C09 Chapter Contents
C09 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentC09ABF

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

C09ABF returns the details of the chosen two-dimensional discrete wavelet filter. For a chosen mother wavelet, discrete wavelet transform type (single-level or multi-level DWT) and end extension method, this routine returns the maximum number of levels of resolution (appropriate to a multi-level transform), the filter length, the total number of approximation, horizontal, vertical and diagonal coefficients and the number of coefficients in the second dimension for the single-level case. This routine must be called before any of the two-dimensional transform routines in this chapter.

## 2  Specification

 SUBROUTINE C09ABF ( WAVNAM, WTRANS, MODE, M, N, NWL, NF, NWCT, NWCN, ICOMM, IFAIL)
 INTEGER M, N, NWL, NF, NWCT, NWCN, ICOMM(180), IFAIL CHARACTER(*) WAVNAM CHARACTER(1) WTRANS, MODE

## 3  Description

Two-dimensional discrete wavelet transforms (DWT) are characterised by the mother wavelet, the end extension method and whether multiresolution analysis is to be performed. For the selected combination of choices for these three characteristics, and for given dimensions ($m×n$) of data matrix $A$, C09ABF returns the dimension details for the transform determined by this combination. The dimension details are: ${l}_{\mathrm{max}}$, the maximum number of levels of resolution that would be computed were a multi-level DWT applied; ${n}_{f}$, the filter length; ${n}_{\mathrm{ct}}$ the total number of approximation, horizontal, vertical and diagonal coefficients (over all levels in the multi-level DWT case); and ${n}_{\mathrm{cn}}$, the number of coefficients in the second dimension for a single-level DWT. These values are also stored in the communication array ICOMM, as are the input choices, so that they may be conveniently communicated to the two-dimensional transform routines in this chapter.

None.

