Linear-phase Paraunitary Filter Banks Research Articles

Embedded distributed arithmetic based quaternions multiplier of paraunitary filter bank for lossless-to-lossy image coding

This paper presents a systematic design of the integer-to-integer invertible quaternionic multiplier based on the block-lifting structure and pipelined embedded processor of the given multiplier using distributed arithmetic (DA) as a block of M-band linear phase paraunitary filter banks (LP PUFB) based on the quaternionic algebra (Q-PUFB) for the lossy-to-lossless image coding. A bank Q-PUFB based on the DA block-lifting structure reduces the number of rounding operations and has a regular layout. Since the block-lifting structures with rounding operations can implement the integer-to-integer transform (Int-Q-PUFB).

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.

Lattice structure for arbitrary-length oversampled linear phase paraunitary filter bank with unequal numbers of symmetric and antisymmetric filters

Oversampled linear phase paraunitary filter bank (OLPPUFB) can be efficiently designed via lattice structure. Xu et al. have studied the lattice structure for arbitrary-length OLPPUFB (ALOLPPUFB), i.e. OLPPUFB with filter length KM+β, where M is the integer decimation factor, K is an integer, and β is an integer between 0 and M. Such work was restricted to be used for the case with equal numbers of symmetric and antisymmetric filters, and cannot be easily generalized for other possible cases. To address this issue, we develop in this letter the lattice structure for ALOLPPUFB with unequal numbers of symmetric and antisymmetric filters. The proposed method is carried out by combining the polyphase matrices of OLPPUFB with filter length KN, where N is the integer decimation factor, K is an integer. The efficiency of the method is shown by design examples.

Lattice Structure for Regular Linear Phase Paraunitary Filter Bank With Odd Decimation Factor

Linear phase paraunitary filter bank (LPPUFB) with regular property, i.e. regular LPPUFB (RLPPUFB), can be efficiently designed via lattice structure. The key to the design of such filter bank is to formulate an easy-to-handle regular condition in terms of the lattice structure of LPPUFB, and the solution for the case with even decimation factor has been published. In this letter, a method is proposed to design RLPPUFB with odd decimation factor. First, small part of the free parameters in the lattice structure of LPPUFB is discarded. Then, the lattice structure is equivalently converted to a new one for which the easy-to-handle regular condition can be easily obtained. In contrast to a simple method that discards about a half of the free parameters, the proposed method only discards small part of the free parameters and it therefore can produce better RLPPUFBs.

A New Design Method of the Starting Block in Lattice Structure of Arbitrary-Length Linear Phase Paraunitary Filter Bank by Combining Two Polyphase Matrices

Constructing the starting block is the core of designing an arbitrary-length linear phase paraunitary filter bank (ALLPPUFB), and lattice structure is an efficient approach of ALLPPUFB design. This brief proposes a new ALLPPUFB starting block design by combining two polyphase matrices of constrained-length linear phase paraunitary filter bank. The proposed design is much easier to be understood than existing ones due to the brief design process. Moreover, it might be modified and used for unaddressed topics in neighboring fields, such as starting block design for arbitrary-length oversampled linear phase paraunitary filter bank.

Directional Lapped Orthogonal Transform: Theory and Design

This paper proposes a directional design method of 2-D nonseparable linear-phase paraunitary filter banks. The proposed method is based on a lattice structure consisting of the 2-D separable DCT block and nonseparable support extension processes. Because of the nonseparability, the bases are allowed to be directional with the critically fixed subsampling, overlapping, orthogonal, symmetric, real-valued, and compact support properties. First, a novel vanishing moment (VM) condition is introduced as a suitable directional constraint, where the moment is referred to as the trend VM. The condition forces wavelet filters, i.e., high-pass and bandpass filters, to annihilate trend-surface components. Second, some theoretical properties of TVMs are discussed for general 2-D paraunitary systems, and then, the properties are applied to the lattice parameters. In order to verify the significance, several design examples are shown, the trend-surface annihilation properties are numerically confirmed, and the denoising capability is evaluated for images through shrinkage. It is shown that our proposed transforms yield perceptually preferable results.

