NAG Library Routine Document

C09EAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

C09EAF computes the two-dimensional discrete wavelet transform (DWT) at a single level. The initialization routine C09ABF must be called first to set up the DWT options.

2 Specification

SUBROUTINE C09EAF (

M, N, A, LDA, CA, LDCA, CH, LDCH, CV, LDCV, CD, LDCD, ICOMM, IFAIL)

INTEGER	M, N, LDA, LDCA, LDCH, LDCV, LDCD, ICOMM(180), IFAIL
REAL (KIND=nag_wp)	A(LDA,N), CA(LDCA,), CH(LDCH,), CV(LDCV,), CD(LDCD,)

3 Description

C09EAF computes the two-dimensional DWT of a given input data array, considered as a matrix

A

, at a single level. For a chosen wavelet filter pair, the output coefficients are obtained by applying convolution and downsampling by two to the input,

A

, first over columns and then to the result over rows. The matrix of approximation (or smooth) coefficients,

C_{a}

, is produced by the low pass filter over columns and rows; the matrix of horizontal coefficients,

C_{h}

, is produced by the high pass filter over columns and the low pass filter over rows; the matrix of vertical coefficients,

C_{v}

, is produced by the low pass filter over columns and the high pass filter over rows; and the matrix of diagonal coefficients,

C_{d}

, is produced by the high pass filter over columns and rows. To reduce distortion effects at the ends of the data array, several end extension methods are commonly used. Those provided are: periodic or circular convolution end extension, half-point symmetric end extension, whole-point symmetric end extension and zero end extension. The total number,

n_{ct}

, of coefficients computed for

C_{a}

C_{h}

C_{v}

, and

C_{d}

together and the number of columns of each coefficients matrix,

n_{cn}

, are returned by the initialization routine C09ABF. These values can be used to calculate the number of rows of each coefficients matrix,

n_{cm}

, using the formula

n_{cm} = n_{ct} / (4 n_{cn})

4 References

Daubechies I (1992) Ten Lectures on Wavelets SIAM, Philadelphia

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 C09EAF is called.
Constraint: $LDA \geq M$ .
5: CA(LDCA, $*$ ) – REAL (KIND=nag_wp) arrayOutput: Note: the second dimension of the array CA must be at least $n_{cn}$ where $n_{cn}$ is the parameter NWCN returned by routine C09ABF.
On exit: contains the $n_{cm}$ by $n_{cn}$ matrix of approximation coefficients, $C_{a}$ .
6: LDCA – INTEGERInput: On entry: the first dimension of the array CA as declared in the (sub)program from which C09EAF is called.
Constraint: $LDCA \geq n_{cm}$ where $n_{cm} = n_{ct} / (4 n_{cn})$ and $n_{cn}$ , $n_{ct}$ are returned by the initialization routine C09ABF.
7: CH(LDCH, $*$ ) – REAL (KIND=nag_wp) arrayOutput: Note: the second dimension of the array CH must be at least $n_{cn}$ where $n_{cn}$ is the parameter NWCN returned by routine C09ABF.
On exit: contains the $n_{cm}$ by $n_{cn}$ matrix of horizontal coefficients, $C_{h}$ .
8: LDCH – INTEGERInput: On entry: the first dimension of the array CH as declared in the (sub)program from which C09EAF is called.
Constraint: $LDCH \geq n_{cm}$ where $n_{cm} = n_{ct} / (4 n_{cn})$ and $n_{cn}$ , $n_{ct}$ are returned by the initialization routine C09ABF.
9: CV(LDCV, $*$ ) – REAL (KIND=nag_wp) arrayOutput: Note: the second dimension of the array CV must be at least $n_{cn}$ where $n_{cn}$ is the parameter NWCN returned by routine C09ABF.
On exit: contains the $n_{cm}$ by $n_{cn}$ matrix of vertical coefficients, $C_{v}$ .
10: LDCV – INTEGERInput: On entry: the first dimension of the array CV as declared in the (sub)program from which C09EAF is called.
Constraint: $LDCV \geq n_{cm}$ where $n_{cm} = n_{ct} / (4 n_{cn})$ and $n_{cn}$ , $n_{ct}$ are returned by the initialization routine C09ABF.
11: CD(LDCD, $*$ ) – REAL (KIND=nag_wp) arrayOutput: Note: the second dimension of the array CD must be at least $n_{cn}$ where $n_{cn}$ is the parameter NWCN returned by routine C09ABF.
On exit: contains the $n_{cm}$ by $n_{cn}$ matrix of diagonal coefficients, $C_{d}$ .
12: LDCD – INTEGERInput: On entry: the first dimension of the array CD as declared in the (sub)program from which C09EAF is called.
Constraint: $LDCD \geq n_{cm}$ where $n_{cm} = n_{ct} / (4 n_{cn})$ and $n_{cn}$ , $n_{ct}$ are returned by the initialization routine C09ABF.
13: 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.
14: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . 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 $- 1 or 1$ 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 $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

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.

$IFAIL = 2$

On entry,

LDA < M

$IFAIL = 3$

On entry,	$LDCA < n_{cm}$ where $n_{cm} = n_{ct} / (4 n_{cn})$ and $n_{ct}$ , $n_{cn}$ are returned by the initialization routine C09ABF,
or	$LDCH < n_{cm}$ ,
or	$LDCV < n_{cm}$ ,
or	$LDCD < n_{cm}$ .

$IFAIL = 6$: On entry, the initialization routine C09ACF has not been called first or it has been called with $WTRANS ='M'$ , 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.

8 Further Comments

None.

9 Example

This example computes the two-dimensional discrete wavelet decomposition for a

6 \times 6

input matrix using the Daubechies wavelet,

WAVNAM ='DB4'

, with half point symmetric end extension.

NAG Library Routine DocumentC09EAF