Orthonormal wavelet basis with arbitrary real dilation factor

2016 ◽  
Vol 14 (03) ◽  
pp. 1650010 ◽  
Author(s):  
Hiroshi Toda ◽  
Zhong Zhang
Keyword(s):  
Infinite Number ◽  
Wavelet Function ◽  
Good Time ◽  
Wavelet Basis ◽  
Orthonormal Wavelet ◽  
Time Frequency ◽  
Time Frequency Localization ◽  
Dilation Factor ◽  
New Type ◽  
Open Question

Daubechies posed the following problem in Ten Lectures on Wavelets (SIAM, Philadelphia, PA, 1992): “It is an open question whether there exist orthonormal wavelet bases (not necessarily associated with a multiresolution analysis), with good time-frequency localization, and with irrational [Formula: see text]” (that is, for an arbitrary irrational dilation factor [Formula: see text], with appropriate wavelet function [Formula: see text] and constant [Formula: see text], whether can [Formula: see text] construct an orthonormal wavelet basis with good time-frequency localization?). Our answer is “Yes”. In this paper, we introduce a new type of orthonormal wavelet basis having an arbitrary real dilation factor greater than 1. This orthonormal wavelet basis requires an infinite number of wavelet shapes when its dilation factor is irrational.

Orthonormal basis of wavelets with customizable frequency bands

2016 ◽  
Vol 14 (06) ◽  
pp. 1650050 ◽  
Author(s):  
Hiroshi Toda ◽  
Zhong Zhang
Keyword(s):  
Frequency Domain ◽  
Infinite Number ◽  
Orthonormal Basis ◽  
The Other ◽  
Just Intonation ◽  
Frequency Bands ◽  
Orthonormal Bases ◽  
Major Scale ◽  
Dilation Factor ◽  
New Type

We already proved the existence of an orthonormal basis of wavelets having an irrational dilation factor with an infinite number of wavelet shapes, and based on its theory, we proposed an orthonormal basis of wavelets with an arbitrary real dilation factor. In this paper, with the development of these fundamentals, we propose a new type of orthonormal basis of wavelets with customizable frequency bands. Its frequency bands can be freely designed with arbitrary bounds in the frequency domain. For example, we show two types of orthonormal bases of wavelets. One of them has an irrational dilation factor, and the other is designed based on the major scale in just intonation.

Surface Roughness Assessment of Micro-EDM by Wavelet Analysis Based on Goodness of Fit Test

2010 ◽  
Vol 450 ◽  
pp. 304-307 ◽  
Author(s):  
Zhi Jie Chen ◽  
Ji Hong Shen ◽  
Li Bin Guo
Keyword(s):  
Surface Roughness ◽  
Surface Topography ◽  
Goodness Of Fit ◽  
Good Time ◽  
Micro Edm ◽  
Goodness Of Fit Test ◽  
Chi Square ◽  
Time Frequency ◽  
Time Frequency Localization ◽  
Roughness Assessment

The surface topography errors of Micro-EDM are mainly composed of surface roughness, surface waveness and so on, which have influence on the workpiece’s functions and performances in various degrees. Thus, how to pick up these errors without distortion is quite important for evaluating the surface topography. It is well known that Wavelet Transform has good time-frequency localization properties, which is especially suitable for image processing. Therefore, in this paper, a wavelet method for surface roughness of Micro-EDM is presented, in which, the time of wavelet decomposition is determined by Pearson Chi-square goodness of fit test. Furthermore, the results of simulation and experiments show that the proposed methods can separate Micro-EDM’s surface roughness well, meanwhile, using the mean and variance of ideal surface as accuracy requirements of the reference surface could help the separation have a certain certainty and precision.

scholarly journals An innovative image fusion algorithm based on wavelet transform and discrete fast curvelet transform

2011 ◽  
Vol 1 (3) ◽  
Author(s):  
T. Sumathi ◽  
M. Hemalatha
Keyword(s):  
Wavelet Transform ◽  
Image Fusion ◽  
Curvelet Transform ◽  
Relevant Information ◽  
Good Time ◽  
Discrete Wavelet ◽  
Fusion Algorithm ◽  
Time Frequency ◽  
Fused Image

AbstractImage fusion is the method of combining relevant information from two or more images into a single image resulting in an image that is more informative than the initial inputs. Methods for fusion include discrete wavelet transform, Laplacian pyramid based transform, curvelet based transform etc. These methods demonstrate the best performance in spatial and spectral quality of the fused image compared to other spatial methods of fusion. In particular, wavelet transform has good time-frequency characteristics. However, this characteristic cannot be extended easily to two or more dimensions with separable wavelet experiencing limited directivity when spanning a one-dimensional wavelet. This paper introduces the second generation curvelet transform and uses it to fuse images together. This method is compared against the others previously described to show that useful information can be extracted from source and fused images resulting in the production of fused images which offer clear, detailed information.

Detection of the Third Heart Sound Based on Nonlinear Signal Decomposition and Time–Frequency Localization

2016 ◽  
Vol 63 (8) ◽  
pp. 1718-1727 ◽  
Author(s):  
Shovan Barma ◽  
Bo-Wei Chen ◽  
Wen Ji ◽  
Seungmin Rho ◽  
Chih-Hung Chou ◽  
...  
Keyword(s):  
Heart Sound ◽  
Signal Decomposition ◽  
Time Frequency ◽  
The Third ◽  
Third Heart Sound ◽  
Time Frequency Localization ◽  
Nonlinear Signal
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