NAG Library Function Document

nag_wfilt_2d (c09abc)

+− 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_2d (c09abc) 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 function 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 function must be called before any of the two-dimensional transform functions in this chapter.

2 Specification

#include <nag.h>

#include <nagc09.h>

void	nag_wfilt_2d (Nag_Wavelet wavnam, Nag_WaveletTransform wtrans, Nag_WaveletMode mode, Integer m, Integer n, Integer nwlmax, Integer nf, Integer nwct, Integer nwcn, Integer icomm[], NagError *fail)

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 \times n

) of data matrix

A

, nag_wfilt_2d (c09abc) 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 would be computed were a multi-level DWT applied;

n_{f}

, the filter length;

n_{ct}

the total number of approximation, horizontal, vertical and diagonal coefficients (over all levels in the multi-level DWT case); and

n_{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 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.

Constraint:

wtrans = Nag_SingleLevel

Nag_MultiLevel

3: mode – Nag_WaveletModeInput

On entry: the end extension method.

$mode = Nag_Periodic$: 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.

Constraint:

mode = Nag_Periodic

Nag_HalfPointSymmetric

Nag_WholePointSymmetric

Nag_ZeroPadded

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:

m \geq 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:

n \geq 2

6: nwlmax – Integer *Output

On exit: the maximum number of levels of resolution,

l_{\max}

, that can be computed if a multi-level discrete wavelet transform is applied (

wtrans = Nag_MultiLevel

). It is such that

2^{l_{\max}} \leq \min (m, n) < 2^{l_{\max} + 1}

, for

l_{\max}

an integer.

wtrans = Nag_SingleLevel

, nwlmax is not set.

7: 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.

8: nwct – Integer *Output

On exit: the total number of wavelet coefficients,

n_{ct}

, that will be generated. When

wtrans = Nag_SingleLevel

the number of rows required in each of the output coefficient matrices can be calculated as

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

. When

wtrans = Nag_MultiLevel

the length of the array used to store all of the coefficient matrices must be at least

n_{ct}

9: nwcn – Integer *Output

On exit: for a single-level transform (

wtrans = Nag_SingleLevel

), the number of coefficients that would be generated in the second dimension,

n_{cn}

, for each coefficient type. For a multi-level transform (

wtrans = Nag_MultiLevel

) this is set to

1

10: icomm[ $180$ ] – IntegerCommunication Array

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

11: 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.
NE_INT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 2$ .
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 two-dimensional multi-level resolution for a

6 \times 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.

NAG Library Function Documentnag_wfilt_2d (c09abc)

+− 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_2d (c09abc)