A MATLAB Suite for Second Generation Wavelets on an Interval and the Corresponding Adaptive Grid
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A MATLAB Suite for Second Generation Wavelets on an Interval and the Corresponding Adaptive Grid Kavita Goyal1 · Deepika Sharma1
Received: 6 June 2019 / Accepted: 30 October 2019 © Springer Nature B.V. 2019
Abstract A collection of Matlab routines for the second generation wavelet transformation and inverse wavelet transformation on the space L2 ([a, b]) is presented. These wavelet transforms are further used for computing the wavelet function and scaling function values (ψ(x) and φ(x) respectively). At the end a procedure for generating an adaptive grid using second generation wavelet is provided. In the collection of the Matlab routines, we are providing three Matlab functions namely, Reconstruction_testing.m, AdaptiveGrid_standard_testing.m and AdaptiveGrid_modified_testing.m for directly testing the results claimed in the paper. Keywords Wavelets · Lifting scheme · Second generation wavelet
1 Introduction Wavelets form a very useful tool for representation of the data sets or general functions. More importantly they allow representations which are both faster and efficient to estimate. The set of traditional wavelets which we call as the first generation wavelets (e.g., Daubechies wavelet) consists of the wavelet and scaling functions which are dyadic translations and dilates of a single mother and father function respectively. Typical settings where a single function can not be translated and dilated incorporate 1. Construction of wavelets on the bounded regions: It involves the wavelets construction on a higher dimensional Euclidean bounded domain or an interval (such as, for the space L2 ([a, b])). 2. Construction of weighted wavelets: Biorthogonal wavelets for a weighted inner product (such as, for the space Lw 2 ([a, b])).
B D. Sharma
[email protected] K. Goyal [email protected]
1
School of Mathematics, Thapar Institute of Engineering and Technology, Patiala, India
K. Goyal, D. Sharma
To construct wavelets in all the above settings and many more the lifting method was proposed by Swelden [18, 45]. The prime idea, which inspired its name, is to begin with a trivial or very simple multiresolution analysis and gradually works one’s way up to a multiresolution analysis (MRA) with specific properties. The proposed method makes it possible to construct many discrete biorthogonal wavelets commencing from an original one. Moreover, with the lifting method all the first generation wavelets can also be constructed. The wavelets so constructed are termed as second generation wavelets. The second generation wavelets are not certainly translates and dilates of a single function, yet these wavelets take the advantage of all the useful features of the first generation wavelets. The third generation wavelets are also designed for the same purpose. Diffusion wavelets and spectral graph wavelets come under the category of third generation wavelets. These wavelets were constructed by Coifman [6] and Hammond [22] respectively. These wavelets are constructed in such a framework that these can be applied on gr
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