelet basis and the rate of decay of the wavelet coefficients. This paper considers how regularity of a desired degree can be structurally imposed onto biorthogonal filter banks (BOFBs) so that they can be designed with exact regularity and fast convergence via unconstrained optimization. The considered design space is a useful class of M-channel causal finite-impulse response (FIR) BOFBs (having anticausal FIR inverses) that are characterized by the dyadic-based structure W(z)=I-UV/sup /spl dagger//+z/sup -1/UV/sup /spl dagger// for which U and V are M/spl times//spl gamma/ parameter matrices satisfying V/sup /spl dagger//U=I/sub /spl gamma//, 1/spl les//spl gamma//spl les/M, for any M/spl ges/2. Structural conditions for regularity are derived, where the Householder transform is found convenient. As a special case, a class of regular linear-phase BOFBs is considered by further imposing linear phase (LP) on the dyadic-based structure. In this way, an alternative and simplified parameterization of the biorthogonal linear-phase filter banks (GLBTs) is obtained, and the general theory of structural regularity is shown to simplify significantly. Regular BOFBs are designed according to the proposed theory and are evaluated using a transform-based image codec. They are found to provide better objective performance and improved perceptual quality of the decompressed images. Specifically, the blocking artifacts are reduced, and texture details are better preserved. For fingerprint images, the proposed biorthogonal transform codec outperforms the FBI scheme by 1-1.6 dB in PSNR.

Variable-length lapped transforms with a combination of multiple synthesis filter banks for image coding

A class of lapped transforms for image coding, which are characterized by variable-length synthesis filters, is introduced. In this class, the synthesis filter bank (FB) is first defined with an arbitrary combination of finite impulse response synthesis filters of perfect reconstruction FBs. An analysis FB is then obtained using direct matrix inversion or iterative implementation of Neumann series expansion. Moreover, to improve compression, we introduce a unitary transform that follows the analysis FB. This class enables a greater freedom of design than previously presented variable-length lapped transforms. We illustrate several design examples and present experimental results for image coding, which indicate that the proposed transforms are promising and comparable with conventional subband transforms including wavelets.

An improved and simplified design of cosine-modulated pseudo-QMF filterbanks

In this paper, a simple and efficient design of prototype filters for cosine-modulated filterbanks is proposed. Variable combinational window function with high side-lobe-fall-off-rate (SLFOR) has been used to design the prototype filter for cosine-modulated filterbanks. Cutoff frequency of the filter is optimized to minimize the reconstruction error, which is also selected as an objective function. Very small values of reconstruction and aliasing errors have been obtained with high SLFOR combinational window filters, resulting in near perfect reconstruction (NPR) filterbanks. However, higher filter order is required to get NPR solution.

On the design of high-complexity cosine-modulated transmultiplexers based on the frequency-response masking approach

Two efficient techniques exploiting the frequency-response masking (FRM) approach are proposed in order to make it feasible to design prototype filters for highly selective nearly perfect-reconstruction cosine-modulated transmultiplexers and filter banks (CMTs and CMFBs) having a very large number of channels. In these design schemes, the number of unknowns is drastically reduced when compared with the corresponding techniques for designing direct-form prototype filters. Furthermore, in the proposed techniques, the main figures of merits, that is, the intersymbol interference and the interchannel interference for CMTs and the overall and aliasing distortions for CMFBs are taken into account in a controlled manner. In order to speed up the convergence of these two optimization techniques, simplifications for computing the resulting nonlinear constraints and the corresponding gradient vectors are proposed. They differ from each other in the sense that the first and second ones utilize the frequency-domain and time-domain constraints for controlling the figures of merit, respectively. Combining these two techniques results in numerically efficient algorithms for designing optimized CMTs (or CMFBs) with a reduced computational complexity (number of arithmetic operations per output sample), particularly when both branches of the FRM structure are required. Design examples are included illustrating the efficiency of the design methods and the high performance of the resulting CMT structures.

Tight and sibling frames originated from discrete splines

We present a new family of frames, which are generated by perfect reconstruction filter banks. The filter banks are based on the discrete interpolatory splines and are related to Butterworth filters. Each filter bank comprises one interpolatory symmetric low-pass filter, one band-pass and one high-pass filters. In the sibling frames case, all the filters are linear phase and generate symmetric scaling functions with analysis and synthesis pairs of framelets. In the tight frame case, all the analysis waveforms coincide with their synthesis counterparts. In the sibling frame, we can vary the framelets making them different for synthesis and analysis cases. This enables us to swap vanishing moments between the synthesis and the analysis framelets or to add smoothness to the synthesis framelets. We construct dual pairs of frames, where all the waveforms are symmetric and the framelets may have any number of vanishing moments. Although most of the designed filters are IIR, they allow fast implementation via recursive procedures. The waveforms are well localized in time domain despite their infinite support.

