NAG Library Function Document

nag_wfilt (c09aac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_wfilt (c09aac) returns the details of the chosen one-dimensional discrete wavelet filter. For a chosen mother wavelet, discrete wavelet transform type (single-level or multi-level DWT or MODWT) and end extension method, this function returns the maximum number of levels of resolution (appropriate to a multi-level transform), the filter length, and the number of approximation coefficients (equal to the number of detail coefficients) for a single-level DWT or MODWT or the total number of coefficients for a multi-level DWT or MODWT. This function must be called before any of the one-dimensional discrete transform functions in this chapter.

2 Specification

#include <nag.h>

#include <nagc09.h>

void	nag_wfilt (Nag_Wavelet wavnam, Nag_WaveletTransform wtrans, Nag_WaveletMode mode, Integer n, Integer nwlmax, Integer nf, Integer nwc, Integer icomm[], NagError fail)

3 Description

One-dimensional discrete wavelet transforms (DWT) or maximum overlap wavelet transforms (MODWT) 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 a given length,

n

, of the input data array,

x

, nag_wfilt (c09aac) returns the dimension details for the transform determined by this combination. The dimension details are:

l_{\max}

, the maximum number of levels of resolution that that could be computed were a multi-level DWT/MODWT applied;

n_{f}

, the filter length;

n_{c}

the number of approximation (or detail) coefficients for a single-level DWT/MODWT or the total number of coefficients generated by a multi-level DWT/MODWT over

l_{\max}

levels. These values are also stored in the communication array icomm, as are the input choices, so that they may be conveniently communicated to the one-dimensional transform functions in this chapter.

4 References

None.

5 Arguments

1: wavnam – Nag_WaveletInput

On entry: the name of the mother wavelet. See the c09 Chapter Introduction for details.

$wavnam = Nag_Haar$: Haar wavelet.
$wavnam = Nag_Daubechies n$ , where $n = 2, 3, \dots, 10$: Daubechies wavelet with $n$ vanishing moments ( $2 n$ coefficients). For example, $wavnam = Nag_Daubechies4$ is the name for the Daubechies wavelet with $4$ vanishing moments ( $8$ coefficients).
$wavnam = Nag_Biorthogonal x_y$ , where $x_y$ 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 $x$ . $y$ . For example $wavnam = Nag_Biorthogonal1_1$ is the name for the Biorthogonal wavelet of order $1.1$ .

Constraint:

wavnam = Nag_Haar

Nag_Daubechies2

Nag_Daubechies3

Nag_Daubechies4

Nag_Daubechies5

Nag_Daubechies6

Nag_Daubechies7

Nag_Daubechies8

Nag_Daubechies9

Nag_Daubechies10

Nag_Biorthogonal1_1

Nag_Biorthogonal1_3

Nag_Biorthogonal1_5

Nag_Biorthogonal2_2

Nag_Biorthogonal2_4

Nag_Biorthogonal2_6

Nag_Biorthogonal2_8

Nag_Biorthogonal3_1

Nag_Biorthogonal3_3

Nag_Biorthogonal3_5

Nag_Biorthogonal3_7

2: wtrans – Nag_WaveletTransformInput

On entry: the type of discrete wavelet transform that is to be applied.

$wtrans = Nag_SingleLevel$: Single-level decomposition or reconstruction by discrete wavelet transform.
$wtrans = Nag_MultiLevel$: Multiresolution, by a multi-level DWT or its inverse.
$wtrans = Nag_MODWTSingle$: Single-level decomposition or reconstruction by maximal overlap discrete wavelet transform.
$wtrans = Nag_MODWTMulti$: Multi-level resolution by a maximal overlap discrete wavelet transform or its inverse.

Constraint:

wtrans = Nag_SingleLevel

Nag_MultiLevel

Nag_MODWTSingle

Nag_MODWTMulti

3: mode – Nag_WaveletModeInput

On entry: the end extension method. Note that only periodic end extension is currently available for the MODWT.

$mode = Nag_Periodic$: Periodic end extension.
$mode = Nag_HalfPointSymmetric$: Half-point symmetric end extension.
$mode = Nag_WholePointSymmetric$: Whole-point symmetric end extension.
$mode = Nag_ZeroPadded$: Zero end extension.

Constraints:

$mode = Nag_Periodic$ , $Nag_HalfPointSymmetric$ , $Nag_WholePointSymmetric$ or $Nag_ZeroPadded$ for DWT;
$mode = Nag_Periodic$ for MODWT.

4: n – IntegerInput

On entry: the number of elements,

n

, in the input data array,

x

Constraint:

n \geq 2

5: nwlmax – Integer *Output

On exit: the maximum number of levels of resolution,

l_{\max}

, that can be computed when a multi-level discrete wavelet transform is applied. It is such that

2^{l_{\max}} \leq n < 2^{l_{\max} + 1}

, for

l_{\max}

an integer.

6: nf – Integer *Output

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.

7: nwc – Integer *Output

On exit: for a single-level transform (

wtrans = Nag_SingleLevel

Nag_MODWTSingle

), the number of approximation coefficients that would be generated for the given problem size, mother wavelet, extension method and type of transform; this is also the corresponding number of detail coefficients. For a multi-level transform (

wtrans = Nag_MultiLevel

Nag_MODWTMulti

) the total number of coefficients that would be generated over

l_{\max}

levels and with

keepa = Nag_StoreAll

for MODWT.

8: icomm[ $100$ ] – IntegerCommunication Array

On exit: contains details of the wavelet transform and the problem dimension which is to be communicated to the one-dimensional discrete discrete transform functions in this chapter.

9: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
On entry, $wtrans = Nag_MODWTSingle$ or $Nag_MODWTMulti$ and $mode \neq Nag_Periodic$ .
Constraint: $mode = Nag_Periodic$ when $wtrans = Nag_MODWTSingle$ or $Nag_MODWTMulti$ .
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 2$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example computes the one-dimensional multi-level resolution for

8

values by a discrete wavelet transform using the Haar wavelet with zero end extensions. The length of the wavelet filter, the number of levels of resolution, the number of approximation coefficients at each level and the total number of wavelet coefficients are printed.

NAG Library Function Documentnag_wfilt (c09aac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_wfilt (c09aac)