c09 Chapter Contents
c09 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_mldwt_2d (c09ecc)

## 1  Purpose

nag_mldwt_2d (c09ecc) computes the two-dimensional multi-level discrete wavelet transform (DWT). The initialization function nag_wfilt_2d (c09abc) must be called first to set up the DWT options.

## 2  Specification

 #include #include
 void nag_mldwt_2d (Integer m, Integer n, const double a[], Integer lda, Integer lenc, double c[], Integer nwl, Integer dwtlvm[], Integer dwtlvn[], Integer icomm[], NagError *fail)

## 3  Description

nag_mldwt_2d (c09ecc) computes the multi-level DWT of two-dimensional data. For a given wavelet and end extension method, nag_mldwt_2d (c09ecc) will compute a multi-level transform of a matrix $A$, using a specified number, ${n}_{\mathrm{fwd}}$, of levels. The number of levels specified, ${n}_{\mathrm{fwd}}$, must be no more than the value ${l}_{\mathrm{max}}$ returned in nwlmax by the initialization function nag_wfilt_2d (c09abc) 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 matrix, $A$. Level 1 consists of the first set of coefficients computed: the vertical (${v}_{1}$), horizontal (${h}_{1}$) and diagonal (${d}_{1}$) coefficients are stored at this level while the approximation (${a}_{1}$) coefficients are used as the input to a repeat of the wavelet transform at the next level. This process is continued until, at level ${n}_{\mathrm{fwd}}$, all four types of coefficients are stored. The output array, $C$, stores these sets of coefficients in reverse order, starting with ${a}_{{n}_{\mathrm{fwd}}}$ followed by ${v}_{{n}_{\mathrm{fwd}}},{h}_{{n}_{\mathrm{fwd}}},{d}_{{n}_{\mathrm{fwd}}},{v}_{{n}_{\mathrm{fwd}}-1},{h}_{{n}_{\mathrm{fwd}}-1},{d}_{{n}_{\mathrm{fwd}}-1},\dots ,{v}_{1},{h}_{1},{d}_{1}$.

None.

