PERFECT TRANSLATION INVARIANCE WITH A WIDE RANGE OF SHAPES OF HILBERT TRANSFORM PAIRS OF WAVELET BASES

2010 ◽  
Vol 08 (04) ◽  
pp. 501-520 ◽  
Author(s):  
HIROSHI TODA ◽  
ZHONG ZHANG
Keyword(s):  
Wavelet Transform ◽  
Hilbert Transform ◽  
Calculation Method ◽  
Discrete Wavelet ◽  
Translation Invariance ◽  
Wavelet Bases ◽  
Discrete Signal ◽  
Wide Range ◽  
Meyer Wavelet ◽  
The Hilbert Transform

It is well known that a Hilbert transform pair of wavelet bases improves the lack of translation invariance of the discrete wavelet transform. However, its shapes and improvement are limited by the difficulty in applying the Hilbert transform pair to a discrete signal. In this paper, novel Hilbert transform pairs of wavelet bases, which are based on a Meyer wavelet and have a wide range of shapes, are proposed to create perfect translation invariance, and their calculation method is designed to apply these wavelet bases to any discrete signal. Therefore, perfect translation invariance is achieved with a wide range of shapes of the Hilbert transform pairs of wavelet bases.

THE DESIGN OF COMPLEX WAVELET PACKET TRANSFORMS BASED ON PERFECT TRANSLATION INVARIANCE THEOREMS

2010 ◽  
Vol 08 (04) ◽  
pp. 537-558 ◽  
Author(s):  
HIROSHI TODA ◽  
ZHONG ZHANG ◽  
TAKASHI IMAMURA
Keyword(s):  
Wavelet Transforms ◽  
Wavelet Packet ◽  
Wavelet Packets ◽  
Discrete Wavelet ◽  
Translation Invariance ◽  
Discrete Wavelet Transforms ◽  
Wide Range ◽  
Different Shapes ◽  
Complex Wavelet ◽  
Meyer Wavelet

The useful theorems for achieving perfect translation invariance have already been proved, and based on these theorems, dual-tree complex discrete wavelet transforms with perfect translation invariance have been proposed. However, due to the complication of frequency divisions with wavelet packets, it is difficult to design complex wavelet packet transforms with perfect translation invariance. In this paper, based on the aforementioned theorems, novel complex wavelet packet transforms are designed to achieve perfect translation invariance. These complex wavelet packet transforms are based on the Meyer wavelet, which has the important characteristic of possessing a wide range of shapes. In this paper, two types of complex wavelet packet transforms are designed with the optimized Meyer wavelet. One of them is based on a single Meyer wavelet and the other is based on a number of different shapes of the Meyer wavelets to create good localization of wavelet packets.

MULTIVARIATE SPECTRAL ANALYSIS USING HILBERT WAVELET PAIRS

2004 ◽  
Vol 02 (04) ◽  
pp. 567-587 ◽  
Author(s):  
BRANDON WHITCHER ◽  
PETER F. CRAIGMILE
Keyword(s):  
Time Series ◽  
Spectral Analysis ◽  
Wavelet Transform ◽  
Discrete Wavelet Transform ◽  
Multivariate Time Series ◽  
Discrete Wavelet ◽  
Pass Filter ◽  
Wavelet Filters ◽  
The Hilbert Transform ◽  
High Pass

We investigate the use of Hilbert wavelet pairs (HWPs) in the non-decimated discrete wavelet transform for the time-varying spectral analysis of multivariate time series. HWPs consist of two high-pass and two low-pass compactly supported filters, such that one high-pass filter is the Hilbert transform (approximately) of the other. Thus, common quantities in the spectral analysis of time series (e.g., power spectrum, coherence, phase) may be estimated in both time and frequency. Compact support of the wavelet filters ensures that the frequency axis will be partitioned dyadically as with the usual discrete wavelet transform. The proposed methodology is used to analyze a bivariate time series of zonal (u) and meridional (v) winds over Truk Island.

scholarly journals Fast Compressed Sensing MRI Based on Complex Double-Density Dual-Tree Discrete Wavelet Transform

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
Shanshan Chen ◽  
Bensheng Qiu ◽  
Feng Zhao ◽  
Chao Li ◽  
Hongwei Du
Keyword(s):  
Wavelet Transform ◽  
Compressed Sensing ◽  
Discrete Wavelet Transform ◽  
Signal To Noise Ratio ◽  
Similarity Index ◽  
Discrete Wavelet ◽  
Translation Invariance ◽  
Signal To Noise ◽  
Reconstruction Methods ◽  
Noise Ratio

Compressed sensing (CS) has been applied to accelerate magnetic resonance imaging (MRI) for many years. Due to the lack of translation invariance of the wavelet basis, undersampled MRI reconstruction based on discrete wavelet transform may result in serious artifacts. In this paper, we propose a CS-based reconstruction scheme, which combines complex double-density dual-tree discrete wavelet transform (CDDDT-DWT) with fast iterative shrinkage/soft thresholding algorithm (FISTA) to efficiently reduce such visual artifacts. The CDDDT-DWT has the characteristics of shift invariance, high degree, and a good directional selectivity. In addition, FISTA has an excellent convergence rate, and the design of FISTA is simple. Compared with conventional CS-based reconstruction methods, the experimental results demonstrate that this novel approach achieves higher peak signal-to-noise ratio (PSNR), larger signal-to-noise ratio (SNR), better structural similarity index (SSIM), and lower relative error.

