NAG CL Interface
c09ecc (dim2_multi_fwd)
1
Purpose
c09ecc computes the twodimensional multilevel discrete wavelet transform (DWT). The initialization function
c09abc must be called first to set up the DWT options.
2
Specification
void 
c09ecc (Integer m,
Integer n,
const double a[],
Integer lda,
Integer lenc,
double c[],
Integer nwl,
Integer dwtlvm[],
Integer dwtlvn[],
Integer icomm[],
NagError *fail) 

The function may be called by the names: c09ecc, nag_wav_dim2_multi_fwd or nag_mldwt_2d.
3
Description
c09ecc computes the multilevel DWT of twodimensional data. For a given wavelet and end extension method,
c09ecc will compute a multilevel 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
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 multilevel 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 ${v}_{{n}_{\mathrm{fwd}}},{h}_{{n}_{\mathrm{fwd}}},{d}_{{n}_{\mathrm{fwd}}},{v}_{{n}_{\mathrm{fwd}}1},{h}_{{n}_{\mathrm{fwd}}1},{d}_{{n}_{\mathrm{fwd}}1},\dots ,{v}_{1},{h}_{1},{d}_{1}$.
4
References
None.
5
Arguments

1:
$\mathbf{m}$ – Integer
Input

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
c09abc.

2:
$\mathbf{n}$ – Integer
Input

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
c09abc.

3:
$\mathbf{a}\left[{\mathbf{lda}}\times {\mathbf{n}}\right]$ – const double
Input

Note: the $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(j1\right)\times {\mathbf{lda}}+i1\right]$.
On entry: the $m$ by $n$ data matrix $A$.

4:
$\mathbf{lda}$ – Integer
Input

On entry: the stride separating matrix row elements in the array
a.
Constraint:
${\mathbf{lda}}\ge {\mathbf{m}}$.

5:
$\mathbf{lenc}$ – Integer
Input

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
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
$3\times \u2308\overline{m}/2\u2309\times \u2308\overline{n}/2\u2309$ for
${\mathbf{mode}}=\mathrm{Nag\_Periodic}$ in
c09abc and
$3\times \u230a\left(\overline{m}+{n}_{f}1\right)/2\u230b\times \u230a\left(\overline{n}+{n}_{f}1\right)/2\u230b$ for
${\mathbf{mode}}=\mathrm{Nag\_HalfPointSymmetric}$,
$\mathrm{Nag\_WholePointSymmetric}$ or
$\mathrm{Nag\_ZeroPadded}$, where the input data is of dimension
$\overline{m}\times \overline{n}$ at that level and
${n}_{f}$ is the filter length
nf provided by the call to
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:
$\mathbf{c}\left[{\mathbf{lenc}}\right]$ – double
Output

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
c09eyc or
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
$q\left(i\right)={\mathbf{dwtlvm}}\left[{n}_{\mathrm{fwd}}i\right]\times {\mathbf{dwtlvn}}\left[{n}_{\mathrm{fwd}}i\right]$. Then, letting
${k}_{1}=q\left({n}_{\mathrm{fwd}}\right)$ and
${k}_{\mathit{j}+1}={k}_{\mathit{j}}+q\left({n}_{\mathrm{fwd}}\u2308\mathit{j}/3\u2309+1\right)$, 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 ${n}_{\mathrm{fwd}}\u2308j/3\u2309+1$ vertical, horizontal and diagonal coefficients. These are:
 vertical coefficients if $j\mathrm{mod}3=1$;
 horizontal coefficients if $j\mathrm{mod}3=2$;
 diagonal coefficients if $j\mathrm{mod}3=0$,
for $j=1,\dots ,3{n}_{\mathrm{fwd}}$.

7:
$\mathbf{nwl}$ – Integer
Input

On entry: the number of levels, ${n}_{\mathrm{fwd}}$, in the multilevel 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
c09abc.

8:
$\mathbf{dwtlvm}\left[{\mathbf{nwl}}\right]$ – Integer
Output

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:
$\mathbf{dwtlvn}\left[{\mathbf{nwl}}\right]$ – Integer
Output

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:
$\mathbf{icomm}\left[180\right]$ – Integer
Communication Array

On entry: contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization function
c09abc.
On exit: contains additional information on the computed transform.

11:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ 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,
${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint:
${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$, the value of
m on initialization (see
c09abc).
On entry,
${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint:
${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$, the value of
n on initialization (see
c09abc).
On entry, ${\mathbf{nwl}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nwl}}\ge 1$.
 NE_INT_2

On entry, ${\mathbf{lda}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{lenc}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{lenc}}\ge \u2329\mathit{\text{value}}\u232a$, the total number of coefficents to be generated.
On entry,
${\mathbf{nwl}}=\u2329\mathit{\text{value}}\u232a$ and
${\mathbf{nwlmax}}=\u2329\mathit{\text{value}}\u232a$ in
c09abc.
Constraint:
${\mathbf{nwl}}\le {\mathbf{nwlmax}}$ in
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.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
7
Accuracy
The accuracy of the wavelet transform depends only on the floatingpoint operations used in the convolution and downsampling and should thus be close to machine precision.
8
Parallelism and Performance
c09ecc is not threaded in any implementation.
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,
${\hat{c}}_{ij}$, 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
${\hat{c}}_{ij}={c}_{ij}+\sigma {\epsilon}_{ij}$ 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
taking
${\overline{c}}_{ij}$ to be an approximation to the required detail coefficient without noise,
${c}_{ij}$. The resulting coefficients can then be used as input to
c09edc in order to reconstruct the denoised signal. See
Section 10 in
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 multilevel resolution transform of a dataset using the Daubechies wavelet (see
${\mathbf{wavnam}}=\mathrm{Nag\_Daubechies2}$ in
c09abc) using halfpoint 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
c09edc.
10.1
Program Text
10.2
Program Data
10.3
Program Results