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.

scholarly journals Non-equispaced B-spline wavelets

2016 ◽  
Vol 14 (06) ◽  
pp. 1650056 ◽  
Author(s):  
Maarten Jansen
Keyword(s):  
Wavelet Transforms ◽  
Orthogonal Wavelet ◽  
B Spline ◽  
Wavelet Representation ◽  
Spline Wavelets ◽  
New Construction ◽  
Irregular Point ◽  
Interpolating Wavelets ◽  
New Framework ◽  
Orthogonal Wavelet Transform

This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen–Daubechies–Feauveau wavelets. The new construction is based on the factorization of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non-orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet transforms on irregular point sets with a focus on their use for smoothing or other applications in statistics.

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.

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.

scholarly journals Orthogonal wavelet frames and vector-valued wavelet transforms

2007 ◽  
Vol 23 (2) ◽  
pp. 215-234 ◽  
Author(s):  
Ghanshyam Bhatt ◽  
Brody Dylan Johnson ◽  
Eric Weber
Keyword(s):  
Wavelet Transforms ◽  
Wavelet Frames ◽  
Orthogonal Wavelet ◽  
Vector Valued

scholarly journals Performance Comparison of Wavelets Generated from Four Different Orthogonal Transforms for Watermarking With Various Attacks

2013 ◽  
Vol 9 (3) ◽  
pp. 1139-1152 ◽  
Author(s):  
H. B. Kekre ◽  
Tanuja Sarode ◽  
Shachi Natu
Keyword(s):  
Wavelet Transform ◽  
Wavelet Transforms ◽  
Image Watermarking ◽  
Histogram Equalization ◽  
Performance Comparison ◽  
Orthogonal Wavelet ◽  
Good Comparison ◽  
Orthogonal Transform ◽  
Contrast Stretching ◽  
Orthogonal Wavelet Transform

This paper proposes a watermarking technique using different orthogonal wavelet transforms like Hartley wavelet, Kekrewavelet, Slant wavelet and Real Fourier wavelet transform generated from corresponding orthogonal transform. Theseorthogonal wavelet transforms have been generated using different sizes of component orthogonal transform matrices.For example 256*256 size orthogonal wavelet transform can be generated using 128*128 and 2*2 size componentorthogonal transform. It can also be generated using 64*64 and 4*4, 32*32 and 8*8, 16*16 and 16*16 size componentorthogonal transform matrices. In this paper the focus is to compare the performance of above mentioned transformsgenerated using 128*128 and 2*2 size component orthogonal transform and 64*64 and 4*4 size component orthogonaltransform in digital image watermarking. The other two combinations are not considered as their performance iscomparatively not as good. Comparison shows that wavelet transforms generated using (128,2) combination of orthogonal transform give better performances than wavelet transforms generated using (64,4) combination of orthogonaltransformfor contrast stretching, cropping, Gaussian noise, histogram equalization and resizing attacks. Real Fourierwavelet and Slant wavelet prove to be better for histogram equalization and resizing attack respectively than DCT waveletand Walsh wavelet based watermarking presented in previous work.

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