## 5  Arguments

1:     mIntegerInput
On entry: number of rows, $m$, of data matrix $A$.
Constraint: this must be the same as the value m passed to the initialization function nag_wfilt_2d (c09abc).
2:     nIntegerInput
On entry: number of columns, $n$, of data matrix $A$.
Constraint: this must be the same as the value n passed to the initialization function nag_wfilt_2d (c09abc).
3:     a[${\mathbf{lda}}×{\mathbf{n}}$]const doubleInput
Note: the $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{lda}}+i-1\right]$.
On entry: the $m$ by $n$ data matrix $A$.
4:     ldaIntegerInput
On entry: the stride separating matrix row elements in the array a.
Constraint: ${\mathbf{lda}}\ge {\mathbf{m}}$.
5:     lencIntegerInput
On entry: the dimension of the array c. c must be large enough to contain, ${n}_{\mathrm{ct}}$, wavelet coefficients. The maximum value of ${n}_{\mathrm{ct}}$ is returned in nwct by the call to the initialization function nag_wfilt_2d (c09abc) and corresponds to the DWT being continued for the maximum number of levels possible for the given data set. When the number of levels, ${n}_{\mathrm{fwd}}$, is chosen to be less than the maximum, ${l}_{\mathrm{max}}$, then ${n}_{\mathrm{ct}}$ is correspondingly smaller and lenc can be reduced by noting that the vertical, horizontal and diagonal coefficients are stored at every level and that in addition the approximation coefficients are stored for the final level only. The number of coefficients stored at each level is given by $3×⌈\stackrel{-}{m}/2⌉×⌈\stackrel{-}{n}/2⌉$ for ${\mathbf{mode}}=\mathrm{Nag_Periodic}$ in nag_wfilt_2d (c09abc) and $3×⌊\left(\stackrel{-}{m}+{n}_{f}-1\right)/2⌋×⌊\left(\stackrel{-}{n}+{n}_{f}-1\right)/2⌋$ for ${\mathbf{mode}}=\mathrm{Nag_HalfPointSymmetric}$, $\mathrm{Nag_WholePointSymmetric}$ or $\mathrm{Nag_ZeroPadded}$, where the input data is of dimension $\stackrel{-}{m}×\stackrel{-}{n}$ at that level and ${n}_{f}$ is the filter length nf provided by the call to nag_wfilt_2d (c09abc). At the final level the storage is $4/3$ times this value to contain the set of approximation coefficients.
Constraint: ${\mathbf{lenc}}\ge {n}_{\mathrm{ct}}$, where ${n}_{\mathrm{ct}}$ is the total number of coefficients that correspond to a transform with nwl levels.
6:     c[lenc]doubleOutput
On exit: the coefficients of the discrete wavelet transform. If you need to access or modify the approximation coefficients or any specific set of detail coefficients then the use of nag_wav_2d_coeff_ext (c09eyc) or nag_wav_2d_coeff_ins (c09ezc) is recommended. For completeness the following description provides details of precisely how the coefficient are stored in c but this information should only be required in rare cases.
Let $q\left(\mathit{i}\right)$ denote the number of coefficients (of each type) at level $\mathit{i}$, for $\mathit{i}=1,2,\dots ,{n}_{\mathrm{fwd}}$, such that $q\left(i\right)={\mathbf{dwtlvm}}\left[{n}_{\mathrm{fwd}}-i\right]×{\mathbf{dwtlvn}}\left[{n}_{\mathrm{fwd}}-i\right]$. Then, letting ${k}_{1}=q\left({n}_{\mathrm{fwd}}\right)$ and ${k}_{\mathit{j}+1}={k}_{\mathit{j}}+q\left({n}_{\mathrm{fwd}}-⌈\mathit{j}/3⌉+1\right)$, for $\mathit{j}=1,2,\dots ,3{n}_{\mathrm{fwd}}$, the coefficients are stored in c as follows:
${\mathbf{c}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{k}_{1}$
Contains the level ${n}_{\mathrm{fwd}}$ approximation coefficients, ${a}_{{n}_{\mathrm{fwd}}}$.
${\mathbf{c}}\left[\mathit{i}-1\right]$, for $\mathit{i}={k}_{j}+1,\dots ,{k}_{j+1}$
Contains the level ${n}_{\mathrm{fwd}}-⌈j/3⌉+1$ vertical, horizontal and diagonal coefficients. These are:
• vertical coefficients if ;
• horizontal coefficients if ;
• diagonal coefficients if ,
for $j=1,\dots ,3{n}_{\mathrm{fwd}}$.
7:     nwlIntegerInput
On entry: the number of levels, ${n}_{\mathrm{fwd}}$, 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 nwlmax (the maximum number of levels) by the call to the initialization function nag_wfilt_2d (c09abc).
8:     dwtlvm[nwl]IntegerOutput
On exit: the number of coefficients in the first dimension for each coefficient type at each level. ${\mathbf{dwtlvm}}\left[\mathit{i}-1\right]$ contains the number of coefficients in the first dimension (for each coefficient type computed) at the (${n}_{\mathrm{fwd}}-\mathit{i}+1$)th level of resolution, for $\mathit{i}=1,2,\dots ,{n}_{\mathrm{fwd}}$. Thus for the first ${n}_{\mathrm{fwd}}-1$ levels of resolution, ${\mathbf{dwtlvm}}\left[{n}_{\mathrm{fwd}}-\mathit{i}\right]$ is the size of the first dimension of the matrices of vertical, horizontal and diagonal coefficients computed at this level; for the final level of resolution, ${\mathbf{dwtlvm}}\left[0\right]$ is the size of the first dimension of the matrices of approximation, vertical, horizontal and diagonal coefficients computed.
9:     dwtlvn[nwl]IntegerOutput
On exit: the number of coefficients in the second dimension for each coefficient type at each level. ${\mathbf{dwtlvn}}\left[\mathit{i}-1\right]$ contains the number of coefficients in the second dimension (for each coefficient type computed) at the (${n}_{\mathrm{fwd}}-\mathit{i}+1$)th level of resolution, for $\mathit{i}=1,2,\dots ,{n}_{\mathrm{fwd}}$. Thus for the first ${n}_{\mathrm{fwd}}-1$ levels of resolution, ${\mathbf{dwtlvn}}\left[{n}_{\mathrm{fwd}}-\mathit{i}\right]$ is the size of the second dimension of the matrices of vertical, horizontal and diagonal coefficients computed at this level; for the final level of resolution, ${\mathbf{dwtlvn}}\left[0\right]$ is the size of the second dimension of the matrices of approximation, vertical, horizontal and diagonal coefficients computed.
10:   icomm[$180$]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_2d (c09abc).
On exit: contains additional information on the computed transform.
11:   failNagError *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.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INITIALIZATION
Either the initialization function has not been called first or icomm has been corrupted.
Either the initialization function was called with ${\mathbf{wtrans}}=\mathrm{Nag_SingleLevel}$ or icomm has been corrupted.
NE_INT
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$, the value of m on initialization (see nag_wfilt_2d (c09abc)).
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$, the value of n on initialization (see nag_wfilt_2d (c09abc)).
On entry, ${\mathbf{nwl}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nwl}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{lda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{lenc}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lenc}}\ge ⟨\mathit{\text{value}}⟩$, the total number of coefficents to be generated.
On entry, ${\mathbf{nwl}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nwlmax}}=⟨\mathit{\text{value}}⟩$ in nag_wfilt_2d (c09abc).
Constraint: ${\mathbf{nwl}}\le {\mathbf{nwlmax}}$ in nag_wfilt_2d (c09abc).
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  Parallelism and Performance

