Reduction of transients during lifting based spatial switching of two-channel filter banks

Time/space varying filter banks (FBs) are useful for non-stationary images. Lifting factorization of FBs results in structural perfect reconstruction even during the transition from one FB to other. This allows spatial switching between arbitrary FBs, avoiding the need to design border FBs. However, we show that lifting based switching between arbitrarily designed FBs induces spurious transients in the subbands during the transition. In this paper, we study the transients in lifting based switching of two-channel FBs. We propose two solutions to overcome the transients. One solution consists of a boundary handling mechanism to switch between any arbitrarily designed FBs, while the other solution proposes to design the FBs with a set of conditions applied on lifting steps. Both solutions maintain good frequency response during the transition and eliminate the transients. Using the proposed methods, we develop a spatial adaptive transform by switching between the long length FBs (either the JPEG2000 9/7 FB or the newly designed 13/11 FB) and the short length FBs (JPEG2000 5/3 FB) for lossy image compression. This adaptive transform shows PSNR improvement for images over JPEG2000 9/7 FB in low bit rate region (up to 0.2 bpp) and subjective improvements with reduced ringing up to medium bit rates (up to 0.6 bpp).     

Dyadic-based factorizations for regular paraunitary filterbanks and M-band orthogonal wavelets with structural vanishing moments

Paraunitary filterbanks (PUFBs) can be designed and implemented using either degree-one or order-one dyadic-based factorization. This work discusses how regularity of a desired degree is structurally imposed on such factorizations for any number of channels M /spl ges/ 2, without necessarily constraining the phase responses. The regular linear-phase PUFBs become a special case under the proposed framework. We show that the regularity conditions are conveniently expressed in terms of recently reported M-channel lifting structures, which allow for fast, reversible, and possibly multiplierless implementations in addition to improved design efficiency, as suggested by numerical experience. M-band orthonormal wavelets with structural vanishing moments are obtained by iterating the resulting regular PUFBs on the lowpass channel. Design examples are presented and evaluated using a transform-based image coder, and they are found to outperform previously reported designs.     

Design and Factorization of Two-Channel Perfect Reconstruction Filter Banks with Causal-Stable IIR Filters

In this paper, new design and factorization methods of two-channel perfect reconstruction (PR) filter banks (FBs) with casual-stable IIR filters are introduced. The polyphase components of the analysis filters are assumed to have an identical denominator in order to simplify the PR condition. A modified model reduction is employed to derive a nearly PR causal-stable IIR FB as the initial guess to obtain a PR IIR FB from a PR FIR FB. To obtain high quality PR FIR FBs for carrying out model reduction, cosine-rolloff FIR filters are used as the initial guess to a nonlinear optimization software for solving to the PR solution. A factorization based on the lifting scheme is proposed to convert the IIR FB so obtained to a structurally PR system. The arithmetic complexity of this FB, after factorization, can be reduced asymptotically by a factor of two. Multiplier-less IIR FB can be obtained by replacing the lifting coefficients with the canonical signal digitals (CSD) or sum of powers of two (SOPOT) coefficients.     

Lapped unimodular transform and its factorization

Two types of lapped transforms have been studied in detail in the literature, namely, the lapped orthogonal transform (LOT) and its extension, the biorthogonal lapped transform (BOLT). In this paper, we will study the lapped unimodular transform (LUT). All three transforms are first-order matrices with finite impulse response (FIR) inverses. We will show that like LOT and BOLT, all LUTs can be factorized into degree-one unimodular matrices. The factorization is both minimal and complete. We will also show that all first-order systems with FIR inverses can be minimally factorized as a cascade of degree-one LOT, BOLT, and LUT building blocks. Two examples will be given to demonstrate that despite having a very small system delay, the LUTs have a satisfactory performance in comparison with LOT and BOLT.     

