NAG Library Routine Document

C09ECF

+− 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

C09ECF computes the two-dimensional multi-level discrete wavelet transform (DWT). The initialization routine C09ABF must be called first to set up the DWT options.

2 Specification

SUBROUTINE C09ECF (

M, N, A, LDA, LENC, C, NWL, DWTLVM, DWTLVN, ICOMM, IFAIL)

INTEGER	M, N, LDA, LENC, NWL, DWTLVM(NWL), DWTLVN(NWL), ICOMM(180), IFAIL
REAL (KIND=nag_wp)	A(LDA,N), C(LENC)

3 Description

C09ECF computes the multi-level DWT of two-dimensional data. For a given wavelet and end extension method, C09ECF will compute a multi-level transform of a matrix

A

, using a specified number,

n_{l}

, of levels. The number of levels specified,

n_{l}

, must be no more than the value

l_{\max}

returned in NWL by the initialization routine C09ABF for the given problem. The transform is returned as a set of coefficients for the different levels (packed into a single array) and a representation of the multi-level structure.

The notation used here assigns level

0

to the input matrix,

A

. Level 1 consists of the first set of coefficients computed: the vertical (

v_{1}

), horizontal (

h_{1}

) and diagonal (

d_{1}

) coefficients are stored at this level while the approximation (

a_{1}

) coefficients are used as the input to a repeat of the wavelet transform at the next level. This process is continued until, at level

n_{l}

, all four types of coefficients are stored. The output array,

C

, stores these sets of coefficients in reverse order, starting with

a_{n_{l}}

followed by

v_{n_{l}}, h_{n_{l}}, d_{n_{l}}, v_{n_{l} - 1}, h_{n_{l} - 1}, d_{n_{l} - 1}, \dots, v_{1}, h_{1}, d_{1}

4 References

None.

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

n

data matrix

A

4: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which C09ECF is called.

Constraint:

LDA \geq M

5: LENC – INTEGERInput

On entry: the dimension of the array C as declared in the (sub)program from which C09ECF is called. C must be large enough to contain,

n_{ct}

, wavelet coefficients. The maximum value of

n_{ct}

is returned in NWCT by the call to the initialization routine C09ABF and corresponds to the DWT being continued for the maximum number of levels possible for the given data set. When the number of levels,

n_{l}

, is chosen to be less than the maximum,

l_{\max}

, then

n_{ct}

is correspondingly smaller and LENC can be reduced by noting that the vertical, horizontal and diagonal coefficients are stored at every level and that in addition the approximation coefficients are stored for the final level only. The number of coefficients stored at each level is given by

3 \times ⌈\bar{m} / 2⌉ \times ⌈\bar{n} / 2⌉

for

MODE ='P'

in C09ABF and

3 \times ⌊(\bar{m} + n_{f} - 1) / 2⌋ \times ⌊(\bar{n} + n_{f} - 1) / 2⌋

for

MODE ='H'

'W'

'Z'

, where the input data is of dimension

\bar{m} \times \bar{n}

at that level and

n_{f}

is the filter length NF provided by the call to C09ABF. At the final level the storage is

4 / 3

times this value to contain the set of approximation coefficients.

Constraint:

LENC \geq n_{ct}

, where

n_{ct}

is the total number of coefficients that correspond to a transform with NWL levels.

6: C(LENC) – REAL (KIND=nag_wp) arrayOutput

On exit: the coefficients of a multi-level wavelet transform of the dataset.

Let

q (i)

denote the number of coefficients (of each type) at level

i

, for

i = 1, 2, \dots, n_{l}

, such that

q (i) = DWTLVM (n_{l} - i + 1) \times DWTLVN (n_{l} - i + 1)

. Then, letting

k_{1} = q (n_{l})

and

k_{j + 1} = k_{j} + q (n_{l} - ⌈j / 3⌉ + 1)

, for

j = 1, 2, \dots, 3 n_{l}

, the coefficients are stored in C as follows:

$C (i)$ , for $i = 1, 2, \dots, k_{1}$

Contains the level

n_{l}

approximation coefficients,

a_{n_{l}}

$C (i)$ , for $i = k_{j} + 1, \dots, k_{j + 1}$

Contains the level

n_{l} - ⌈j / 3⌉ + 1

vertical, horizontal and diagonal coefficients. These are:

vertical coefficients if $j mod 3 = 1$ ;
horizontal coefficients if $j mod 3 = 2$ ;
diagonal coefficients if $j mod 3 = 0$ ,

for

j = 1, \dots, 3 n_{l}

7: NWL – INTEGERInput

On entry: the number of levels,

n_{l}

, in the multi-level resolution to be performed.

