NAG Library Routine Document
C09FCF
1 Purpose
C09FCF computes the threedimensional multilevel discrete wavelet transform (DWT). The initialization routine
C09ACF must be called first to set up the DWT options.
2 Specification
SUBROUTINE C09FCF ( 
M, N, FR, A, LDA, SDA, LENC, C, NWL, DWTLVM, DWTLVN, DWTLVFR, ICOMM, IFAIL) 
INTEGER 
M, N, FR, LDA, SDA, LENC, NWL, DWTLVM(NWL), DWTLVN(NWL), DWTLVFR(NWL), ICOMM(260), IFAIL 
REAL (KIND=nag_wp) 
A(LDA,SDA,FR), C(LENC) 

3 Description
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,
${n}_{\mathrm{fwd}}$, 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
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.
The notation used here assigns level $0$ to the input data, $A$. Level 1 consists of the first set of coefficients computed: the seven sets of detail coefficients are stored at this level while the approximation 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 eight types of coefficients are stored. All coefficients are packed into a single array.
4 References
Wang Y, Che X and Ma S (2012) Nonlinear filtering based on 3D wavelet transform for MRI denoising URASIP Journal on Advances in Signal Processing 2012:40
5 Parameters
 1: $\mathrm{M}$ – INTEGERInput

On entry: 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: $\mathrm{N}$ – INTEGERInput

On entry: 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: $\mathrm{FR}$ – INTEGERInput

On entry: the number of twodimensional frames.
Constraint:
this must be the same as the value
FR passed to the initialization routine
C09ACF.
 4: $\mathrm{A}\left({\mathbf{LDA}},{\mathbf{SDA}},{\mathbf{FR}}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the $m$ by $n$ by $\mathit{fr}$ threedimensional input data $A$, where with ${A}_{ijk}$ stored in ${\mathbf{A}}\left(i,j,k\right)$.
 5: $\mathrm{LDA}$ – 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: $\mathrm{SDA}$ – 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: $\mathrm{LENC}$ – 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: $\mathrm{C}\left({\mathbf{LENC}}\right)$ – REAL (KIND=nag_wp) arrayOutput

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
C09FYF or
C09FZF is recommended. For completeness the following description provides details of precisely how the coefficients 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
$q\left(i\right)={\mathbf{DWTLVM}}\left({n}_{\mathrm{fwd}}i+1\right)\times {\mathbf{DWTLVN}}\left({n}_{\mathrm{fwd}}i+1\right)\times {\mathbf{DWTLVFR}}\left({n}_{\mathrm{fwd}}i+1\right)$. Then, letting
${k}_{1}=q\left({n}_{\mathrm{fwd}}\right)$ and
${k}_{\mathit{j}+1}={k}_{\mathit{j}}+q\left({n}_{\mathrm{fwd}}\u2308\mathit{j}/7\u2309+1\right)$, for
$\mathit{j}=1,2,\dots ,7{n}_{\mathrm{fwd}}$, 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, ${a}_{{n}_{\mathrm{fwd}}}$. 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 ${n}_{\mathrm{fwd}}\u2308j/7\u2309+1$ 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 $j=1,\dots ,7{n}_{\mathrm{fwd}}$. See Section 2.1 in the C09 Chapter Introduction for a description of how these coefficients are produced.
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.
 9: $\mathrm{NWL}$ – INTEGERInput

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
C09ACF.
 10: $\mathrm{DWTLVM}\left({\mathbf{NWL}}\right)$ – 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 (${n}_{\mathrm{fwd}}\mathit{i}+1$)th level of resolution, for $\mathit{i}=1,2,\dots ,{n}_{\mathrm{fwd}}$.
 11: $\mathrm{DWTLVN}\left({\mathbf{NWL}}\right)$ – 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 (${n}_{\mathrm{fwd}}\mathit{i}+1$)th level of resolution, for $\mathit{i}=1,2,\dots ,{n}_{\mathrm{fwd}}$.
 12: $\mathrm{DWTLVFR}\left({\mathbf{NWL}}\right)$ – 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 (${n}_{\mathrm{fwd}}\mathit{i}+1$)th level of resolution, for $\mathit{i}=1,2,\dots ,{n}_{\mathrm{fwd}}$.
 13: $\mathrm{ICOMM}\left(260\right)$ – 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: $\mathrm{IFAIL}$ – 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).
6 Error Indicators and Warnings
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).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

On entry,
${\mathbf{FR}}=\u2329\mathit{\text{value}}\u232a$.
Constraint:
${\mathbf{FR}}=\u2329\mathit{\text{value}}\u232a$, the value of
FR on initialization (see
C09ACF).
On entry,
${\mathbf{M}}=\u2329\mathit{\text{value}}\u232a$.
Constraint:
${\mathbf{M}}=\u2329\mathit{\text{value}}\u232a$, the value of
M on initialization (see
C09ACF).
On entry,
${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint:
${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$, the value of
N on initialization (see
C09ACF).
 ${\mathbf{IFAIL}}=2$

On entry, ${\mathbf{LDA}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{M}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{M}}$.
On entry, ${\mathbf{SDA}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{N}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{SDA}}\ge {\mathbf{N}}$.
 ${\mathbf{IFAIL}}=3$

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
C09ACF.
Constraint:
${\mathbf{NWL}}\le {\mathbf{NWLMAX}}$ in
C09ACF.
 ${\mathbf{IFAIL}}=6$

Either the communication array
ICOMM has been corrupted or there has not been a prior call to the initialization routine
C09ACF.
The initialization routine was called with ${\mathbf{WTRANS}}=\text{'S'}$.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction 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
Not applicable.
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
Section 10 in C09FZF 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 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.
10.1 Program Text
Program Text (c09fcfe.f90)
10.2 Program Data
Program Data (c09fcfe.d)
10.3 Program Results
Program Results (c09fcfe.r)