Design of perfect-reconstruction nonuniform recombination filter banks with flexible rational sampling factors

This paper studies the design of a class of perfect-reconstruction (PR) nonuniform filter banks (FBs) called recombination nonuniform FBs (RNFBs). They are constructed by merging subbands in a uniform FB with sets of transmultiplexers (TMUXs). It generalizes the RNFBs previously proposed by the authors to allow more general choice of the sampling factors. The spectral inversion and spurious response suppression problems of these new RNFBs using cosine modulation are analyzed, and a simple design method based on a matching condition is proposed. It is also found that the FB and the TMUX in the recombination structure can be designed separately to satisfy the matching condition. In addition, real-time adaptive merging of the channels to provide dynamic nonuniform frequency partitioning is feasible. Another advantage of the RNFBs is that the recombination and processing of the subband signal can be done at the decimated domain of the uniform FB, which greatly reduces its implementation complexity. Design examples show that high quality nonuniform PR FBs with low implementation complexity and variable time-frequency resolution can be obtained by the proposed method.

Performance analysis and prototype filter design for perfect-reconstruction cosine-modulated filter banks with fixed-point implementation

Cosine modulated filter banks have gained popularity for their ability to provide perfect reconstruction (PR) while maintaining an efficient design and implementation. However, this effectiveness is hindered if the filter bank is implemented in the fixed-point domain where quantization, rounding, and overflow occur, and result in reconstruction errors. In this article we demonstrate how to maintain PR of the filter bank when implementing it in fixed-point number format with constant wordlength. We explore how the frequency selectivity of the analysis and synthesis filters changes from the floating point ones due to fixed-point errors and present new design criteria for filter banks that will be implemented in fixed-point number format.

A new approach for estimation of statistically matched wavelet

This paper presents a new approach for the estimation of wavelets that is matched to a given signal in the statistical sense. Based on this approach, a number of new methods to estimate statistically matched wavelets are proposed. The paper first proposes a new method for the estimation of statistically matched two-band compactly supported biorthogonal wavelet system. Second, a new method is proposed to estimate statistically matched semi-orthogonal two-band wavelet system that results in compactly supported or infinitely supported wavelet. Next, the proposed method of estimating two-band wavelet system is generalized to M-band wavelet system. Here, the key idea lies in the estimation of analysis wavelet filters from a given signal. This is similar to a sharpening filter used in image enhancement. The output of analysis highpass filter branch is viewed to be equivalent to an error in estimating the middle sample from the neighborhood. To minimize this error, a minimum mean square error (MMSE) criterion is employed. Since wavelet expansion acts like Karhunen-Loe/spl grave/ve-type expansion for generalized 1/f/sup /spl beta// processes, it is assumed that the given signal is a sample function of an mth-order fractional Brownian motion. Therefore, the autocorrelation structure of a generalized 1/f/sup /spl beta// process is used in the estimation of analysis filters using the MMSE criterion. We then present methods to design a finite impulse response/infinite impulse response (FIR/IIR) biorthogonal perfect reconstruction filterbank, leading to the estimation of a compactly supported/infinitely supported statistically matched wavelet. The proposed methods are very simple. Simulation results to validate the proposed theory are presented for different synthetic self-similar signals as well as music and speech clips. Estimated wavelets for different signals are compared with standard biorthogonal 9/7 and 5/3 wavelets for the application of compression and are shown to have better results.

A two-channel nonuniform perfect reconstruction filter Bank with irrational down-sampling factors

Most of the existing methods designed to implement fractional interpolation and decimation are limited by rational scaling factors such as L/M, where L and M are positive integers. The general procedure is usually done with up-sampling by L first, and then followed by down-sampling by M. This scheme, however, requires two steps and higher sampling rates to fulfill and is not capable in dealing with irrational scaling factors, which cannot be represented by L/M. We propose a new method to solve the above two difficulties by a single step. Furthermore, a new structure of a two-channel nonuniform perfect reconstruction filter bank is derived using this new method, and the experimental results are presented.

