# NAG Library Function Documentnag_mldwt (c09ccc)

## 1  Purpose

nag_mldwt (c09ccc) computes the one-dimensional multi-level discrete wavelet transform (DWT). The initialization function nag_wfilt (c09aac) must be called first to set up the DWT options.

## 2  Specification

#include <nag.h>
#include <nagc09.h>

void  nag_mldwt (Integer n, const double x[], Integer lenc, double c[], Integer nwl, Integer dwtlev[], Integer icomm[],  NagError *fail)

## 3  Description

nag_mldwt (c09ccc) computes the multi-level DWT of one-dimensional data. For a given wavelet and end extension method, nag_mldwt (c09ccc) will compute a multi-level transform of a data array, ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, using a specified number, ${n}_{l}$, of levels. The number of levels specified, ${n}_{l}$, must be no more than the value ${l}_{\mathrm{max}}$ returned in nwl by the initialization function nag_wfilt (c09aac) for the given problem. The transform is returned as a set of coefficients for the different levels (packed into a single array) and a representation of the multi-level structure.
The notation used here assigns level $0$ to the input dataset, $x$, with level $1$ being the first set of coefficients computed, with the detail coefficients, ${d}_{1}$, being stored while the approximation coefficients, ${a}_{1}$, are used as the input to a repeat of the wavelet transform. This process is continued until, at level ${n}_{l}$, both the detail coefficients, ${d}_{{n}_{l}}$, and the approximation coefficients, ${a}_{{n}_{l}}$ are retained. The output array, $C$, stores these sets of coefficients in reverse order, starting with ${a}_{{n}_{l}}$ followed by ${d}_{{n}_{l}},{d}_{{n}_{l}-1},\dots ,{d}_{1}$.

None.

## 5  Arguments

1:     n – IntegerInput
On entry: the number of elements, $n$, in the data array $x$.
Constraint: this must be the same as the value n passed to the initialization function nag_wfilt (c09aac).
2:     x[n] – const doubleInput
On entry: x contains the one-dimensional input dataset ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
3:     lenc – IntegerInput
On entry: the dimension of the array c. c must be large enough to contain the number, ${n}_{c}$, of wavelet coefficients. The maximum value of ${n}_{c}$ is returned in nwc by the call to the initialization function nag_wfilt (c09aac) and corresponds to the DWT being continued for the maximum number of levels possible for the given data set. When the number of levels, ${n}_{l}$, is chosen to be less than the maximum, then ${n}_{c}$ is correspondingly smaller and lenc can be reduced by noting that the number of coefficients at each level is given by $⌈\stackrel{-}{n}/2⌉$ for ${\mathbf{mode}}=\mathrm{Nag_Periodic}$ in nag_wfilt (c09aac) and $⌊\left(\stackrel{-}{n}+{n}_{f}-1\right)/2⌋$ for ${\mathbf{mode}}=\mathrm{Nag_HalfPointSymmetric}$, $\mathrm{Nag_WholePointSymmetric}$ or $\mathrm{Nag_ZeroPadded}$, where $\stackrel{-}{n}$ is the number of input data at that level and ${n}_{f}$ is the filter length provided by the call to nag_wfilt (c09aac). At the final level the storage is doubled to contain the set of approximation coefficients.
Constraint: ${\mathbf{lenc}}\ge {n}_{c}$, where ${n}_{c}$ is the number of approximation and detail coefficients that correspond to a transform with nwl levels.
4:     c[lenc] – doubleOutput
On exit: Let $q\left(\mathit{i}\right)$ denote the number of coefficients (of each type) produced by the wavelet transform at level $\mathit{i}$, for $\mathit{i}={n}_{l},{n}_{l}-1,\dots ,1$. These values are returned in dwtlev. Setting ${k}_{1}=q\left({n}_{l}\right)$ and ${k}_{\mathit{j}+1}={k}_{\mathit{j}}+q\left({n}_{l}-\mathit{j}+1\right)$, for $\mathit{j}=1,2,\dots ,{n}_{l}$, the coefficients are stored as follows:
${\mathbf{c}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{k}_{1}$
Contains the level ${n}_{l}$ approximation coefficients, ${a}_{{n}_{l}}$.
${\mathbf{c}}\left[\mathit{i}-1\right]$, for $\mathit{i}={k}_{1}+1,\dots ,{k}_{2}$
Contains the level ${n}_{l}$ detail coefficients ${d}_{{n}_{l}}$.
${\mathbf{c}}\left[\mathit{i}-1\right]$, for $\mathit{i}={k}_{j}+1,\dots ,{k}_{j+1}$
Contains the level ${n}_{l}-\mathit{j}+1$ detail coefficients, for $\mathit{j}=2,3,\dots ,{n}_{l}$.
5:     nwl – IntegerInput
On entry: the number of levels, ${n}_{l}$, in the multi-level resolution to be performed.
Constraint: $1\le {\mathbf{nwl}}\le {l}_{\mathrm{max}}$, where ${l}_{\mathrm{max}}$ is the value returned in nwl (the maximum number of levels) by the call to the initialization function nag_wfilt (c09aac).
6:     dwtlev[${\mathbf{nwl}}+1$] – IntegerOutput
On exit: the number of transform coefficients at each level. ${\mathbf{dwtlev}}\left[0\right]$ and ${\mathbf{dwtlev}}\left[1\right]$ contain the number, $q\left({n}_{l}\right)$, of approximation and detail coefficients respectively, for the final level of resolution (these are equal); ${\mathbf{dwtlev}}\left[\mathit{i}-1\right]$ contains the number of detail coefficients, $q\left({n}_{l}-\mathit{i}+2\right)$, for the (${n}_{l}-\mathit{i}+2$)th level, for $\mathit{i}=3,4,\dots ,{n}_{l}+1$.
7:     icomm[$100$] – IntegerCommunication Array
On entry: contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization function nag_wfilt (c09aac).
On exit: contains additional information on the computed transform.
8:     fail – NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_ARRAY_DIM_LEN
On entry, lenc is set too small: ${\mathbf{lenc}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lenc}}\ge 〈\mathit{\text{value}}〉$.
NE_BAD_PARAM
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INITIALIZATION
Either the initialization function has not been called first or array icomm has been corrupted.
Either the initialization function was called with ${\mathbf{wtrans}}=\mathrm{Nag_SingleLevel}$ or array icomm has been corrupted.
On entry, n is inconsistent with the value passed to the initialization function: ${\mathbf{n}}=〈\mathit{\text{value}}〉$, n should be $〈\mathit{\text{value}}〉$.
On entry, nwl is larger than the maximum number of levels returned by the initialization function: ${\mathbf{nwl}}=〈\mathit{\text{value}}〉$, maximum $\text{}=〈\mathit{\text{value}}〉$.
NE_INT
On entry, ${\mathbf{nwl}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nwl}}\ge 1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