## 5  Parameters

1:     WAVNAM – CHARACTER(*)Input
On entry: the name of the mother wavelet. See the C09 Chapter Introduction for details.
${\mathbf{WAVNAM}}=\text{'HAAR'}$
Haar wavelet.
${\mathbf{WAVNAM}}=\text{'DB}\mathbit{n}\text{'}$, where $\mathbit{n}=2,3,\dots ,10$
Daubechies wavelet with $\mathbit{n}$ vanishing moments ($2\mathbit{n}$ coefficients). For example, ${\mathbf{WAVNAM}}=\text{'DB4'}$ is the name for the Daubechies wavelet with $4$ vanishing moments ($8$ coefficients).
${\mathbf{WAVNAM}}=\text{'BIOR}\mathbit{x}$.$\mathbit{y}\text{'}$, where $\mathbit{x}$.$\mathbit{y}$ can be one of 1.1, 1.3, 1.5, 2.2, 2.4, 2.6, 2.8, 3.1, 3.3, 3.5 or 3.7
Biorthogonal wavelet of order $\mathbit{x}$.$\mathbit{y}$. For example ${\mathbf{WAVNAM}}=\text{'BIOR3.1'}$ is the name for the biorthogonal wavelet of order $3.1$.
Constraint: ${\mathbf{WAVNAM}}=\text{'HAAR'}$, $\text{'DB2'}$, $\text{'DB3'}$, $\text{'DB4'}$, $\text{'DB5'}$, $\text{'DB6'}$, $\text{'DB7'}$, $\text{'DB8'}$, $\text{'DB9'}$, $\text{'DB10'}$, $\text{'BIOR1.1'}$, $\text{'BIOR1.3'}$, $\text{'BIOR1.5'}$, $\text{'BIOR2.2'}$, $\text{'BIOR2.4'}$, $\text{'BIOR2.6'}$, $\text{'BIOR2.8'}$, $\text{'BIOR3.1'}$, $\text{'BIOR3.3'}$, $\text{'BIOR3.5'}$ or $\text{'BIOR3.7'}$.
2:     WTRANS – CHARACTER(1)Input
On entry: the type of discrete wavelet transform that is to be applied.
${\mathbf{WTRANS}}=\text{'S'}$
Single-level decomposition or reconstruction by discrete wavelet transform.
${\mathbf{WTRANS}}=\text{'M'}$
Multiresolution, by a multi-level DWT or its inverse.
Constraint: ${\mathbf{WTRANS}}=\text{'S'}$ or $\text{'M'}$.
3:     MODE – CHARACTER(1)Input
On entry: the end extension method.
${\mathbf{MODE}}=\text{'P'}$
Periodic end extension.
${\mathbf{MODE}}=\text{'H'}$
Half-point symmetric end extension.
${\mathbf{MODE}}=\text{'W'}$
Whole-point symmetric end extension.
${\mathbf{MODE}}=\text{'Z'}$
Zero end extension.
Constraint: ${\mathbf{MODE}}=\text{'P'}$, $\text{'H'}$, $\text{'W'}$ or $\text{'Z'}$.
4:     M – INTEGERInput
On entry: the number of elements, $m$, in the first dimension (number of rows of data matrix $A$) of the input data.
Constraint: ${\mathbf{M}}\ge 2$.
5:     N – INTEGERInput
On entry: the number of elements, $n$, in the second dimension (number of columns of data matrix $A$) of the input data.
Constraint: ${\mathbf{N}}\ge 2$.
6:     NWL – INTEGEROutput
On exit: the maximum number of levels of resolution, ${l}_{\mathrm{max}}$, that can be computed if a multi-level discrete wavelet transform is applied (${\mathbf{WTRANS}}=\text{'M'}$). It is such that ${2}^{{l}_{\mathrm{max}}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)<{2}^{{l}_{\mathrm{max}}+1}$, for ${l}_{\mathrm{max}}$ an integer.
If ${\mathbf{WTRANS}}=\text{'S'}$, NWL is not set.
7:     NF – INTEGEROutput
On exit: the filter length, ${n}_{f}$, for the supplied mother wavelet. This is used to determine the number of coefficients to be generated by the chosen transform.
8:     NWCT – INTEGEROutput
On exit: the total number of wavelet coefficients, ${n}_{\mathrm{ct}}$, that will be generated. When ${\mathbf{WTRANS}}=\text{'S'}$ the number of rows required in each of the output coefficient matrices can be calculated as ${n}_{\mathrm{cm}}={n}_{\mathrm{ct}}/\left(4{n}_{\mathrm{cn}}\right)$. When ${\mathbf{WTRANS}}=\text{'M'}$ the length of the array used to store all of the coefficient matrices must be at least ${n}_{\mathrm{ct}}$.
9:     NWCN – INTEGEROutput
On exit: for a single-level transform (${\mathbf{WTRANS}}=\text{'S'}$), the number of coefficients that would be generated in the second dimension, ${n}_{\mathrm{cn}}$, for each coefficient type. For a multi-level transform (${\mathbf{WTRANS}}=\text{'M'}$) this is set to $1$.
10:   ICOMM($180$) – INTEGER arrayCommunication Array
On exit: contains details of the wavelet transform and the problem dimension which is to be communicated to the two-dimensional discrete transform routines in this chapter.
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, ${\mathbf{WAVNAM}}\ne \text{'HAAR'}$, $\text{'DB2'}$, $\text{'DB3'}$, $\text{'DB4'}$, $\text{'DB5'}$, $\text{'DB6'}$, $\text{'DB7'}$, $\text{'DB8'}$, $\text{'DB9'}$, $\text{'DB10'}$, $\text{'BIOR1.1'}$, $\text{'BIOR1.3'}$, $\text{'BIOR1.5'}$, $\text{'BIOR2.2'}$, $\text{'BIOR2.4'}$, $\text{'BIOR2.6'}$, $\text{'BIOR2.8'}$, $\text{'BIOR3.1'}$, $\text{'BIOR3.3'}$, $\text{'BIOR3.5'}$ or $\text{'BIOR3.7'}$.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{WTRANS}}\ne \text{'S'}$ or $\text{'M'}$.
${\mathbf{IFAIL}}=3$
 On entry, ${\mathbf{MODE}}\ne \text{'P'}$, $\text{'H'}$, $\text{'W'}$ or $\text{'Z'}$.
${\mathbf{IFAIL}}=4$
 On entry, ${\mathbf{M}}<2$, or ${\mathbf{N}}<2$.

Not applicable.

None.

## 9  Example

This example computes the two-dimensional multi-level resolution for a $6×6$ matrix by a discrete wavelet transform using the Haar wavelet with whole-point symmetric end extensions. The number of levels of transformation actually performed is one less than the maximum possible. This number of levels, the length of the wavelet filter, the total number of coefficients and the number of coefficients in each dimension for each level are printed along with the vertical detail coefficients from the first level, before a reconstruction is performed.

### 9.1  Program Text

Program Text (c09abfe.f90)

### 9.2  Program Data

Program Data (c09abfe.d)

### 9.3  Program Results

Program Results (c09abfe.r)