CORDIC-lifting factorization of paraunitary filter banks based on the quaternionic multipliers for lossless image coding

Quaternions have offered a new paradigm to the signal processing community: to operate directly in a multidimensional domain. We have recently introduced the quaternionic approach to the design and implementation of paraunitary filter banks: four- and eight-channel linear-phase paraunitary filter banks, including those with pairwise-mirror-image symmetric frequency responses. The hypercomplex number theory is utilized to derive novel lattice structures in which quaternion multipliers replace Givens (planar) rotations. Unlike the conventional algorithms, the proposed computational schemes maintain losslessness regardless of their coefficient quantization. Moreover, the one regularity conditions can be expressed directly in terms of the quaternion lattice coefficients and thus easily satisfied even in finite-precision arithmetic. In this paper, a novel approach to realizing CORDIC-lifting factorization of paraunitary filter banks is presented, which is based on the embedding of the CORDIC algorithm inside the lifting scheme. Lifting allows for making multiplications invertible. The 2D CORDIC engine using sparse iterations and asynchronous pipeline processor architecture based on the embedded CORDIC engine as stage of processor is reported. Also it is necessary to notice, that the quaternion multiplier lifting scheme based on the 2D CORDIC algorithm is the structural decision for the lossless digital signal processing. This approach applies to very practical filter banks, which are essential for image processing, and addresses interesting theoretical questions.     

离散子波提升算法研究及其性能分析

论文基于矩阵变换和变换矩阵级联分解的思想,提出一种新的多相矩阵表示形式,对离散子波提升算法的机理进行了完整的理论分析,对子波提升算法和子波变换双通道滤波实现的理想重构条件进行了等价性证明,并利用互补滤波器组的对偶性提出一利新的子波提升分解算法的级联矩阵分解形式,使提升算法的机理解释更加完善,然后基于文中提出的矩阵级联分解形式,以(2,2)双正交子波变换为例说明了离散子波提升分解算法的实现,并就算法的可逆性、运算量和原位实现等问题进行了简要讨论。     

Orthonormal FBs With Rational Sampling Factors and Oversampled DFT-Modulated FBs: A Connection and Filter Design

Methods widely used to design filters for uniformly sampled filter banks (FBs) are not applicable for FBs with rational sampling factors and oversampled discrete Fourier transform (DFT)-modulated FBs. In this paper, we show that the filter design problem (with regularity factors/vanishing moments) for these two types of FBs is the same. Following this, we propose two finite-impulse-response (FIR) filter design methods for these FBs. The first method describes a parameterization of FBs with a single regularity factor/vanishing moment. The second method, which can be used to design FBs with an arbitrary number of regularity factors/vanishing moments, uses results from frame theory. We also describe how to modify this method so as to obtain linear phase filters. Finally, we discuss and provide a motivation for iterated DFT-modulated FBs.     

A Lifting Scheme of Symmetric-antisymmetric Multiwavelet Transform for Image Coding

A symmetric-antisymmetric (SA) multiwavelet lifting factorization of matrix filter banks is presented as an extension of Sweldens’ traditional scalar wavelet lifting scheme. A prefilter absorbed (PA) multiwavelet lifting factorization is also presented to reduce the redundant computations in prefiltering. Then an algorithm of multiwavelet lifting is designed and used for signal decomposition. The experimental results show this lifting scheme can achieve lower complexity while preserve high quality for image coding.     

A class of regular biorthogonal linear-phase filterbanks: theory, structure, and application in image coding

This paper discusses a method of regularity imposition onto biorthogonal linear-phase M-band filterbanks using the lattice structure. A lifting structure is proposed for lattice matrix parameterization where regularity constraints can be imposed. The paper focuses on cases with analysis and synthesis filterbanks having up to two degrees of regularity. Necessary and sufficient conditions for regular filterbanks in terms of the filter impulse response, frequency response, scaling function, and wavelets are revisited and are derived in terms of the lattice matrices. This also leads to a constraint on the minimum filter length. Presented design examples are optimized for the purpose of image coding, i.e., the main objectives are coding gain and frequency selectivity. Simulation results from an image coding application also show that these transforms yield improvement in the perceptual quality in the reconstruction images. The approach has also been extended to the case of integer/rational lifting coefficients, which are desirable in many practical applications.     

