C09FCF computes the threedimensional multilevel discrete wavelet transform (DWT). The initialization routine
C09ACF must be called first to set up the DWT options.
C09FCF computes the multilevel DWT of threedimensional data. For a given wavelet and end extension method, C09FCF will compute a multilevel transform of a threedimensional array
$A$, 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 routine
C09ACF 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.
None.
 1: M – INTEGERInput
On entry: the first dimension of the input data: the number of rows of each twodimensional frame.
Constraint:
this must be the same as the value
M passed to the initialization routine
C09ACF.
 2: N – INTEGERInput
On entry: the second dimension of the input data: the number of columns of each twodimensional frame.
Constraint:
this must be the same as the value
N passed to the initialization routine
C09ACF.
 3: FR – INTEGERInput
On entry: the third dimension of the input data: the number of twodimensional frames.
Constraint:
this must be the same as the value
FR passed to the initialization routine
C09ACF.
 4: A(LDA,SDA,FR) – REAL (KIND=nag_wp) arrayInput

On entry: the $m$ by $n$ by $\mathit{fr}$ input threedimensional array $A$.
 5: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which C09FCF is called.
Constraint:
${\mathbf{LDA}}\ge {\mathbf{M}}$.
 6: SDA – INTEGERInput
On entry: the second dimension of the array
A as declared in the (sub)program from which C09FCF is called.
Constraint:
${\mathbf{SDA}}\ge {\mathbf{N}}$.
 7: LENC – INTEGERInput
On entry: the dimension of the array
C as declared in the (sub)program from which C09FCF is called.
Constraint:
${\mathbf{LENC}}\ge {n}_{\mathrm{ct}}$, where
${n}_{\mathrm{ct}}$ is the total number of wavelet coefficients that correspond to a transform with
NWL levels.
 8: C(LENC) – REAL (KIND=nag_wp) arrayOutput

On exit: the coefficients of the discrete wavelet transform.
Let
$q\left(\mathit{i}\right)$ denote the number of coefficients (of each type) at level
$\mathit{i}$, for
$\mathit{i}=1,2,\dots ,{n}_{l}$, such that
$q\left(i\right)={\mathbf{DWTLVM}}\left({n}_{l}i+1\right)\times {\mathbf{DWTLVN}}\left({n}_{l}i+1\right)\times {\mathbf{DWTLVFR}}\left({n}_{l}i+1\right)$. Then, letting
${k}_{1}=q\left({n}_{l}\right)$ and
${k}_{\mathit{j}+1}={k}_{\mathit{j}}+q\left({n}_{l}\u2308\mathit{j}/7\u2309+1\right)$, for
$\mathit{j}=1,2,\dots ,7{n}_{l}$, 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 ${n}_{l}$ approximation coefficients, ${a}_{{n}_{l}}$. Note that for computational efficiency reasons these coefficients are stored as ${\mathbf{DWTLVM}}\left(1\right)\times {\mathbf{DWTLVN}}\left(1\right)\times {\mathbf{DWTLVFR}}\left(1\right)$ in C.
 ${\mathbf{C}}\left(\mathit{i}\right)$, for $\mathit{i}={k}_{j}+1,\dots ,{k}_{j+1}$
 Contains the level ${n}_{l}\u2308j/7\u2309+1$ detail coefficients. These are:
 LLH coefficients if $j\mathrm{mod}7=1$;
 LHL coefficients if $j\mathrm{mod}7=2$;
 LHH coefficients if $j\mathrm{mod}7=3$;
 HLL coefficients if $j\mathrm{mod}7=4$;
 HLH coefficients if $j\mathrm{mod}7=5$;
 HHL coefficients if $j\mathrm{mod}7=6$;
 HHH coefficients if $j\mathrm{mod}7=0$,
for $j=1,\dots ,7{n}_{l}$.
Note that for computational efficiency reasons these coefficients are stored as
${\mathbf{DWTLVFR}}\left(\u2308j/7\u2309\right)\times {\mathbf{DWTLVM}}\left(\u2308j/7\u2309\right)\times {\mathbf{DWTLVN}}\left(\u2308j/7\u2309\right)$ in
C.
See
Section 9 for details of how to access each set of coefficients in order to perform extraction from C following a call to this routine, or insertion into
C before a call to the threedimensional multilevel inverse routine
C09FDF.
 9: NWL – INTEGERInput
On entry: the number of levels, ${n}_{l}$, 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
NWL (the maximum number of levels) by the call to the initialization routine
C09ACF.
 10: DWTLVM(NWL) – INTEGER arrayOutput
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 (${n}_{l}\mathit{i}+1$)th level of resolution, for $\mathit{i}=1,2,\dots ,{n}_{l}$.
 11: DWTLVN(NWL) – INTEGER arrayOutput
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 (${n}_{l}\mathit{i}+1$)th level of resolution, for $\mathit{i}=1,2,\dots ,{n}_{l}$.
 12: DWTLVFR(NWL) – INTEGER arrayOutput
On exit: the number of coefficients in the third dimension for each coefficient type at each level.
${\mathbf{DWTLVFR}}\left(\mathit{i}\right)$ contains the number of coefficients in the third dimension (for each coefficient type computed) at the (${n}_{l}\mathit{i}+1$)th level of resolution, for $\mathit{i}=1,2,\dots ,{n}_{l}$.
 13: ICOMM($260$) – INTEGER arrayCommunication Array
On entry: contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization routine
C09ACF.
On exit: contains additional information on the computed transform.
 14: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ 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).
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
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.
The example program shows how the wavelet coefficients at each level can be extracted from the output array
C. Denoising can be carried out by applying a thresholding operation to the detail coefficients at every level. If
${c}_{ij}$ is a detail coefficient then
${\hat{c}}_{ij}={c}_{ij}+\sigma {\epsilon}_{ij}$ 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
${\stackrel{}{c}}_{ij}$ to be an approximation to the required detail coefficient without noise,
${c}_{ij}$. The resulting coefficients can then be used as input to
C09FDF 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.
This example computes the threedimensional multilevel discrete wavelet decomposition for
$7\times 6\times 5$ input data using the biorthogonal wavelet of order
$1.1$ (set
${\mathbf{WAVNAM}}=\text{'BIOR1.1'}$ in
C09ACF) with periodic end extension, prints a selected set of wavelet coefficients and then reconstructs and verifies that the reconstruction matches the original data.