Boundary Operation of 2-D Nonseparable Linear-Phase Paraunitary Filter Banks

This paper proposes a boundary operation technique of 2-D nonseparable linear-phase paraunitary filter banks (NS-LPPUFBs) for size limitation. The proposed technique is based on a lattice structure consisting of the 2-D separable block discrete cosine transform and nonseparable support-extension processes. The bases are allowed to be anisotropic with the fixed critically subsampling, overlapping, orthogonal, symmetric, real-valued, and compact-support properties. First, the blockwise implementation is developed so that the basis images can be locally controlled. The local control of basis images is shown to maintain orthogonality. This property leads a basis termination (BT) technique as a boundary operation. The technique overcomes the drawback of NS-LPPUFBs that the popular symmetric extension method is invalid. Through some experimental results of diagonal texture coding, the significance of the BT is verified.

A Simplified Lattice Structure of Two Dimensional Generalized Lapped Orthogonal Transform

In this paper, we propose a novel lattice structure of two dimensional (2D) nonseparable linear-phase paraunitary filter banks (LPPUFBs) called 2D GenLOT. Muramatsu et al. have previously proposed a lattice structure of 2D nonseparable LPPUFBs which have efficient frequency response. However, the proposed structure requires less number of design parameters and computational costs than the conventional one. Through some design examples and simulation results, we show that both filter banks have comparable frequency response and coding gain.

Reversible design of the starting block in lattice structure of arbitrary-length linear phase paraunitary filter banks

An important theory concerned with lattice structure is completeness, which is most determined by the completeness of its starting block. This letter provides an alternate process to design the starting block of arbitrary-length linear phase paraunitary filter banks (ALLPPUFB). The design is reversible thus guaranteeing the completeness of the starting block directly. In contrast, existing designs of ALLPPUFB starting block are non-reversible, hence an extra proof is required to show the completeness of the starting block.

On the Arbitrary-Length $M$-Channel Linear Phase Perfect Reconstruction Filter Banks

This correspondence extends the theory and lattice factorizations for M-channel linear phase perfect reconstruction filter banks (LPPRFBs). We deal with FIR FBs with real-valued filter coefficients in which all filters have the same arbitrary length L = KM + beta (0 les beta < M) and same symmetry center, in contrast to traditional constrained length profile L=KM. First, refined existence conditions on this larger class of FBs are given. Then, lattice structures are developed for both odd and even-channel FBs, and their relationship with time-domain lapped transforms is explained. These structures are more general compared to conventional design methods, and cover them as special cases. Furthermore, we discuss how to structurally impose the regularity onto the lattice structure to ensure the smoothness of the basis function, which is very desirable in high performance image compression. Finally, these lattice structures are proven to be minimal in terms of the number of delay elements used in implementation, and to completely span the class of LPPRFBs with length L les 2M and the whole class of linear phase paraunitary filter banks (LPPUFBs).

Constraints of Second-Order Vanishing Moments on Lattice Structures for Non-separable Orthogonal Symmetric Wavelets

In this paper, a design method of two-dimensional (2-D) orthogonal symmetric wavelets is proposed by using a lattice structure for multi-dimensional (M-D) linear-phase paraunitary filter banks (LPPUFB), which the authors have proposed as a previous work and then modified by Lu Gan et al. The derivation process for the constraints on the second-order vanishing moments is shown and some design examples obtained through optimization with the constraints are exemplified. In order to verify the significance of the constraints, some experimental results are shown for Lena and Barbara image.

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.

A simplified lattice factorization for linear-phase paraunitary filter banks with pairwise mirror image frequency responses

In this paper, by extending our previous work on general linear-phase paraunitary filter banks even-channel (LPPUFBs), we develop a new structure for LPPUFBs with the pairwise mirror image (PMI) frequency responses, which is a simplified version of the lattice proposed by Nguyen et al. Our simplification is achieved through trivial matrix manipulations and the cosine-sine (C-S) decomposition of a general orthogonal matrix. The resulting new structure covers the same class of PMI-LPPUFBs as the original lattice, while substantially reducing the number of free parameters involved in the nonlinear optimization. A design example is presented to demonstrate the effectiveness of the new structure.