On minimal lattice factorizations of symmetric-antisymmetric multifilterbanks

This paper introduces two minimal lattice structures for symmetric-antisymmetric multiwavelets (SAMWTs) and symmetric-antisymmetric multifilterbanks (SAMFBs). First, by exploring the relation of the symmetric-antisymmetric property in multifilterbanks and the linear-phase property in traditional scalar filterbanks, we show that the implementation and design of an SAMFB can be converted into that of a four-channel scalar linear-phase perfect reconstruction filterbank (LPPRFB). Then, based on the lattice factorization for LPPRFBs, we propose two fast, modular, minimal structures for SAMFBs. To demonstrate the effectiveness of the proposed lattice structures, several rational or dyadic-coefficient SAMWT design examples are presented along with their application in image coding.

Development of new filter bank for detection of nodular patterns and linear patterns in medical images

Abnormal shadows in medical images are described as isolated nodular structures or linear structures. However, the size and shape of the shadows usually vary from case to case. Consequently, in order to detect abnormal shadows with high accuracy, we need to use a filter that can deal with various sizes and shapes, rather than a fixed filter assuming a typical shadow shape. The purpose of this study is to detect abnormal shadows with high accuracy, and a new perfect reconstruction filter bank is developed, introducing multiresolution based on wavelet analysis. The filter bank is designed so that the subimages generated the elements of a Hessian matrix at each resolution level. By calculating the small and large eigenvalues, a new filter bank has the following three properties. (1) Nodular patterns of various sizes can be enhanced. (2) Both nodular and linear patterns of various sizes can be enhanced. (3) The original image can be reconstructed with these patterns removed. The filter bank is applied to enhance microcalcifications in mammograms and to enhance/remove bone tissue in chest radiographs. The effectiveness of the method is demonstrated. © 2005 Wiley Periodicals, Inc. Syst Comp Jpn, 36(13): 81–91, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.20171

A new method of estimating wavelet with desired features from a given signal

This paper proposes a new method of estimating both biorthogonal compactly supported as well as semi-orthogonal infinitely/compactly supported wavelet from a given signal. The method is based on maximizing projection of the given signal onto successive scaling subspace. This results in minimization of energy of signal in the wavelet subspace. The idea used to estimate analysis wavelet filter is similar to a sharpening filter used in image enhancement. First, a new method is proposed that helps in the design of 2-band FIR biorthogonal perfect reconstruction filter bank from a given signal. This leads to the design of biorthogonal compactly supported wavelet. It is also shown that a wavelet with desired support as well as desired number of vanishing moments can be designed with the proposed method. Next, a method is proposed to design semi-orthogonal wavelets that are usually infinitely supported wavelets. Here, corresponding to FIR analysis filters, the resulting synthesis filters are IIR filters that satisfy the property of perfect reconstruction.

Efficient Wideband Channelizer for Software Radio Systems Using Modulated PR Filterbanks

An efficient method is proposed for channelizing frequency division multiplexed (FDM) channels in wideband software radio (SWR) received signals that do not satisfy the conditions required for polyphase decomposition of the discrete filterbank (DFB) channelizer. The proposed method, which uses modulated perfect reconstruction (PR) filterbanks, requires fewer computations than DFBs for channelizing wideband signals that are composed of FDM channels of nonequal bandwidths, especially when a large number of channels are extracted. The proposed channelizer, if applied in the reverse direction, can be used to synthesize a set of channels with nonequal bandwidths into a single wideband signal in SWR transmitters. A method is also proposed for efficiently designing the modulated PR filterbanks, which have a large number of subchannels and prototype filters with high stopband attenuations that are used in the proposed channelizer. The computational complexity of the proposed channelizer is compared with the complexity of the DFB channelizer for channelizing the wideband and high-dynamic-range signals that are typical of SWR systems, and simulation results of the proposed channelization method are discussed.

Time-Domain Oversampled Lapped Transforms: Theory, Structure, and Application in Image Coding

This paper generalizes time-domain lapped transforms (TDLTs) proposed by Tran et al. to oversampled systems, thus leading to time-domain oversampled lapped transforms (TDOLTs). These new transforms correspond to a subclass of oversampled linear-phase perfect reconstruction filterbanks (OLPPRFBs), which can be implemented by adding a prefilter before the discrete cosine transform (DCT) and a post-filter after the inverse discrete cosine transform (IDCT). Structures of the pre- and post-filters are developed, and the frame-theoretic properties of TDOLTs are analyzed. A new parameterization of lattice matrices through the Givens-QR factorization is proposed for unconstrained optimization. Comparisons with other parameterization methods are also included. Several design examples, along with some image coding results, are presented to demonstrate the validity of the theory and the potential of TDOLTs in image coding, especially in error-resilient coding.