Constraint:

1 \leq NWL \leq l_{\max}

, where

l_{\max}

is the value returned in NWL (the maximum number of levels) by the call to the initialization routine C09ABF.

8: DWTLVM(NWL) – INTEGER arrayOutput

On exit: the number of coefficients in the first dimension for each coefficient type at each level.

DWTLVM (i)

contains the number of coefficients in the first dimension (for each coefficient type computed) at the (

n_{l} - i + 1

)th level of resolution, for

i = 1, 2, \dots, n_{l}

. Thus for the first

n_{l} - 1

levels of resolution,

DWTLVM (n_{l} - i + 1)

is the size of the first dimension of the matrices of vertical, horizontal and diagonal coefficients computed at this level; for the final level of resolution,

DWTLVM (1)

is the size of the first dimension of the matrices of approximation, vertical, horizontal and diagonal coefficients computed.

9: DWTLVN(NWL) – INTEGER arrayOutput

On exit: the number of coefficients in the second dimension for each coefficient type at each level.

DWTLVN (i)

contains the number of coefficients in the second dimension (for each coefficient type computed) at the (

n_{l} - i + 1

)th level of resolution, for

i = 1, 2, \dots, n_{l}

. Thus for the first

n_{l} - 1

levels of resolution,

DWTLVN (n_{l} - i + 1)

is the size of the second dimension of the matrices of vertical, horizontal and diagonal coefficients computed at this level; for the final level of resolution,

DWTLVN (1)

is the size of the second dimension of the matrices of approximation, vertical, horizontal and diagonal coefficients computed.

10: 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.

On exit: contains additional information on the computed transform.

11: 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,

LENC < n_{ct}^{*}

, where

n_{ct}^{*}

is the required number of wavelet coefficients for NWL, the number of levels requested (for the maximum possible number of levels this value is returned in NWCT by the call to the initialization routine C09ABF).

$IFAIL = 5$

On entry,	$NWL < 1$ ,
or	$NWL > l_{\max}$ , where $l_{\max}$ is the value returned in NWL by the call to the initialization routine C09ABF.

$IFAIL = 7$: On entry, the initialization routine C09ABF has not been called first or it has been called with $WTRANS ='S'$ , 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

The wavelet coefficients at each level can be extracted from the output array C using the information contained in DWTLVM and DWTLVN on exit (see the descriptions of C, DWTLVM and DWTLVN in Section 5). For example, given an input data set,

A

, denoising can be carried out by applying a thresholding operation to the detail (vertical, horizontal and diagonal) coefficients at every level. The elements

C (k_{1} + 1)

C (k_{n_{l} + 1})

, as described in Section 5, contain the detail coefficients,

{\hat{c}}_{i j}

, for

i = n_{l}, n_{l} - 1, \dots, 1

and

j = 1, 2, \dots, 3 q (i)

, where

q (i)

is the number of each type of coefficient at level

i

and

{\hat{c}}_{i j} = c_{i j} + σ ε_{i j}

and

σ ε_{i j}

is the transformed noise term. If some threshold parameter

α

is chosen, a simple hard thresholding rule can be applied as

{\bar{c}}_{i j} = \{\begin{cases} 0, & if ​ |{\hat{c}}_{i j}| \leq α \\ {\hat{c}}_{i j}, & if ​ |{\hat{c}}_{i j}| > α, \end{cases}

taking

{\bar{c}}_{i j}

to be an approximation to the required detail coefficient without noise,

c_{i j}

. The resulting coefficients can then be used as input to C09EDF in order to reconstruct the denoised signal.

See the references given in the introduction to this chapter for a more complete account of wavelet denoising and other applications.

9 Example

This example performs a multi-level resolution transform of a dataset using the Daubechies wavelet (see

WAVNAM ='DB2'

in C09ABF) using half-point symmetric end extensions, the maximum possible number of levels of resolution, where the number of coefficients in each level and the coefficients themselves are not changed. The original dataset is then reconstructed using C09EDF.

NAG Library Routine DocumentC09ECF

+− 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

NAG Library Routine Document

C09ECF