DCT-based general structure for linear-phase paraunitary filterbanks

In this paper, we present a general structure for linear phase paraunitary filterbanks (LPPUFB) via both time-domain pre-processing and frequency-domain post-processing of the discrete cosine transform (DCT). The proposed structure not only covers existing frameworks with either pre-processing or post-processing of the DCT but enables the design of DCT-based LPPUFB with both partial-block overlapping and variable-length filters as well. Some high-performance and low-complexity transforms can be obtained by fine-tuning the structure parameters. Furthermore, a DCT-oriented initialization method is developed to improve the convergence and to simplify,the parameterization in the filterbank optimization process. The application in image compression is demonstrated.

Design of 2D filter banks based on 1D lattice structure

AbstractThis paper proposes a design procedure for the 2D linear‐phase para‐unitary filter bank, based on the 1D lattice structure. As the first step, the perfect reconstruction conditions of 1D and 2D cases are compared, and it is shown that the 2D filter bank satisfying the perfect reconstruction condition can be realized by the 1D lattice structure with a constraint. It is known, on the other hand, that the 1D perfect reconstruction filter bank can be converted to the lattice structure, using only half of the filter coefficients, and a simple optimization procedure can derive the whole filter coefficients. Based on those investigations, it is shown that the 2D filter bank can be designed by directly designing only half of the channels of the 2D filter bank in the time domain. © 2002 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 85(12): 38–46, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10005

Lattice structures parameterization of linear phase paraunitary matrices with pairwise mirror-image symmetry in the frequency domain with an odd number of rows

Much attention has been placed on the design of digital filter banks with pairwise mirror-image symmetry in the frequency domain because such filter bank structure requires fewer multipliers and less time to design than most other structures. Furthermore, the designed subband filters have better attenuation in most cases. In this paper, the polyphase matrix of linear phase paraunitary filter banks with an odd number of channels and pairwise mirror-image symmetry frequency responses are completely characterized and parameterized by a lattice structure. The parameterization is shown to be complete and minimal. Design examples are presented.

A new algorithm for linear-phase paraunitary filter banks with pairwise mirror-image frequency responses

Subband filter banks have attracted much attention during the past few years. In this paper, an efficient design algorithm, which leads to linear-phase paraunitary filter banks with pairwise mirror-image frequency responses, is revisited and further studied. New lattice structures are presented to extend the algorithm to the case where the number ( M) of channels is odd. Design examples of filter banks with 3 and 5 channels are presented.

Fast initialisation of linear-phase paraunitaryfilter banksusing adjacent near-power-complementary constraints

An efficient initialisation algorithm is presented for linear-phase paraunitary filter banks by incorporating near-power-complementary constraints on spectrally adjacent filters. Experiments show that the proposed initialisation helps the design parameters in the lattice structures quickly converge to a set of initial values to be used in further re-optimisation of filter banks.

Complex-valued linear-phase paraunitary filterbanks

Complex-valued linear-phase paraunitary filter banks An algorithm for complex-valued paraunitary filter banks with conjugate symmetry is proposed which leads to the filters having linear-phase frequency responses. Lattice structures are proposed that can be treated as a generalisation of the factorisation of real-valued linear-phase filter banks.

A design method of multidimensional linear-phase paraunitary filter banks with a lattice structure

A lattice structure of multidimensional (MD) linear-phase paraunitary filter banks (LPPUFBs) is proposed, which makes it possible to design such systems in a systematic manner. Our proposed structure can produce MD-LPPUFBs whose filters all have the region of support /spl Nscr/(M/spl Xi/), where M and /spl Xi/ are the decimation and positive integer diagonal matrices, respectively, and /spl Nscr/(N) denotes the set of integer vectors in the fundamental parallelepiped of a matrix N. It is shown that if /spl Nscr/(M) is reflection invariant with respect to some center, then the reflection invariance of /spl Nscr/(M/spl Xi/) is guaranteed. This fact is important in constructing MD linear-phase filter banks because the reflection invariance is necessary for any linear-phase filter. Since our proposed system structurally restricts both the paraunitary and linear-phase properties, an unconstrained optimization process can be used to design MD-LPPUFBs. Our proposed structure is developed for both an even and an odd number of channels and includes the conventional 1-D system as a special case. It is also shown to be minimal, and the no-DC-leakage condition is presented. Some design examples show the significance of our proposed structure for both the rectangular and nonrectangular decimation cases.