Orthogonal wavelet transforms and filter banks

2003 ◽  
Author(s):  
G. Evangelista
Keyword(s):  
Wavelet Transforms ◽  
Filter Banks ◽  
Orthogonal Wavelet

REFINEMENT INDEPENDENT WAVELETS FOR USE IN ADAPTIVE MULTIRESOLUTION SCHEMES

2008 ◽  
Vol 06 (04) ◽  
pp. 521-539 ◽  
Author(s):  
MAARTEN JANSEN
Keyword(s):  
Wavelet Transforms ◽  
Point Sets ◽  
Orthogonal Wavelet ◽  
Multiscale Decomposition ◽  
Irregular Point ◽  
Multiresolution Schemes ◽  
Linear Interpolating

This paper constructs a class of semi-orthogonal and bi-orthogonal wavelet transforms on possibly irregular point sets with the property that the scaling coefficients are independent from the order of refinement. That means that scaling coefficients at a given scale can be constructed with the configuration at that scale only. This property is of particular interest when the refinement operation is data dependent, leading to adaptive multiresolution analyses. Moreover, the proposed class of wavelet transforms are constructed using a sequence of just two lifting steps, one of which contains a linear interpolating prediction operator. This operator easily allows extensions towards directional offsets from predictions, leading to an edge-adaptive nonlinear multiscale decomposition.

Pattern Changes of Time-Shifted Vibration Signals on Wavelet Time-Scale Maps

1995 ◽  
Author(s):  
Da Jun Chen ◽  
Wei Ji Wang
Keyword(s):  
Wavelet Transform ◽  
Time Scale ◽  
Wavelet Transforms ◽  
Vibration Signal ◽  
Continuous Wavelet ◽  
Time Axis ◽  
Orthogonal Wavelet ◽  
Signal Component ◽  
Analysis Technique ◽  
Map Analysis

Abstract As a multi-resolution signal decomposition and analysis technique, the wavelet transforms have been already introduced to vibration signal processing. In this paper, a comparison on the time-scale map analysis is made between the discrete and the continuous wavelet transform. The orthogonal wavelet transform decomposes the vibration signal onto a series of orthogonal wavelet functions and the number of wavelets on one wavelet level is different from those on the other levels. Since the grids are unevenly distributed on the time-scale map, it is shown that a representation pattern of a vibration component on the map may be significantly altered or even be broken down into pieces when the signal has a shift along the time axis. On contrary, there is no such uneven distribution of grids on the continuous wavelet time-scale map, so that the representation pattern of a vibration signal component will not change its shape when the signal component shifts along the time axis. Therefore, the patterns in the continuous wavelet time-scale map are more easily recognised by human visual inspection or computerised automatic diagnosis systems. Using a Gaussian enveloped oscillation wavelet, the wavelet transform is capable of retaining the frequency meaning used in the spectral analysis, while making the interpretation of patterns on the time-scale maps easier.

Perfect-translation-invariant variable-density complex discrete wavelet transform

2014 ◽  
Vol 12 (04) ◽  
pp. 1460001 ◽  
Author(s):  
Hiroshi Toda ◽  
Zhong Zhang ◽  
Takashi Imamura
Keyword(s):  
Frequency Domain ◽  
Frequency Band ◽  
Time Domain ◽  
Wavelet Transforms ◽  
Filter Banks ◽  
Wavelet Packet ◽  
Discrete Wavelet ◽  
Translation Invariance ◽  
Logarithmic Scale ◽  
The Time Domain

The theorems giving the conditions for discrete wavelet transforms (DWTs) to achieve perfect translation invariance (PTI) have already been proven, and based on these theorems, the dual-tree complex DWT and the complex wavelet packet transform, achieving PTI, have already been proposed. However, there is not so much flexibility in their wavelet density. In the frequency domain, the wavelet density is fixed by octave filter banks, and in the time domain, each wavelet is arrayed on a fixed coordinate, and the wavelet packet density in the frequency domain can be only designed by dividing an octave frequency band equally in linear scale, and its density in the time domain is constrained by the division number of an octave frequency band. In this paper, a novel complex DWT is proposed to create variable wavelet density in the frequency and time domains, that is, an octave frequency band can be divided into N filter banks in logarithmic scale, where N is an integer larger than or equal to 3, and in the time domain, a distance between wavelets can be varied in each level, and its transform achieves PTI.

Wavelet Transforms and Multirate Filtering

2002 ◽  
pp. 86-104 ◽  
Author(s):  
Raghuveer Rao
Keyword(s):  
Image Processing ◽  
Wavelet Transform ◽  
Wavelet Transforms ◽  
Wavelet Transformation ◽  
Discrete Wavelet ◽  
Orthogonal Wavelet ◽  
Seminal Paper ◽  
Wavelet Expansions ◽  
Square Integrable Function ◽  
The Relationship

One of the most fascinating developments in the field of multirate signal processing has been the establishment of its link to the discrete wavelet transform. Indeed, it is precisely this link that has been responsible for the rapid application of wavelets in fields such as image compression. The objective of this chapter is to provide an overview of the wavelet transform and develop its link to multirate filtering. The birth of the field of wavelet transforms is now attributed to the seminal paper by Grossman and Morlet (1984) detailing the continuous wavelet transform or CWT. The CWT of a square integrable function is obtained by integrating it over regions defined by translations and dilations of a windowing function called the mother wavelet. The idea of representing functions or signals in terms of dilations can be found even in engineering articles dating back by several years, for example, Helstrom (1966). However, Grossman and Morlet’s formulation was more complete and was motivated by potential application to modeling seismic data. The next step of significance was the discovery of orthogonal wavelet basis functions and their role in defining multi-resolution representations (Daubechies 1988; Meyer 1992). Daubechies also provided a method for constructing compactly supported wavelets. Mallat (1989) established the fact that coefficients of orthogonal wavelet expansions can be obtained through multirate filtering which paved the way for widespread investigation of using wavelet transforms in signal and image processing applications. The objective of the chapter is to provide an overview of the relationship between multirate filtering and wavelet transformation. We begin with a brief account of the CWT, then go through the discrete wavelet transformation (DWT) followed by derivation of the relationship between the DWT and multirate filtering. The chapter concludes with an account of selected applications in digital image processing.

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