Subsampling Matrices for Wavelet Decompositions on Body Centered Cubic Lattices

This work derives a family of dilation matrices for the body-centered cubic (BCC) lattice, which is optimal in the sense of spectral sphere packing. While satisfying the necessary conditions for dilation, these matrices are all cube roots of an integer scalar matrix. This property offers theoretical advantages for construction of wavelet functions in addition to the practical advantages when iterating through a perfect reconstruction filter bank based on BCC downsampling. Lastly, we factor the BCC matrix into two matrices that allow us to cascade two two-channel perfect reconstruction filter banks in order to construct a four-channel perfect reconstruction filter bank based on BCC downsampling.

Characterization of Nonuniform Perfect-Reconstruction Filterbanks Using Unit-Step Signal

An algebraic characterization of nonuniform perfect reconstruction (PR) filterbanks with integer decimation factors is presented. The PR property is formulated in the z domain based on the response of the linear multirate systems to the delayed unit-step signals. This leads to a unique class of characterizing formulas that are necessary and sufficient conditions for the PR property and free from the complex roots of unity. Two related characterizations of nonuniform PR systems, in the form of necessary conditions, are also developed based on these formulas. As a concrete example, the results are then used to derive necessary and sufficient conditions for PR nonuniform delay chain filterbanks. The conditions show that nonuniform delay chain filterbanks are the signal processing realizations of the mathematical notion of the exact covering systems of congruence relations. Important results from the mathematics literature on the exact covering systems are introduced. The results elucidate the admissible factors of decimation for the nonuniform PR delay chain systems in settings with maximally distinct decimation factors. A simple test of the PR property for delay chain systems is also presented. The test is based on the divisibility of certain polynomials by the cyclotomic polynomials. Finally, multirate systems based on the Beatty sequences, which are the irrational generalization of the exact covering systems, are briefly discussed.

Filter-Bank-Based Narrowband Interference Detection and Suppression in Spread Spectrum Systems

A filter-bank-based narrowband interference detection and suppression method is developed and its performance is studied in a spread spectrum system. The use of an efficient, complex, critically decimated perfect reconstruction filter bank with a highly selective subband filter prototype, in combination with a newly developed excision algorithm, offers a solution with efficient implementation and performance close to the theoretical limit derived as a function of the filter bank stopband attenuation. Also methods to cope with the transient effects in case of frequency hopping interference are developed and the resulting performance shows only minor degradation in comparison to the stationary case.

New Design and Realization Techniques for a Class of Perfect Reconstruction Two-Channel FIR Filterbanks and Wavelets Bases

This paper proposes two new methods for designing a class of two-channel perfect reconstruction (PR) finite impulse response (FIR) filterbanks (FBs) and wavelets with K-regularity of high order and studies its multiplier-less implementation. It is based on the two-channel structural PR FB proposed by Phoong et al (1995). The basic principle is to represent the K-regularity condition as a set of linear equality constraints in the design variables so that the least square and minimax design problems can be solved, respectively, as a quadratic programming problem with linear equality constraints (QPLC) and a semidefinite programming (SDP) problem. We also demonstrate that it is always possible to realize such FBs with sum-of-powers-of-two (SOPOT) coefficients while preserving the regularity constraints using Bernstein polynomials. However, this implementation usually requires long coefficient wordlength and another direct-form implementation, which can realize multiplier-less wavelets with K-regularity condition up to fifth order, is proposed. Several design examples are given to demonstrate the effectiveness of the proposed methods.

Eigenstructure Approach for Complete Characterization of Linear-Phase FIR Perfect Reconstruction Analysis Length<tex>$2M$</tex>Filterbanks

The eigenstructure based characterization of M-channel finite impulse response perfect reconstruction (FIR PR) filterbanks in a previous paper by the authors is extended here to the linear-phase case. Some results relating to linear-phase filterbanks is derived by finding appropriate restrictions on the eigenstructure of the analysis polyphase matrix. Consequently, a complete and minimal characterization for such filterbanks with all analysis length 2M and any synthesis length is developed. Parameterization and design examples are also presented.

A Novel and Efficient Design of Multidimensional PR Two-Channel Filter Banks With Hourglass-Shaped Passband Support

In this letter, we propose a new design technique for two-dimensional two-channel maximally decimated perfect reconstruction (PR) filter banks with hourglass-shaped spectral support. The proposed approach is different from the conventional design techniques in the sense that it does not require any modulator. Furthermore, the proposed design retains the separable filtering property, while ensuring that the overall system is structurally PR and maximally decimated. Moreover, we extend such structure to higher dimensional case and discuss several interesting properties. A design example is given to demonstrate the effectiveness of the proposed approach.