scholarly journals Assessing the Contribution of the Oscillatory Potentials to the Genesis of the Photopic ERG with the Discrete Wavelet Transform

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Mathieu Gauvin ◽  
Allison L. Dorfman ◽  
Nataly Trang ◽  
Mercedes Gauthier ◽  
John M. Little ◽  
...  
Keyword(s):  
Diabetic Retinopathy ◽  
Wavelet Transform ◽  
Discrete Wavelet Transform ◽  
High Frequency ◽  
Normal Subjects ◽  
Oscillatory Potentials ◽  
Discrete Wavelet ◽  
Wide Range ◽  
Frequency Components ◽  
Photopic Erg

The electroretinogram (ERG) is composed of slow (i.e., a-, b-waves) and fast (i.e., oscillatory potentials: OPs) components. OPs have been shown to be preferably affected in some diseases (such as diabetic retinopathy), while the a- and b-waves remain relatively intact. The purpose of this study was to determine the contribution of OPs to the building of the ERG and to examine whether a signal mostly composed of OPs could also exist. DWT analyses were performed on photopic ERGs (flash intensities: −2.23 to 2.64 log cd·s·m−2in 21 steps) obtained from normal subjects (n=40) and patients (n=21) affected with a retinopathy. In controls, the %OP value (i.e., OPs energy/ERG energy) is stimulus- and amplitude-independent (range: 56.6–61.6%; CV = 6.3%). In contrast, the %OPs measured from the ERGs of our patients varied significantly more (range: 35.4%–89.2%;p<0.05) depending on the pathology, some presenting with ERGs that are almost solely composed of OPs. In conclusion, patients may present with a wide range of %OP values. Findings herein also support the hypothesis that, in certain conditions, the photopic ERG can be mostly composed of high-frequency components.

scholarly journals Image Forgery Recognition using SWT Method

2019 ◽  
Vol 8 (6) ◽  
pp. 293-296
Keyword(s):  
Wavelet Transform ◽  
Discrete Wavelet Transform ◽  
Digital Image ◽  
Discrete Wavelet ◽  
Translation Invariance ◽  
Feature Transformation ◽  
Scale Invariant ◽  
Image Forgery ◽  
Invariant Feature ◽  
Scale Invariant Feature

Using SWT (Stationary Wavelet Change) & SIFT (Scale Invariant Feature Transformation) we attempted to increase the number of features recognized & matched with digital image for forgery identification. Digital image received preferable match for forged area. We collected the forgery area using SIFT& SURF for identification of forgery. We used DWT (Discrete Wavelet Transform) w.r.t. SIFT & SW to subdue absence of translation invariance..

scholarly journals Dual tree complex wavelet transform-based signal denoising method exploiting neighbourhood dependencies and goodness-of-fit test

2018 ◽  
Vol 5 (9) ◽  
pp. 180436 ◽  
Author(s):  
Khuram Naveed ◽  
Bisma Shaukat ◽  
Naveed ur Rehman
Keyword(s):  
Wavelet Transform ◽  
Goodness Of Fit ◽  
Signal Denoising ◽  
Discrete Wavelet ◽  
Translation Invariance ◽  
Complex Wavelet Transform ◽  
Goodness Of Fit Test ◽  
Denoising Method ◽  
Complex Wavelet

A novel signal denoising method is proposed whereby goodness-of-fit (GOF) test in combination with a majority classifications-based neighbourhood filtering is employed on complex wavelet coefficients obtained by applying dual tree complex wavelet transform (DT-CWT) on a noisy signal. The DT-CWT has proven to be a better tool for signal denoising as compared to the conventional discrete wavelet transform (DWT) owing to its approximate translation invariance. The proposed framework exploits statistical neighbourhood dependencies by performing the GOF test locally on the DT-CWT coefficients for their preliminary classification/detection as signal or noise. Next, a deterministic neighbourhood filtering approach based on majority noise classifications is employed to detect false classification of signal coefficients as noise (via the GOF test) which are subsequently restored. The proposed method shows competitive performance against the state of the art in signal denoising.

PERFECT-TRANSLATION-INVARIANT CUSTOMIZABLE COMPLEX DISCRETE WAVELET TRANSFORM

2013 ◽  
Vol 11 (04) ◽  
pp. 1360003 ◽  
Author(s):  
HIROSHI TODA ◽  
ZHONG ZHANG ◽  
TAKASHI IMAMURA
Keyword(s):  
Wavelet Transform ◽  
Discrete Wavelet Transform ◽  
Frequency Domain ◽  
Wavelet Transforms ◽  
Wavelet Packet ◽  
Variable Density ◽  
Discrete Wavelet ◽  
Translation Invariance ◽  
Translation Invariant ◽  
High Degree

The theorems, giving the condition of perfect translation invariance for discrete wavelet transforms, have already been proven. Based on these theorems, the dual-tree complex discrete wavelet transform, the 2-dimensional discrete wavelet transform, the complex wavelet packet transform, the variable-density complex discrete wavelet transform and the real-valued discrete wavelet transform, having perfect translation invariance, were proposed. However, their customizability of wavelets in the frequency domain is limited. In this paper, also based on these theorems, a new type of complex discrete wavelet transform is proposed, which achieves perfect translation invariance with high degree of customizability of wavelets in the frequency domain.

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