On ladder structures and linear phase conditions for bi-orthogonal filter banks

Shah, Imran; Kalker, Ton

doi:10.1109/icassp.1994.390060

Cited by 8 publications

(2 citation statements)

References 10 publications

(5 reference statements)

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“…Lifting also leads to a filter bank implementation known as ladder structures [2]. Moreover, it is known that all 1-D FIR filter banks fit into lifting [15], [31], [43], [53].…”

Section: Introductionmentioning

confidence: 99%

Wavelet families of increasing order in arbitrary dimensions

Kovačević¹,

Sweldens²

2000

IEEE Trans. on Image Process.

168

133

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Abstract-We build discrete-time compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The associated scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its advantages: fast transform, in-place calculation, and integer-to-integer transforms. We show that two lifting steps suffice: predict and update. The predict step can be built using multivariate polynomial interpolation, while update is a multiple of the adjoint of predict. While we concentrate on the discrete-time case, some discussion of convergence and stability issues together with examples is given.

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“…Lifting also leads to a filter bank implementation known as ladder structures [2]. Moreover, it is known that all 1-D FIR filter banks fit into lifting [15], [31], [43], [53].…”

Section: Introductionmentioning

confidence: 99%

Wavelet families of increasing order in arbitrary dimensions

Kovačević¹,

Sweldens²

2000

IEEE Trans. on Image Process.

168

133

View full text Add to dashboard Cite

show abstract

“…Next, consider the matrices U (z), V (z). These are left-extension matrices [9] that lengthen the wavelet filters by introducing free parameters u, v into the analysis filter bank while preserving linear phase. However, orthogonality of the wavelet filters is not preserved by these matrices.…”

Section: D Ncwtc For Complex-valued Signalsmentioning

confidence: 99%

Non-redundant, linear-phase, semi-orthogonal, directional complex wavelets [image/video processing applications]

Fernandes

Wakin

Baraniuk

2004 IEEE International Conference on Acoustics, Speech, and Signal Processing

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The directionality and phase information provided by nonredundant complex wavelet transforms (NCWTs) provide significant potential benefits for image/video processing and compression applications. However, because existing NCWTs are created by downsampling filtered wavelet coefficients, the finest scale of these transforms has resolution 4× lower than the real input signal. In this paper, we propose a linear-phase, semi-orthogonal, directional NCWT design using a novel triband filter bank. At the finest scale, the resulting transform has resolution 3× lower than the real input signal. We provide a design example to demonstrate three important properties for image/video processing applications: directionality, magnitude coherency, and phase coherency.

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Factoring wavelet transforms into lifting steps

Daubechies

Sweldens

1998

The Journal of Fourier Analysis and Applications

1,718

946

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This article is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formula SL(n; R[Z, z-I]) = E(n; R[z, z-l])); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e., non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers.Math Subject Classifications. 42C15, 42C05, 19-02.

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On ladder structures and linear phase conditions for bi-orthogonal filter banks

Cited by 8 publications

References 10 publications

Wavelet families of increasing order in arbitrary dimensions

Wavelet families of increasing order in arbitrary dimensions

Non-redundant, linear-phase, semi-orthogonal, directional complex wavelets [image/video processing applications]

Factoring wavelet transforms into lifting steps

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