Not applicable.

The wavelet coefficients at each level can be extracted from the output array c using the information contained in dwtlvm and dwtlvn on exit (see the descriptions of c, dwtlvm and dwtlvn in Section 5). For example, given an input data set, $A$, denoising can be carried out by applying a thresholding operation to the detail (vertical, horizontal and diagonal) coefficients at every level. The elements ${\mathbf{c}}\left[{k}_{1}\right]$ to ${\mathbf{c}}\left[{k}_{{n}_{\mathrm{fwd}}+1}-1\right]$, as described in Section 5, contain the detail coefficients, ${\stackrel{^}{c}}_{ij}$, for $\mathit{i}={n}_{\mathrm{fwd}},{n}_{\mathrm{fwd}}-1,\dots ,1$ and $\mathit{j}=1,2,\dots ,3q\left(i\right)$, where $q\left(i\right)$ is the number of each type of coefficient at level $i$ and ${\stackrel{^}{c}}_{ij}={c}_{ij}+\sigma {\epsilon }_{ij}$ 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
 $c- ij = 0, if ​ c^ij ≤ α c^ij , if ​ c^ij > α,$
taking ${\stackrel{-}{c}}_{ij}$ to be an approximation to the required detail coefficient without noise, ${c}_{ij}$. The resulting coefficients can then be used as input to nag_imldwt_2d (c09edc) in order to reconstruct the denoised signal. See Section 10 in nag_wav_2d_coeff_ins (c09ezc) for a simple example of denoising.
See the references given in the introduction to this chapter for a more complete account of wavelet denoising and other applications.

## 10  Example

This example performs a multi-level resolution transform of a dataset using the Daubechies wavelet (see ${\mathbf{wavnam}}=\mathrm{Nag_Daubechies2}$ in nag_wfilt_2d (c09abc)) using half-point symmetric end extensions, the maximum possible number of levels of resolution, where the number of coefficients in each level and the coefficients themselves are not changed. The original dataset is then reconstructed using nag_imldwt_2d (c09edc).

### 10.1  Program Text

Program Text (c09ecce.c)

### 10.2  Program Data

Program Data (c09ecce.d)

### 10.3  Program Results

Program Results (c09ecce.r)