NAG FL Interface
c09eaf (dim2_​sngl_​fwd)

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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

Fortran Interface
Subroutine c09eaf ( m, n, a, lda, ca, ldca, ch, ldch, cv, ldcv, cd, ldcd, icomm, ifail)
Integer, Intent (In) :: m, n, lda, ldca, ldch, ldcv, ldcd
Integer, Intent (Inout) :: icomm(180), ifail
Real (Kind=nag_wp), Intent (In) :: a(lda,n)
Real (Kind=nag_wp), Intent (Inout) :: ca(ldca,*), ch(ldch,*), cv(ldcv,*), cd(ldcd,*)
C Header Interface
#include <nag.h>
void  c09eaf_ (const Integer *m, const Integer *n, const double a[], const Integer *lda, double ca[], const Integer *ldca, double ch[], const Integer *ldch, double cv[], const Integer *ldcv, double cd[], const Integer *ldcd, Integer icomm[], Integer *ifail)
The routine may be called by the names c09eaf or nagf_wav_dim2_sngl_fwd.

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, Ca, is produced by the low pass filter over columns and rows; the matrix of horizontal coefficients, Ch, is produced by the high pass filter over columns and the low pass filter over rows; the matrix of vertical coefficients, Cv, is produced by the low pass filter over columns and the high pass filter over rows; and the matrix of diagonal coefficients, Cd, 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, nct, of coefficients computed for Ca, Ch, Cv, and Cd together and the number of columns of each coefficients matrix, ncn, are returned by the initialization routine c09abf. These values can be used to calculate the number of rows of each coefficients matrix, ncm, using the formula ncm=nct/(4ncn).

4 References

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

5 Arguments

1: m Integer Input
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 Integer Input
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) array Input
On entry: the m×n data matrix A.
4: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which c09eaf is called.
Constraint: ldam.
5: ca(ldca,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array ca must be at least ncn where ncn is the argument nwcn returned by routine c09abf.
On exit: contains the ncm×ncn matrix of approximation coefficients, Ca.
6: ldca Integer Input
On entry: the first dimension of the array ca as declared in the (sub)program from which c09eaf is called.
Constraint: ldcancm where ncm=nct/(4ncn) and ncn, nct are returned by the initialization routine c09abf.
7: ch(ldch,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array ch must be at least ncn where ncn is the argument nwcn returned by routine c09abf.
On exit: contains the ncm×ncn matrix of horizontal coefficients, Ch.
8: ldch Integer Input
On entry: the first dimension of the array ch as declared in the (sub)program from which c09eaf is called.
Constraint: ldchncm where ncm=nct/(4ncn) and ncn, nct are returned by the initialization routine c09abf.
9: cv(ldcv,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array cv must be at least ncn where ncn is the argument nwcn returned by routine c09abf.
On exit: contains the ncm×ncn matrix of vertical coefficients, Cv.
10: ldcv Integer Input
On entry: the first dimension of the array cv as declared in the (sub)program from which c09eaf is called.
Constraint: ldcvncm where ncm=nct/(4ncn) and ncn, nct are returned by the initialization routine c09abf.
11: cd(ldcd,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array cd must be at least ncn where ncn is the argument nwcn returned by routine c09abf.
On exit: contains the ncm×ncn matrix of diagonal coefficients, Cd.
12: ldcd Integer Input
On entry: the first dimension of the array cd as declared in the (sub)program from which c09eaf is called.
Constraint: ldcdncm where ncm=nct/(4ncn) and ncn, nct are returned by the initialization routine c09abf.
13: icomm(180) Integer array Communication 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 Integer Input/Output
On entry: ifail must be set to 0, −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 −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 −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 -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 or −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=value.
Constraint: m=value, the value of m on initialization (see c09abf).
On entry, n=value.
Constraint: n=value, the value of n on initialization (see c09abf).
ifail=2
On entry, lda=value and m=value.
Constraint: ldam.
ifail=3
On entry, ldca=value.
Constraint: ldcavalue, the number of wavelet coefficients in the first dimension.
On entry, ldcd=value.
Constraint: ldcdvalue, the number of wavelet coefficients in the first dimension.
On entry, ldch=value.
Constraint: ldchvalue, the number of wavelet coefficients in the first dimension.
On entry, ldcv=value.
Constraint: ldcvvalue, the number of wavelet coefficients in the first dimension.
ifail=6
Either the initialization routine has not been called first or icomm has been corrupted.
Either the initialization routine was called with wtrans='M' or icomm has been corrupted.
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.
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.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
c09eaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

This example computes the two-dimensional discrete wavelet decomposition for a 6×6 input matrix using the Daubechies wavelet, wavnam='DB4', with half point symmetric end extension.

10.1 Program Text

Program Text (c09eafe.f90)

10.2 Program Data

Program Data (c09eafe.d)

10.3 Program Results

Program Results (c09eafe.r)