## 7  Accuracy

The accuracy of the wavelet transform depends only on the floating point operations used in the convolution and downsampling and should thus be close to machine precision.

## 8  Further Comments

The wavelet coefficients at each level can be extracted from the output array c using the information contained in dwtlev on exit (see the descriptions of c and dwtlev in Section 5). For example, given an input data set, $x$, denoising can be carried out by applying a thresholding operation to the detail coefficients at every level. The elements ${\mathbf{C}}\left({k}_{1}+1:{k}_{{n}_{l}+1}\right)$, as described in Section 5, contain the detail coefficients, ${\stackrel{^}{d}}_{\mathit{i}\mathit{j}}$, for $\mathit{i}={n}_{l},{n}_{l}-1,\dots ,1$ and $\mathit{j}=1,2,\dots ,q\left(i\right)$, where ${\stackrel{^}{d}}_{ij}={d}_{ij}+\sigma {\epsilon }_{ij}$ and $\sigma {\epsilon }_{ij}$ is the transformed noise term. If some threshold parameter $\alpha $ is chosen, a simple hard thresholding rule can be applied as

$d- ij = 0, if ​ d^ij ≤ α d^ij , if ​ d^ij > α,$
taking ${\stackrel{-}{d}}_{ij}$ to be an approximation to the required detail coefficient without noise, ${d}_{ij}$. The resulting coefficients can then be used as input to nag_imldwt (c09cdc) in order to reconstruct the denoised signal.
See the references given in the introduction to this chapter for a more complete account of wavelet denoising and other applications.

## 9  Example

This example performs a multi-level resolution of a dataset using the Daubechies wavelet (see ${\mathbf{wavnam}}=\mathrm{Nag_Daubechies4}$ in nag_wfilt (c09aac)) using zero end extensions, the number of levels of resolution, the number of coefficients in each level and the coefficients themselves are reused. The original dataset is then reconstructed using nag_imldwt (c09cdc).

### 9.1  Program Text

Program Text (c09ccce.c)

### 9.2  Program Data

Program Data (c09ccce.d)

### 9.3  Program Results

Program Results (c09ccce.r)
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