Linear-phase Perfect Reconstruction Filter Banks Research Articles

Multidimensional Nonseparable Oversampled Lapped Transforms: Theory and Design

This paper extends the theory of the one-dimensional oversampled linear-phase perfect reconstruction filter banks (OLPPRFBs) developed by Gan et al. to multidimensional (MD) cases and proposes MD nonseparable oversampled lapped transforms (NSOLTs). NSOLTs allow us to achieve an overcomplete analysis-synthesis system with nonseparable, symmetric, real-valued, overlapping, and compact-supported filters. The proposed systems are based on lattice structures and the redundancy is flexibly controlled by the number of channels and downsampling ratio. The proposed structure is shown to be capable of constructing Parseval tight frames in any number of dimensions. The number of design parameters are examined under the Parseval tight frame constraint. In order to design NSOLTs specified for sparse approximation of image or volume data, an example-based design procedure is introduced. The effectiveness of this method is verified by examining design samples and evaluating their sparse approximation performance using the iterative hard thresholding algorithm for a natural image and MRI volume data patch.

A method to initialize free parameters in lattice structure of arbitrary-length linear phase perfect reconstruction filter bank

The optimization of the free parameters in the lattice structure of linear phase perfect reconstruction filter bank (LPPRFB) is commonly highly nonlinear, therefore the free parameters should be carefully initialized. Such methods have been published for constrained-length LPPRFB (CLLPPRFB), i.e. LPPRFB with filter length MK, where M is the decimation factor and K is an integer. In contrast to CLLPPRFB, arbitrary-length LPPRFB (ALLPPRFB) [i.e. LPPRFB with filter length MK+β, where β is an integer between 0 and M] provides more choices of filter banks, offering better trade-off between filter length and filter performance. Nevertheless, as far as we know there is no systematic study on initializing the free parameters in the lattice structure of ALLPPRFB, and the required method cannot be easily generalized from those for CLLPPRFB. To address this issue, an initialization method will be described in this paper that initializes a longer ALLPPRFB by a shorter one, and the length difference between the longer and the shorter filter banks can be any number allowed by an ALLPPRFB system. The efficiency of the proposed method will be shown by design examples and image compression experiments.

A method for initializing free parameters in lattice structure of linear phase perfect reconstruction filter bank

Since the optimization of the free parameters in the lattice structure of linear phase perfect reconstruction filter bank (LPPRFB) is commonly highly nonlinear, the initialization of the free parameters is important before starting the optimization. We systematically study the initialization of the free parameters, following the way that initializes the free parameters in a higher-order LPPRFB by those in a lower-order one. The proposed method is carried out in two steps: formulating the relation between the higher-order and the lower-order LPPRFBs in terms of polyphase matrices, and then implementing the initialization based on the relation. Design examples show that, compared with published LPPRFBs, the filter banks yielded by our method show comparable or even better performance.

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).

On the closeness of the space spanned by the lattice structures for a class of linear phase perfect reconstruction filter banks

The incompleteness of the existing lattice structures has been well established for M -channel FIR linear phase perfect reconstruction filter banks (LPPRFBs) with filter length L > 2 M in the literature, and even the nonexistence of complete order-one lattice has been reported recently. Thus, a question arises naturally as to how large the space spanned by the existing lattice structure is, and about its closeness over some polynomial transformations. The study for such issue can reveal what sense of optimality the lattice based design for LPPRFBs possesses. Inspired from this perspective, this paper firstly studies the closeness of the space spanned by the existing lattice structures under the polynomial transformations for arbitrary equal-length LPPRFBs. We have shown that this space is closed under the popular polynomial transforms widely used in FB design, which establishes the suboptimality of the lattice based design methods for LPPRFBs. Furthermore, the explicit relationship between the lattice parameters before and after transformations has been shown for describing the closeness of the space spanned by those lattice structures.

Higher-order feasible building blocks for lattice structure of oversampled linear-phase perfect reconstruction filter banks

This paper proposes new building blocks for the lattice structure of oversampled linear-phase perfect reconstruction filter banks (OLPPRFBs). The structure is an extended version of higher-order feasible building blocks for critically sampled LPPRFBs. It uses fewer number of building blocks and design parameters than those of traditional OLPPRFBs, whereas frequency characteristics of the new OLPPRFBs are comparable to those of traditional one. Furthermore, the building block structures for arbitrary number of channels and downsampling factors are presented.