Design of Time–Frequency Localized Filter Banks: Transforming Non-convex Problem into Convex Via Semidefinite Relaxation Technique

We present a method for designing optimal biorthogonal wavelet filter banks (FBs). Joint time–frequency localization of the filters has been chosen as the optimality criterion. The design of filter banks has been cast as a constrained optimization problem. We design the filter either with the objective of minimizing its frequency spread (variance) subject to the constraint of prescribed time spread or with the objective of minimizing the time spread subject to the fixed frequency spread. The optimization problems considered are inherently non-convex quadratic constrained optimization problems. The non-convex optimization problems have been transformed into convex semidefinite programs (SDPs) employing the semidefinite relaxation technique. The regularity constraints have also been incorporated along with perfect reconstruction constraints in the optimization problem. In certain cases, the relaxed SDPs are found to be tight. The zero duality gap leads to the global optimal solutions. The design examples demonstrate that reasonably smooth wavelets can be designed from the proposed filter banks. The optimal filter banks have been compared with popular filter banks such as Cohen–Daubechies–Feauveau biorthogonal wavelet FBs, time–frequency optimized half-band pair FBs and maximally flat half-band pair FBs. The performance of optimal filter banks has been found better in terms of joint time–frequency localization.     

On simplified order-one factorizations of paraunitary filterbanks

This paper revisits the order-one factorization of causal finite impulse response (FIR) paraunitary filterbanks (PU FBs). The basic form of the factorization was proposed by Vaidyanathan et al. in 1987, which is a cascade of general unitary matrices separated by diagonal delay matrices with arbitrary number of delay elements. Recently, Gao et al. have proved the completeness of this factorization and developed a more efficient structure that only uses approximately half number of free parameters. In this paper, by briefly analyzing Gao et al.'s derivation, we first point out that Gao et al.'s factorization contains redundant free parameters. Two simplified structures of Vaidyanathan's factorization are then developed, i.e., a post-filtering-based structure and a prefiltering-based structure. Our simplification relies on consecutive removal of extra degrees of freedom in adjacent stages, which is accomplished through the C-S decomposition of a general unitary matrix. Since the conventional C-S decomposition leads to a redundant representation, a new C-S decomposition is developed to minimize the number of free parameters by further incorporating the Givens rotation factorization. The proposed structures can maintain the completeness and the minimality of the original lattice. Compared with Gao et al.'s factorization, our derivations are much simpler, while the resulting structures contain fewer free parameters and less implementation cost. Besides, these new factorizations indicate that for a PU FB with a given filter length, the symmetric-delay factorization offers the largest degrees of design freedom. Several design examples are presented to confirm the validity of the theory.     

Generalized block-lifting factorization of M-channel biorthogonal filter banks for lossy-to-lossless image coding

Generalized block-lifting factorization of M-channel (M > 2) biorthogonal filter banks (BOFBs) for lossy-to-lossless image coding is presented in this paper. Since the proposed block-lifting structure is more general than the conventional lifting factorizations and does NOT require many restrictions such as paraunitary, number of channels, and McMillan degree in each building block unlike the conventional lifting factorizations, its coding gain is higher than that of the previous methods. Several proposed BOFBs are designed and applied to image coding. Comparing the results with conventional lossy-to-lossless image coding structures, including the 5/3- and 9/7-tap discrete wavelet transforms in JPEG 2000 and a 4 × 8 hierarchical lapped biorthogonal transform in JPEG XR, the proposed BOFBs achieve better result in both objective measure and perceptual visual quality for the images with a lot of high-frequency components.     

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.     

The generalized lapped pseudo-biorthogonal transform: oversampled linear-phase perfect reconstruction filterbanks with lattice structures

We investigate a lattice structure for a special class of N-channel oversampled linear-phase perfect reconstruction filterbanks with a decimation factor M smaller than N. We deal with systems in which all analysis and synthesis filters have the same finite impulse response (FIR) length and share the same center of symmetry. We provide the minimal lattice factorization of a polyphase matrix of a particular class of these oversampled filterbanks (FBs). All filter coefficients are parameterized by rotation angles and positive values. The resulting lattice structure is able to provide fast implementation and allows us to determine the filter coefficients by solving an unconstrained optimization problem. We consider next the case where we are given the generalized lapped pseudo-biorthogonal transform (GLPBT) lattice structure with specific parameters, and we a priori know the correlation matrix of noise that is added in the transform domain. In this case, we provide an alternative lattice structure that suppress the noise. We show that the proposed systems with the lattice structure cover a wide range of linear-phase perfect reconstruction FBs. We also introduce a new cost function for oversampled FB design that can be obtained by generalizing the conventional coding gain. Finally, we exhibit several design examples and their properties.     

