library.wav Submodule

Module Summary

Interfaces for the NAG Mark 30.3 wav Chapter.

wav - Wavelet Transforms

This module is concerned with the analysis of datasets (or functions or operators) in terms of frequency and scale components using wavelet transforms. Wavelet transforms have been applied in many fields from time series analysis to image processing and the localization in either frequency or scale that they provide is useful for data compression or denoising. In general the standard wavelet transform uses dilation and scaling of a chosen function, $\psi \left(t\right)$, (called the mother wavelet) such that

$${\psi }_{a,b}\left(t\right)=\frac{1}{\sqrt{a}}\psi \left(\frac{t-b}{a}\right)$$

where $a$ gives the scaling and $b$ determines the translation. Wavelet methods can be divided into continuous transforms and discrete transforms. In the continuous case, the pair $a$ and $b$ are real numbers with $a>0$. For the discrete transform, $a$ and $b$ can be chosen as $a=2^{-j}$, $b=k{2}^{-j}$ for integers $j$, $k$

$${\psi }_{j,k}\left(t\right)=2^{j/2}\psi (2^{j}t-k)\text{.}$$

The continuous real valued, one-dimensional wavelet transform (CWT) is included in this module. The discrete wavelet transform (DWT) at a single level together with its inverse and the multi-level DWT with inverse are also provided for one, two and three dimensions. The Maximal Overlap DWT (MODWT) together with its inverse and the multi-level MODWT with inverse are provided for one dimension. The choice of wavelet for CWT includes the Morlet wavelet and derivatives of a Gaussian while the DWT and MODWT offer the orthogonal wavelets of Daubechies and a selection of biorthogonal wavelets.

Functionality Index

One-dimensional

continuous

real wavelet transform: dim1_cont()

discrete

multi-level

inverse wavelet transform: dim1_multi_inv()

wavelet transform: dim1_multi_fwd()

single level

inverse wavelet transform: dim1_sngl_inv()

wavelet transform: dim1_sngl_fwd()

maximal overlap discrete

multi-level

inverse wavelet transform: dim1_mxolap_multi_inv()

wavelet transform: dim1_mxolap_multi_fwd()

single level

inverse wavelet transform: dim1_mxolap_inv()

wavelet transform: dim1_mxolap_fwd()

wavelet filter details: dim1_init()

Three-dimensional

coefficient extraction: dim3_coeff_ext()

coefficient insertion: dim3_coeff_ins()

discrete

multi-level

inverse wavelet transform: dim3_mxolap_multi_inv()

wavelet transform: dim3_multi_fwd()

single level

inverse wavelet transform: dim3_sngl_inv()

wavelet transform: dim3_sngl_fwd()

wavelet filter details: dim3_init()

Two-dimensional

coefficient extraction: dim2_coeff_ext()

coefficient insertion: dim2_coeff_ins()

discrete

multi-level

inverse wavelet transform: dim2_multi_inv()

wavelet transform: dim2_multi_fwd()

single level

inverse wavelet transform: dim2_sngl_inv()

wavelet transform: dim2_sngl_fwd()

wavelet filter details: dim2_init()

For full information please refer to the NAG Library document

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/c09/c09intro.html