c09ecf computes the two-dimensional multi-level discrete wavelet transform (DWT). The initialization routine c09abf must be called first to set up the DWT options.
The routine may be called by the names c09ecf or nagf_wav_dim2_multi_fwd.
3Description
c09ecf computes the multi-level DWT of two-dimensional data. For a given wavelet and end extension method, c09ecf 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 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 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 ${v}_{{n}_{\mathrm{fwd}}},{h}_{{n}_{\mathrm{fwd}}},{d}_{{n}_{\mathrm{fwd}}},{v}_{{n}_{\mathrm{fwd}}-1},{h}_{{n}_{\mathrm{fwd}}-1},{d}_{{n}_{\mathrm{fwd}}-1},\dots ,{v}_{1},{h}_{1},{d}_{1}$.
4References
None.
5Arguments
1: $\mathbf{m}$ – IntegerInput
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: $\mathbf{n}$ – IntegerInput
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: $\mathbf{a}({\mathbf{lda}},{\mathbf{n}})$ – Real (Kind=nag_wp) arrayInput
On entry: the $m\times n$ data matrix $A$.
4: $\mathbf{lda}$ – IntegerInput
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: $\mathbf{lenc}$ – IntegerInput
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 \lceil \overline{m}/2\rceil \times \lceil \overline{n}/2\rceil $ for ${\mathbf{mode}}=\text{'P'}$ in c09abf and $3\times \lfloor (\overline{m}+{n}_{f}-1)/2\rfloor \times \lfloor (\overline{n}+{n}_{f}-1)/2\rfloor $ 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) 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 c09eyforc09ezf 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
${k}_{\mathit{j}+1}={k}_{\mathit{j}}+q({n}_{\mathrm{fwd}}-\lceil \mathit{j}/3\rceil +1)$, 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 ${n}_{\mathrm{fwd}}-\lceil j/3\rceil +1$ 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: $\mathbf{nwl}$ – IntegerInput
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 routine c09abf.
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.
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.
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: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ 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 $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ 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).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, 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, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$, the value of m on initialization (see c09abf).
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$, the value of n on initialization (see c09abf).
${\mathbf{ifail}}=2$
On entry, ${\mathbf{lda}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{m}}$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{lenc}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lenc}}\ge \u27e8\mathit{\text{value}}\u27e9$, the total number of coefficents to be generated.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{nwl}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{nwl}}\ge 1$.
On entry, ${\mathbf{nwl}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nwlmax}}=\u27e8\mathit{\text{value}}\u27e9$ 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.
7Accuracy
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.
8Parallelism and Performance
c09ecf is not threaded in any implementation.
9Further Comments
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.
10Example
This example performs a multi-level resolution transform of a dataset using the Daubechies wavelet (see ${\mathbf{wavnam}}=\text{'DB2'}$ in c09abf) 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 c09edf.