C09 Chapter Contents
C09 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentC09FCF

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

C09FCF computes the three-dimensional multi-level 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 multi-level DWT of three-dimensional data. For a given wavelet and end extension method, C09FCF will compute a multi-level transform of a three-dimensional 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 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 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 ${n}_{l}$, all eight types of coefficients are stored. All coefficients are packed into a single array.

None.

## 5  Parameters

1:     M – INTEGERInput
On entry: the first dimension of the input data: the number of rows of each two-dimensional 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 two-dimensional 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 two-dimensional 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 three-dimensional 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 $q\left(i\right)={\mathbf{DWTLVM}}\left({n}_{l}-i+1\right)×{\mathbf{DWTLVN}}\left({n}_{l}-i+1\right)×{\mathbf{DWTLVFR}}\left({n}_{l}-i+1\right)$. Then, letting ${k}_{1}=q\left({n}_{l}\right)$ and ${k}_{\mathit{j}+1}={k}_{\mathit{j}}+q\left({n}_{l}-⌈\mathit{j}/7⌉+1\right)$, 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 ${\mathbf{DWTLVM}}\left(1\right)×{\mathbf{DWTLVN}}\left(1\right)×{\mathbf{DWTLVFR}}\left(1\right)$ in C.
${\mathbf{C}}\left(\mathit{i}\right)$, for $\mathit{i}={k}_{j}+1,\dots ,{k}_{j+1}$
Contains the level ${n}_{l}-⌈j/7⌉+1$ detail coefficients. These are:
• LLH coefficients if ;
• LHL coefficients if ;
• LHH coefficients if ;
• HLL coefficients if ;
• HLH coefficients if ;
• HHL coefficients if ;
• HHH coefficients if ,
for $j=1,\dots ,7{n}_{l}$.
Note that for computational efficiency reasons these coefficients are stored as ${\mathbf{DWTLVFR}}\left(⌈j/7⌉\right)×{\mathbf{DWTLVM}}\left(⌈j/7⌉\right)×{\mathbf{DWTLVN}}\left(⌈j/7⌉\right)$ 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 three-dimensional multi-level inverse routine C09FDF.
9:     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 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).

## 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, ${\mathbf{FR}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{FR}}=⟨\mathit{\text{value}}⟩$, the value of FR on initialization (see C09ACF).
On entry, ${\mathbf{M}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{M}}=⟨\mathit{\text{value}}⟩$, the value of M on initialization (see C09ACF).
On entry, ${\mathbf{N}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{N}}=⟨\mathit{\text{value}}⟩$, the value of N on initialization (see C09ACF).
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{LDA}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{M}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{M}}$.
On entry, ${\mathbf{SDA}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{N}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{SDA}}\ge {\mathbf{N}}$.
${\mathbf{IFAIL}}=3$
On entry, ${\mathbf{LENC}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{LENC}}\ge ⟨\mathit{\text{value}}⟩$, the total number of coefficents to be generated.
${\mathbf{IFAIL}}=5$
On entry, ${\mathbf{NWL}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{NWL}}\ge 1$.
On entry, ${\mathbf{NWL}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{NWL}}=⟨\mathit{\text{value}}⟩$ in C09ACF.
Constraint: ${\mathbf{NWL}}\le {\mathbf{NWL}}$ in C09ACF.
${\mathbf{IFAIL}}=6$
Either the initialization routine has not been called first or the communication array ICOMM has been corrupted.
The initialization routine was called with ${\mathbf{WTRANS}}=\text{'S'}$.
${\mathbf{IFAIL}}=-999$
Dynamic memory allocation failed.

## 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 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 ${\stackrel{^}{c}}_{ij}={c}_{ij}+\sigma {\epsilon }_{ij}$ 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 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.

## 9  Example

This example computes the three-dimensional multi-level discrete wavelet decomposition for $7×6×5$ 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.

### 9.1  Program Text

Program Text (c09fcfe.f90)

### 9.2  Program Data

Program Data (c09fcfe.d)

### 9.3  Program Results

Program Results (c09fcfe.r)