C09 Chapter Contents
C09 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentC09ECF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

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.

## 2  Specification

 SUBROUTINE C09ECF ( M, N, A, LDA, LENC, C, NWL, DWTLVM, DWTLVN, ICOMM, IFAIL)
 INTEGER M, N, LDA, LENC, NWL, DWTLVM(NWL), DWTLVN(NWL), ICOMM(180), IFAIL REAL (KIND=nag_wp) A(LDA,N), C(LENC)

## 3  Description

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, ${n}_{l}$, 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 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 ${n}_{l}$, all four types of coefficients are stored. The output array, $C$, stores these sets of coefficients in reverse order, starting with ${a}_{{n}_{l}}$ followed by ${v}_{{n}_{l}},{h}_{{n}_{l}},{d}_{{n}_{l}},{v}_{{n}_{l}-1},{h}_{{n}_{l}-1},{d}_{{n}_{l}-1},\dots ,{v}_{1},{h}_{1},{d}_{1}$.

None.

## 5  Parameters

1:     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:     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:     A(LDA,N) – REAL (KIND=nag_wp) arrayInput
On entry: the $m$ by $n$ data matrix $A$.
4:     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:     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, ${n}_{l}$, 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 $3×⌈\stackrel{-}{m}/2⌉×⌈\stackrel{-}{n}/2⌉$ for ${\mathbf{MODE}}=\text{'P'}$ in C09ABF and $3×⌊\left(\stackrel{-}{m}+{n}_{f}-1\right)/2⌋×⌊\left(\stackrel{-}{n}+{n}_{f}-1\right)/2⌋$ for ${\mathbf{MODE}}=\text{'H'}$, $\text{'W'}$ or $\text{'Z'}$, where the input data is of dimension $\stackrel{-}{m}×\stackrel{-}{n}$ 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:     C(LENC) – REAL (KIND=nag_wp) arrayOutput
On exit: the coefficients of a multi-level wavelet transform of the dataset.
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 $q\left(i\right)={\mathbf{DWTLVM}}\left({n}_{l}-i+1\right)×{\mathbf{DWTLVN}}\left({n}_{l}-i+1\right)$. Then, letting ${k}_{1}=q\left({n}_{l}\right)$ and ${k}_{\mathit{j}+1}={k}_{\mathit{j}}+q\left({n}_{l}-⌈\mathit{j}/3⌉+1\right)$, for $\mathit{j}=1,2,\dots ,3{n}_{l}$, 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, ${a}_{{n}_{l}}$.
${\mathbf{C}}\left(\mathit{i}\right)$, for $\mathit{i}={k}_{j}+1,\dots ,{k}_{j+1}$
Contains the level ${n}_{l}-⌈j/3⌉+1$ vertical, horizontal and diagonal coefficients. These are:
• vertical coefficients if ;
• horizontal coefficients if ;
• diagonal coefficients if ,
for $j=1,\dots ,3{n}_{l}$.
7:     NWL – INTEGERInput
On entry: the number of levels, ${n}_{l}$, 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 NWL (the maximum number of levels) by the call to the initialization routine C09ABF.
8:     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 $\mathit{i}=1,2,\dots ,{n}_{l}$. Thus for the first ${n}_{l}-1$ levels of resolution, ${\mathbf{DWTLVM}}\left({n}_{l}-\mathit{i}+1\right)$ 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:     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 $\mathit{i}=1,2,\dots ,{n}_{l}$. Thus for the first ${n}_{l}-1$ levels of resolution, ${\mathbf{DWTLVN}}\left({n}_{l}-\mathit{i}+1\right)$ 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:   ICOMM($180$) – 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 C09ABF.
On exit: contains additional information on the computed transform.
11:   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, M is inconsistent with the value passed to the initialization routine C09ABF, or N is inconsistent with the value passed to the initialization routine C09ABF.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{LDA}}<{\mathbf{M}}$.
${\mathbf{IFAIL}}=3$
 On entry, ${\mathbf{LENC}}<{n}_{\mathrm{ct}}^{*}$, where ${n}_{\mathrm{ct}}^{*}$ is the required number of wavelet coefficients for NWL, the number of levels requested (for the maximum possible number of levels this value is returned in NWCT by the call to the initialization routine C09ABF).
${\mathbf{IFAIL}}=5$
 On entry, ${\mathbf{NWL}}<1$, or ${\mathbf{NWL}}>{l}_{\mathrm{max}}$, where ${l}_{\mathrm{max}}$ is the value returned in NWL by the call to the initialization routine C09ABF.
${\mathbf{IFAIL}}=7$
On entry, the initialization routine C09ABF has not been called first or it has been called with ${\mathbf{WTRANS}}=\text{'S'}$, or the communication array ICOMM has become corrupted.

## 7  Accuracy

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 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 ${\mathbf{C}}\left({k}_{{n}_{l}+1}\right)$, as described in Section 5, contain the detail coefficients, ${\stackrel{^}{c}}_{ij}$, for $\mathit{i}={n}_{l},{n}_{l}-1,\dots ,1$ 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 ${\stackrel{^}{c}}_{ij}={c}_{ij}+\sigma {\epsilon }_{ij}$ 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
 $c- ij = 0, if ​ c^ij ≤ α c^ij , if ​ c^ij > α,$
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 C09EDF 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.

## 9  Example

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.

### 9.1  Program Text

Program Text (c09ecfe.f90)

### 9.2  Program Data

Program Data (c09ecfe.d)

### 9.3  Program Results

Program Results (c09ecfe.r)