On the Existence of Complete Order-One Lattice for Linear Phase Perfect Reconstruction Filter Banks

In this letter, we revisit the completeness of the lattice factorization for -channel linear phase perfect reconstruction filter banks (LPPRFBs) with equal length and further investigate a more fundamental problem, i.e., the existence of complete lattice factorization by using LPPR propagating blocks of order-one causal FIR with anticausal FIR inverse (CAFACAFI) matrices. Reviewing the previous works, we point out the limitation of the existing LPPR propagating blocks and then show its consequence for incompleteness of the existing lattice factorizations. Furthermore, we show the nonexistence of any order-one LPPR propagating block by using CAFACAFI matrices. In addition, the completeness of lattice factorizations has been re-examined for generalized lapped orthogonal transforms (GenLOTs) and lapped biorthogonal transforms (LBTs) with linear phase based on the analysis developed in this letter.

Two-Dimensional Antisymmetric Linear Phase Filter Bank Construction Using Symmetric Completion

For a given antisymmetric linear phase (LP) filter in two dimensions, an approach that can parameterize the counterpart filter such that they can form a perfect reconstruction filter bank (FB) is proposed. This approach is motivated by a previous work and remedies the incomplete solution there. To illustrate its incompleteness, classification of LP perfect reconstruction FBs is introduced. Possibilities of extension of the proposed scheme are also discussed

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.

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.

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.

Oversampled linear-phase perfect reconstruction filterbanks: theory, lattice structure and parameterization

The paper presents the theory, lattice structure, and parameterization for a general class of P-channel oversampled linear-phase perfect reconstruction filterbanks (OLPPRFBs) - systems with sampling factor M (P/spl ges/M) and filter length of L=KM (K/spl ges/1) each. For these OLPPRFBs, the necessary existence conditions on the number of symmetric filters, n/sub /spl beta//, and antisymmetric filters, n/sub /spl alpha//, (i.e., symmetry polarity) are first investigated. VLSI-friendly lattice structures are then developed for two types of OLPPRFBs, type I system (n/sub /spl beta//=n/sub /spl alpha//) and type II system (n/sub /spl beta///spl ne/n/sub /spl alpha//). The completeness and minimality of each type of lattice are also analyzed. Compared with existing work, the proposed lattices are the most general and efficient ones for OLPPRFBs. Besides, through the lattice structures, the sufficiency of the existence conditions is also verified. Next, lifting-based structures are proposed to parameterize a left invertible matrix and all of its left inverses, which leads to unconstrained optimization as well as robust implementation of OLPPRFBs. Finally, several design examples are presented to confirm the validity of the theory and demonstrate the versatility of synthesis filterbanks in the oversampled system.

On the completeness of the lattice factorization for linear-phase perfect reconstruction filter banks

In this letter, we re-examine the completeness of the lattice factorization for M-channel linear-phase perfect reconstruction filter bank (LPPRFB) with filters of the same length L=KM as discussed by Tran et al. (see IEEE Trans. Signal Processing, vol.48, p.133-47, Jan. 2000). We point out that the assertion of completeness is incorrect. Examples are presented to show that the proposed lattice structure of Tran et al. is not complete when K>2. In addition, we verify that the lattice structure is complete only when K/spl les/2.

A simplified lattice factorization for linear-phase perfect reconstruction filter bank

We propose a simplified version of lattice factorization for linear-phase perfect reconstruction filter bank (LPPRFB) derived by T.D. Tran et al. (see IEEE Trans. Signal Processing. vol.48, no.1, p.133-47, Jan. 2000). The proposed new lattice structure spans the same class of LPPRFB, while substantially reducing free parameters in nonlinear optimization and saving computation cost in hardware implementation. To further address the importance of our proposed structure, we generalize our factorization to multidimensional LPPRFB (MD-LPPRFB), and show its effectiveness.

