Hilbert transform pairs of wavelet bases

Selesnick, Ivan

doi:10.1109/97.923042

Cited by 303 publications

(172 citation statements)

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“…Here we summarize the discussion of Selesnick [14] and extend it. By using θ i : R � → (−π/2, +π/2) , i ∈ B, we assume that the phase difference is…”

Section: Extending Selesnick's Discussionmentioning

confidence: 99%

“…The downsampling causes the lack of shiftinvariance [10]. Many proposed methods have been reported for overcoming the problem, including the undecimated wavelet transform (stationary wavelet transform; SWT) [11] and the complex DWT (CDWT) [12,13], by constructing a Hilbert transform pair of wavelet functions [14]. In this paper, we choose the CDWT as a target of research.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Here we summarize the discussion of Selesnick [14] and extend it. By using θ i : R � → (−π/2, +π/2) , i ∈ B, we assume that the phase difference is…”

Section: Extending Selesnick's Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Development of Quasi-Shift-Invariant Complex Discrete Wavelet Transform

Kobayashi¹,

Nakano

2017

Journal of Signal Processing

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“…Среди таких недостатков следует отметить отсутствие инвариантности относительно сдвига базисной функции, осциллирующий характер ко-эффициентов разложения в окрестности сингулярностей и появление артефактов в сигнале, восстановленном после коррекции вейвлет-коэф-фициентов [9]. В работах [10,11] было обосновано, что для устранения данных недостатков целесообразно применение комплексных функций, у которых действительная и мнимая часть обладают свойством со-пряжения по Гильберту, т. е. аналитических или почти аналитических вейвлетов. Однако выбор " хорошего" базиса не гарантирует того, что проводимая с его помощью фильтрация обеспечит снижение ошибки, так как качество очистки сигналов от помех существенно зависит от параметров фильтров, таких как пороговый уровень, задаваемый при фильтрации, и от отношения сигнал/шум.…”

Section: поступило в редакцию 14 марта 2017 гunclassified

Фильтрация Зашумленных Сигналов С Использованием Комплексных Вейвлет-Базисов

Ясин¹,

Павлова²,

Павлов³

2017

Письма В Журнал Технической Физики

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Сопоставлены методы фильтрации, применяющие дискретное вейвлет-преобразование с вещественными базисами семейства Добеши и комплексное вейвлет-преобразование двойной плотности с избыточными (неортонормированными) базисами. Приведены рекомендации по выбору параметров фильтров для минимизации среднеквадратичной ошибки фильтрации зашумленных сигналов. DOI: 10.21883/PJTF.2017.14.44831.16770

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“…The Complex Wavelet Transform (CWT) differs from the DWT in having complex-valued scaling and wavelet functions, viz., ɸ 1 + jɸ 2 and ψ 1 + jψ 2 , respectively, and form Hilbert pairs [13], [14]. Various methods have been proposed for obtaining CWT coefficients [8], however, due to the simplicity of implementation and lower redundancy, the dual-tree CWT (DT-CWT) that has been proposed by Kingsbury [8] and later generalized by Selesnick [13], is becoming popular.…”

Section: Denoising Using Complex Wavelet Transformmentioning

confidence: 99%

Noise Removal From Microarray Images Using Maximum a Posteriori Based Bivariate Estimator

Agnal¹,

Mala²

2013

IJIGSP

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Abstract-Microarray Image contains information about thousands of genes in an organism and these images are affected by several types of noises. They affect the circular edges of spots and thus degrade the image quality. Hence noise removal is the first step of cDNA microarray image analysis for obtaining gene expression level and identifying the infected cells. The Dual Tree Complex Wavelet Transform (DT-CWT) is preferred for denoising microarray images due to its properties like improved directional selectivity and near shift-invariance. In this paper, bivariate estimators namely Linear Minimum Mean Squared Error (LMMSE) and Maximum A Posteriori (MAP) derived by applying DT-CWT are used for denoising microarray images. Experimental results show that MAP based denoising method outperforms existing denoising techniques for microarray images.

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Hilbert transform pairs of wavelet bases

Cited by 303 publications

References 6 publications

Development of Quasi-Shift-Invariant Complex Discrete Wavelet Transform

Development of Quasi-Shift-Invariant Complex Discrete Wavelet Transform

Фильтрация Зашумленных Сигналов С Использованием Комплексных Вейвлет-Базисов

Noise Removal From Microarray Images Using Maximum a Posteriori Based Bivariate Estimator

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