Phase Perfect Reconstruction Filter Banks Research Articles

Time–frequency localized three-band biorthogonal wavelet filter bank using semidefinite relaxation and nonlinear least squares with epileptic seizure EEG signal classification

In this paper, we design time–frequency localized three-band biorthogonal linear phase wavelet filter bank for epileptic seizure electroencephalograph (EEG) signal classification. Time–frequency localized analysis and synthesis low-pass filters (LPF) are designed using convex semidefinite programming (SDP) by transforming a nonconvex problem into a convex SDP using semidefinite relaxation technique. Three-band parameterized lattice biorthogonal linear phase perfect reconstruction filter bank (BOLPPRFB) is chosen and nonlinear least squares algorithm is used to determine its parameters values that generate the designed analysis and synthesis LPF such that the band-pass and high-pass filters are also well localized in time and frequency domain. The designed analysis and synthesis three-band wavelet filter banks are compared with the standard two-band filter banks like Daubechies maximally regular filter banks, Cohen–Daubechies–Feauveau (CDF) biorthogonal filter banks and orthogonal time–frequency localized filter banks. Kruskal–Wallis statistical test is employed to measure the statistical significance of the subband features obtained from the various two and three-band filter banks for epileptic seizure EEG signal classification. The results show that the designed three-band analysis and synthesis filter banks both outperform two-band filter banks in the classification of seizure and seizure-free EEG signals. The designed three-band filter banks and multi-layer perceptron neural network (MLPNN) are further used together to implement a signal classifier that provides classification accuracy better than the recently reported results for epileptic seizure EEG signal classification.

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.

Group-Theoretic Structure of Linear Phase Multirate Filter Banks

Unique lifting factorization results for group lifting structures are used to characterize the group-theoretic structure of two-channel linear phase FIR perfect reconstruction filter bank groups. For D-invariant, order-increasing group lifting structures, it is shown that the associated lifting cascade group C is isomorphic to the free product of the upper and lower triangular lifting matrix groups. Under the same hypotheses, the associated scaled lifting group S is the semidirect product of C by the diagonal gain scaling matrix group D. These results apply to the group lifting structures for the two principal classes of linear phase perfect reconstruction filter banks, the whole- and half-sample symmetric classes. Since the unimodular whole-sample symmetric class forms a group, W, that is in fact equal to its own scaled lifting group, W=S_W, the results of this paper characterize the group-theoretic structure of W up to isomorphism. Although the half-sample symmetric class H does not form a group, it can be partitioned into cosets of its lifting cascade group, C_H, or, alternatively, into cosets of its scaled lifting group, S_H. Homomorphic comparisons reveal that scaled lifting groups covered by the results in this paper have a structure analogous to a "noncommutative vector space."

Lattice Structure for Generalized-Support Multidimensional Linear Phase Perfect Reconstruction Filter Bank

Multidimensional linear phase perfect reconstruction filter bank (MDLPPRFB) can be designed and implemented via lattice structure. The lattice structure for the MDLPPRFB with filter support N(MΞ) has been published by Muramatsu , where M is the decimation matrix, Ξ is a positive integer diagonal matrix, and N(N) denotes the set of integer vectors in the fundamental parallelepiped of the matrix N. Obviously, if Ξ is chosen to be other positive diagonal matrices instead of only positive integer ones, the corresponding lattice structure would provide more choices of filter banks, offering better trade-off between filter support and filter performance. We call such resulted filter bank as generalized-support MDLPPRFB (GSMDLPPRFB). The lattice structure for GSMDLPPRFB, however, cannot be designed by simply generalizing the process that Muramatsu employed. Furthermore, the related theories to assist the design also become different from those used by Muramatsu . Such issues will be addressed in this paper. To guide the design of GSMDLPPRFB, the necessary and sufficient conditions are established for a generalized-support multidimensional filter bank to be linear-phase. To determine the cases we can find a GSMDLPPRFB, the necessary conditions about the existence of it are proposed to be related with filter support and symmetry polarity (i.e., the number of symmetric filters ns and antisymmetric filters na). Based on a process (different from the one Muramatsu used) that combines several polyphase matrices to construct the starting block, one of the core building blocks of lattice structure, the lattice structure for GSMDLPPRFB is developed and shown to be minimal. Additionally, the result in this paper includes Muramatsu's as a special case.

Group Lifting Structures for Multirate Filter Banks II: Linear Phase Filter Banks

The theory of group lifting structures is applied to linear phase lifting factorizations for the two nontrivial classes of two-channel linear phase perfect reconstruction filter banks, the whole- and half-sample symmetric classes. Group lifting structures defined for the reversible and irreversible classes of whole- and half-sample symmetric filter banks are shown to satisfy the hypotheses of the uniqueness theorem for group lifting structures. It follows that linear phase group lifting factorizations of whole- and half-sample symmetric filter banks are therefore independent of the factorization methods used to construct them. These results cover the specification of whole-sample symmetric filter banks in the ISO/IEC JPEG 2000 image coding standard.

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.

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.

M-channel linear phase perfect reconstruction filter bank with rational coefficients

This paper introduces a general class of M-channel linear phase perfect reconstruction filter banks (FBs) with rational coefficients. A subset of the presented solutions has dyadic coefficients, leading to multiplierless implementations suitable for low-power mobile computing. All of these FBs are constructed from a lattice structure that is VLSI-friendly, employs the minimum number of delay elements, and robustly enforces both linear phase and perfect reconstruction property. The lattice coefficients are parameterized as a series of zero-order lifting steps, providing fast, efficient, in-place computation of the subband coefficients. Despite the tight rational or integer constraint, image coding experiments show that these novel FBs are very competitive with current popular transforms such as the 8/spl times/8 discrete cosine transform and the wavelet transform with 9/7-tap biorthogonal irrational-coefficient filters.

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.