Linear-phase perfect reconstruction filter bank: lattice structure, design, and application in image coding

A lattice structure for an M-channel linear-phase perfect reconstruction filter bank (LPPRFB) based on the singular value decomposition (SVD) is introduced. The lattice can be proven to use a minimal number of delay elements and to completely span a large class of LPPRFBs: all analysis and synthesis filters have the same FIR length, sharing the same center of symmetry. The lattice also structurally enforces both linear-phase and perfect reconstruction properties, is capable of providing fast and efficient implementation, and avoids the costly matrix inversion problem in the optimization process. From a block transform perspective, the new lattice can be viewed as representing a family of generalized lapped biorthogonal transform (GLBT) with an arbitrary number of channels M and arbitrarily large overlap. The relaxation of the orthogonal constraint allows the GLBT to have significantly different analysis and synthesis basis functions, which can then be tailored appropriately to fit a particular application. Several design examples are presented along with a high-performance GLBT-based progressive image coder to demonstrate the potential of the new transforms.

Discrete Lagrangian Methods for Designing Multiplierless Two-Channel PR-LP Filter Banks

In this paper, we present a new search method based on the theory of discrete Lagrange multipliers for designing multiplierless PR (perfect reconstruction) LP (linear phase) filter banks. To satisfy the PR constraints, we choose a lattice structure that, under certain conditions, can guarantee the resulting two filters to be a PR pair. Unlike the design of multiplierless QMF filter banks that represents filter coefficients directly using PO2 (powers-of-two) form (also called Canonical Signed Digit or CSD representation), we use PO2 forms to represent the parameters associated with the lattice structure. By representing these parameters as sums or differences of powers of two, multiplications can be carried out as additions, subtractions, and shifts. Using the lattice representation, we decompose the design problem into a sequence of four subproblems. The first two subproblems find a good starting point with continuous parameters using a single-objective, multi-constraint formulation. The last two subproblems first transform the continuous solution found by the second subproblem into a PO2 form, then search for a design in a mixed-integer space. We propose a new search method based on the theory of discrete Lagrange multipliers for finding good designs, and study methods to improve its convergence speed by adjusting dynamically the relative weights between the objective and the Lagrangian part. We show that our method can find good designs using at most four terms in PO2 form in each lattice parameter. Our approach is unique because our results are the first successful designs of multiplierless PR-LP filter banks. It is general because it is applicable to the design of other types of multiplierless filter banks.

A progressive transmission image coder using linear phase uniform filterbanks as block transforms

This paper presents a novel image coding scheme using M-channel linear phase perfect reconstruction filterbanks (LPPRFBs) in the embedded zerotree wavelet (EZW) framework introduced by Shapiro (1993). The innovation here is to replace the EZWs dyadic wavelet transform by M-channel uniform-band maximally decimated LPPRFBs, which offer finer frequency spectrum partitioning and higher energy compaction. The transform stage can now be implemented as a block transform which supports parallel processing and facilitates region-of-interest coding/decoding. For hardware implementation, the transform boasts efficient lattice structures, which employ a minimal number of delay elements and are robust under the quantization of lattice coefficients. The resulting compression algorithm also retains all the attractive properties of the EZW coder and its variations such as progressive image transmission, embedded quantization, exact bit rate control, and idempotency. Despite its simplicity, our new coder outperforms some of the best image coders published previously in the literature, for almost all test images (especially natural, hard-to-code ones) at almost all bit rates.

On M-channel linear phase FIR filter banks and application in image compression

This paper investigates the theory and structure of a large subclass of M-channel linear-phase perfect-reconstruction FIR filter banks-systems with analysis and synthesis filters of length L/sub i/=K/sub i/M+/spl beta/, where /spl beta/ is an arbitrary integer, 0/spl les//spl beta/<M, and K/sub i/ is any positive integer. For this subclass of systems, we first investigate the necessary conditions for the existence of linear-phase perfect-reconstruction filter banks (LPPRFBs). Next, we develop a complete and minimal factorization for all even-channel linear-phase paraunitary systems (the most general lapped orthogonal transforms to date). Finally, several design examples as well as comparisons with previous generalized lapped orthogonal transforms (GenLOTs) in image compression are presented to confirm the validity of the theory.

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.

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. >