Structures, factorizations, and design criteria for oversampledparaunitary filterbanks yielding linear-phase filters

We present here a special class of oversampled filterbanks (FBs), namely, paraunitary FBs with linear-phase filters. We propose some necessary conditions for the existence of such banks, based on the repartition between type I/II and type II/IV linear-phase filters in the bank. For a subset of these FBs, we develop a factorization that leads to a minimal implementation, as well as a direct parameterization of the FBs in terms of elementary rotation angles. This factorization is applied to some design examples, with two different optimization criteria: coding gain and reconstructibility of lost coefficients     

Frame-theoretic analysis of oversampled filter banks

We provide a frame-theoretic analysis of oversampled finite impulse response (FIR) and infinite impulse response (FIR) uniform filter banks (FBs). Our analysis is based on a new relationship between the FBs polyphase matrices and the frame operator corresponding to an FB. For a given oversampled analysis FB, we present a parameterization of all synthesis FBs providing perfect reconstruction. We find necessary and sufficient conditions for an oversampled FB to provide a frame expansion. A new frame-theoretic procedure for the design of paraunitary FBs from given nonparaunitary FBs is formulated. We show that the frame bounds of an FB can be obtained by an eigen-analysis of the polyphase matrices. The relevance of the frame bounds as a characterization of important numerical properties of an FB is assessed by means of a stochastic sensitivity analysis. We consider special cases in which the calculation of the frame bounds and synthesis filters is simplified. Finally, simulation results are presented     

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.     

Lattice structure for regular paraunitary linear-phase filterbanksand M-band orthogonal symmetric wavelets

Parauninary linear-phase (PULP) M-channel uniform filterbanks, which are also known as the generalized lapped orthogonal transforms (GenLOTs), can be designed and implemented using lattice structures. This paper discusses how to impose regularity constraints onto the lattice structure of PULP filterbanks. These conditions are expressed in term of the rotation angles of the lattice components by which the resulting filterbanks are guaranteed to have one or two degrees of regularity, iterating these new regular filterbanks on the lowpass subband generates a large family of symmetric M-band orthonormal wavelets. Design procedures with many design examples are presented. Smooth interpolation using regular PULP filterbanks is illustrated through image coding experiments where the novel M-band wavelets consistently yield smoother reconstructed images and better perceptual quality     

RNS-FPL merged architectures for orthogonal DWT

Novel, regular, compact and easily scalable residue number system (RNS) field-programmable logic (FPL) merged architectures for the orthogonal 1D discrete wavelet transform (DWT) and 1D inverse discrete wavelet transform (1DWT) are presented. These structures halve the number of look-up tables (LUTs) required per octave, providing a sustained throughput independent of the input data and filter coefficient precision. They are suitable to be considered as the core of 2D DWT processors for high data rate image processing applications     

Two Classes of Cosine-Modulated IIR/IIR and IIR/FIR NPR Filter Banks

This paper introduces two classes of cosine-modulated causal and stable filter banks (FBs) with near perfect reconstruction (NPR) and low implementation complexity. Both classes have the same infinite-length impulse response (IIR) analysis FB but different synthesis FBs utilizing IIR and finite-length impulse response (FIR) filters, respectively. The two classes are preferable for different types of specifications. The IIR/FIR FBs are preferred if small phase errors relative to the magnitude error are desired, and vice versa. The paper provides systematic design procedures so that PR can be approximated as closely as desired. It is demonstrated through several examples that the proposed FB classes, depending on the specification, can have a lower implementation complexity compared to existing FIR and IIR cosine-modulated FBs (CMFBs). The price to pay for the reduced complexity is generally an increased delay. Furthermore, two additional attractive features of the proposed FBs are that they are asymmetric in the sense that one of the analysis and synthesis banks has a lower computational complexity compared to the other, which can be beneficial in some applications, and that the number of distinct coefficients is small, which facilitates the design of FBs with large numbers of channels.