NAG FL Interface
c09ecf (dim2_multi_fwd)
1
Purpose
c09ecf computes the twodimensional multilevel discrete wavelet transform (DWT). The initialization routine
c09abf must be called first to set up the DWT options.
2
Specification
Fortran Interface
Subroutine c09ecf ( 
m, n, a, lda, lenc, c, nwl, dwtlvm, dwtlvn, icomm, ifail) 
Integer, Intent (In) 
:: 
m, n, lda, lenc, nwl 
Integer, Intent (Inout) 
:: 
icomm(180), ifail 
Integer, Intent (Out) 
:: 
dwtlvm(nwl), dwtlvn(nwl) 
Real (Kind=nag_wp), Intent (In) 
:: 
a(lda,n) 
Real (Kind=nag_wp), Intent (Inout) 
:: 
c(lenc) 

C Header Interface
#include <nag.h>
void 
c09ecf_ (const Integer *m, const Integer *n, const double a[], const Integer *lda, const Integer *lenc, double c[], const Integer *nwl, Integer dwtlvm[], Integer dwtlvn[], Integer icomm[], Integer *ifail) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
c09ecf_ (const Integer &m, const Integer &n, const double a[], const Integer &lda, const Integer &lenc, double c[], const Integer &nwl, Integer dwtlvm[], Integer dwtlvn[], Integer icomm[], Integer &ifail) 
}

The routine may be called by the names c09ecf or nagf_wav_dim2_multi_fwd.
3
Description
c09ecf computes the multilevel DWT of twodimensional data. For a given wavelet and end extension method,
c09ecf 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 routine
c09abf 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 routine
c09abf.

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 routine
c09abf.

3:
$\mathbf{a}\left({\mathbf{lda}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) array
Input

On entry: the $m$ by $n$ data matrix $A$.

4:
$\mathbf{lda}$ – Integer
Input

On entry: the first dimension of the array
a as declared in the (sub)program from which
c09ecf is called.
Constraint:
${\mathbf{lda}}\ge {\mathbf{m}}$.

5:
$\mathbf{lenc}$ – Integer
Input

On entry: the dimension of the array
c as declared in the (sub)program from which
c09ecf is called.
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 routine
c09abf 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
${\mathbf{mode}}=\text{'P'}$ in
c09abf 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
${\mathbf{mode}}=\text{'H'}$,
$\text{'W'}$ or
$\text{'Z'}$, 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
c09abf. 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:
$\mathbf{c}\left({\mathbf{lenc}}\right)$ – Real (Kind=nag_wp) array
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
c09eyf or
c09ezf 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
$q\left(i\right)={\mathbf{dwtlvm}}\left({n}_{\mathrm{fwd}}i+1\right)\times {\mathbf{dwtlvn}}\left({n}_{\mathrm{fwd}}i+1\right)$. 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:
 ${\mathbf{c}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{k}_{1}$
 Contains the level ${n}_{\mathrm{fwd}}$ approximation coefficients, ${a}_{{n}_{\mathrm{fwd}}}$.
 ${\mathbf{c}}\left(\mathit{i}\right)$, 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 routine
c09abf.

8:
$\mathbf{dwtlvm}\left({\mathbf{nwl}}\right)$ – Integer array
Output

On exit: the number of coefficients in the first dimension for each coefficient type at each level.
${\mathbf{dwtlvm}}\left(\mathit{i}\right)$ 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, ${\mathbf{dwtlvm}}\left({n}_{\mathrm{fwd}}\mathit{i}+1\right)$ 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, ${\mathbf{dwtlvm}}\left(1\right)$ 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 array
Output

On exit: the number of coefficients in the second dimension for each coefficient type at each level.
${\mathbf{dwtlvn}}\left(\mathit{i}\right)$ 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, ${\mathbf{dwtlvn}}\left({n}_{\mathrm{fwd}}\mathit{i}+1\right)$ 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, ${\mathbf{dwtlvn}}\left(1\right)$ 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 array
Communication Array

On entry: contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization routine
c09abf.
On exit: contains additional information on the computed transform.

11:
$\mathbf{ifail}$ – Integer
Input/Output

On entry:
ifail must be set to
$0$,
$1$ or
$1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value
$1$ or
$1$ is recommended. If message printing is undesirable, then the value
$1$ is recommended. Otherwise, the value
$0$ is recommended.
When the value $\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

On entry,
${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint:
${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$, the value of
m on initialization (see
c09abf).
On entry,
${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint:
${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$, the value of
n on initialization (see
c09abf).
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{lda}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{m}}$.
 ${\mathbf{ifail}}=3$

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.
 ${\mathbf{ifail}}=5$

On entry, ${\mathbf{nwl}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nwl}}\ge 1$.
On entry,
${\mathbf{nwl}}=\u2329\mathit{\text{value}}\u232a$ and
${\mathbf{nwlmax}}=\u2329\mathit{\text{value}}\u232a$ in
c09abf.
Constraint:
${\mathbf{nwl}}\le {\mathbf{nwlmax}}$ in
c09abf.
 ${\mathbf{ifail}}=7$

Either the initialization routine has not been called first or
icomm has been corrupted.
Either the initialization routine was called with
${\mathbf{wtrans}}=\text{'S'}$ or
icomm has been corrupted.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 7 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 9 in the Introduction to the NAG Library FL 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
c09ecf 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
${\mathbf{c}}\left({k}_{1}+1\right)$
to
${\mathbf{c}}\left({k}_{{n}_{\mathrm{fwd}}+1}\right)$, 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
c09edf in order to reconstruct the denoised signal. See
Section 10 in
c09ezf 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
${\mathbf{wavnam}}=\text{'DB2'}$ in
c09abf) 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
c09edf.
10.1
Program Text
10.2
Program Data
10.3
Program Results