Exact reconstruction techniques for tree-structured subband coders

Smith, M.J.T.; Barnwell, T.P.

doi:10.1109/tassp.1986.1164832

Cited by 628 publications

(203 citation statements)

References 14 publications

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“…One of these is that h(n) must be the impulse response of a so-called quadrature mirror filter [14,29]. The simplest example of such a q, h pair is the Haar approximation in which…”

Section: N=---00mentioning

confidence: 99%

Recursive and iterative estimation algorithms for multiresolution stochastic processes

Chou

Willsky

Benveniste³

et al.

Proceedings of the 28th IEEE Conference on Decision and Control

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A current topic of great interest is the multi-resolution analysis of signals and the development of multi-scale or multigrid algorithms. In this paper we describe part of a research effort aimed at developing a corresponding theory for stochastic processes described at multiple scales and for their efficient estimation or reconstruction given partial and/or noisy measurements which may also be at several scales. The theories of multi-scale signal representations and wavelet transforms lead naturally to models of signals(in one or several dimensions) on trees and lattices. In this paper we focus on one particular class of processes defined on dyadic trees. The central results of the paper are three algorithms for optimal estimation/reconstruction for such processes: one reminiscent of the Laplacian pyramid and the efficient use of Haar transforms, a second that is iterative in nature and can be viewed as a multigrid relaxation algorithm, and a third that represents an extension of the Rauch-Tung-Striebel algorithm to processes on dyadic trees and involves a new discrete Riccati equation, which in this case has three steps: predict, merge, and measurement update.Related work and extensions are also briefly discussed.

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“…One of these is that h(n) must be the impulse response of a so-called quadrature mirror filter [14,29]. The simplest example of such a q, h pair is the Haar approximation in which…”

Section: N=---00mentioning

confidence: 99%

Recursive and iterative estimation algorithms for multiresolution stochastic processes

Chou

Willsky

Benveniste³

et al.

Proceedings of the 28th IEEE Conference on Decision and Control

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“…4 was utilized for the refinement of the three CQ filters of lengths 8, 16, 32 reported in [2]. Each of these filters was used as initial point for the algorithm.…”

Section: Design Examplesmentioning

confidence: 99%

“…The coefficients of H 0 (z) with N = 32 are shown in Table 1, where for comparison purposes the filter coefficients reported in [2] are also listed. The stopband edge as well as minimum attenuation (set to 40 dB) of each refined CQ filter were identical to that of the filter of same length from [2], and the performance of the filters are evaluated in terms of minimum stopband attenuation and satisfaction of the PR condition. The satisfaction of the PR condition was measured by the largest magnitude error among all N /2 equations in (3).…”

Section: Design Examplesmentioning

confidence: 99%

“…Because all filters involved satisfy 40 dB minimum stopband attenuation, the evaluation results in Table 2 only list the largest equation errors of the three pairs of designs. Figure 2 depicts the magnitude responses of the CQ filters H 0 and H 1 of length 32 from [2] (dashed lines) and their refined versions (solid lines). It is observed that the magnitude responses are practically identical, but the refined filter offers considerably improved satisfaction of the PR condition (see Table 2).…”

Section: Design Examplesmentioning

confidence: 99%

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Direct design of orthogonal filter banks and wavelets

Hinamoto

2009

2009 IEEE International Symposium on Circuits and Systems

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Abstract-This paper presents a new method for the design of two-channel conjugate quadrature (CQ) filter banks in which halfband filter and spectrum factorization are not required. Instead, a CQ filter is directly optimized subject to the perfect reconstruction and possibly other constraints (such as number of vanishing moments (VM)). We develop a design strategy in that the solution is approached sequentially with each update confined to within a small vicinity of the current iterate where the problem at hand behaves like a convex one, thus the update can be obtained as a solution of a convex problem. Four design scenarios are considered, namely the least squares designs with or without VM requirement, and equiripple designs with or without VM requirement. The simulation studies demonstrate that the proposed method is reliable to design high-order CQ filters with improved performance.

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“…The DWT targets discrete-time signals, and is the same as the two-channel filter bank [6,7]. The DWT using a bi-/orthogonal wavelet corresponds to a bi-/orthogonal filter bank, respectively, and both can reconstruct a signal from decomposed signals correctly.…”

Section: Introductionmentioning

confidence: 99%

Development of Quasi-Shift-Invariant Complex Discrete Wavelet Transform

Kobayashi¹,

Nakano

2017

Journal of Signal Processing

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The downsampling of a discrete wavelet transform (DWT) has a side effect of the lack of shiftinvariance. There are two main solutions for this effect: one is the stationary wavelet transform (SWT), which does not apply downsampling. The other is the complex DWT (CDWT), which uses dual multiresolution analysis (MRA). We choose the CDWT as a target of research. It is well known that wavelet functions become Hilbert transform pairs if the low-pass filters (LPFs) on the reconstruction side have half-sample shifts. In this paper, we propose a quasi-shift-invariant (QSI) CDWT for bi-/orthogonal wavelets as a new CDWT. We report three new works (W1-W3) on it: (W1) we generalized the condition of Hilbert transform pairs and employed a complex wavelet function as a conjugate analytical signal. (W2) We defined a structure that achieves shift-invariance. The structure requires half-sample delays between the inputs of real and imaginary parts. (W3) We proposed an implementation of the QSI-CDWT and confirmed that our method has higher shift-invariance than the conventional CDWT. However, two problems (P1, P2) remain unsolved: (P1) our method requires more resources, such as memory and calculation time, than the conventional CDWT. (P2) Our theory cannot make all packets shift-invariant in a classical wavelet packet transform tree.

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Exact reconstruction techniques for tree-structured subband coders

Cited by 628 publications

References 14 publications

Recursive and iterative estimation algorithms for multiresolution stochastic processes

Recursive and iterative estimation algorithms for multiresolution stochastic processes

Direct design of orthogonal filter banks and wavelets

Development of Quasi-Shift-Invariant Complex Discrete Wavelet Transform

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