NAG Library Function Document

nag_mldwt (c09ccc)

+− 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_mldwt (c09ccc) computes the one-dimensional multi-level discrete wavelet transform (DWT). The initialization function nag_wfilt (c09aac) must be called first to set up the DWT options.

2 Specification

#include <nag.h>

#include <nagc09.h>

void	nag_mldwt (Integer n, const double x[], Integer lenc, double c[], Integer nwl, Integer dwtlev[], Integer icomm[], NagError *fail)

3 Description

nag_mldwt (c09ccc) computes the multi-level DWT of one-dimensional data. For a given wavelet and end extension method, nag_mldwt (c09ccc) will compute a multi-level transform of a data array,

x_{i}

, for

i = 1, 2, \dots, n

, using a specified number,

n_{fwd}

, of levels. The number of levels specified,

n_{fwd}

, must be no more than the value

l_{\max}

returned in nwlmax by the initialization function nag_wfilt (c09aac) 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 dataset,

x

, with level

1

being the first set of coefficients computed, with the detail coefficients,

d_{1}

, being stored while the approximation coefficients,

a_{1}

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

n_{fwd}

, both the detail coefficients,

d_{n_{fwd}}

, and the approximation coefficients,

a_{n_{fwd}}

are retained. The output array,

C

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

a_{n_{fwd}}

followed by

d_{n_{fwd}}, d_{n_{fwd} - 1}, \dots, d_{1}

4 References

None.

5 Arguments

1: n – IntegerInput

On entry: the number of elements,

n

, in the data array

x

Constraint: this must be the same as the value n passed to the initialization function nag_wfilt (c09aac).

2: x[n] – const doubleInput

On entry: x contains the one-dimensional input dataset

x_{i}

, for

i = 1, 2, \dots, n

3: lenc – IntegerInput

On entry: the dimension of the array c. c must be large enough to contain the number,

n_{c}

, of wavelet coefficients. The maximum value of

n_{c}

is returned in nwc by the call to the initialization function nag_wfilt (c09aac) 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_{fwd}

, is chosen to be less than the maximum, then

n_{c}

is correspondingly smaller and lenc can be reduced by noting that the number of coefficients at each level is given by

⌈\bar{n} / 2⌉

for

mode = Nag_Periodic

in nag_wfilt (c09aac) and

⌊(\bar{n} + n_{f} - 1) / 2⌋

for

mode = Nag_HalfPointSymmetric

Nag_WholePointSymmetric

Nag_ZeroPadded

, where

\bar{n}

is the number of input data at that level and

n_{f}

is the filter length provided by the call to nag_wfilt (c09aac). At the final level the storage is doubled to contain the set of approximation coefficients.

Constraint:

lenc \geq n_{c}

, where

n_{c}

is the number of approximation and detail coefficients that correspond to a transform with nwlmax levels.

4: c[lenc] – doubleOutput

On exit: let

q (i)

denote the number of coefficients (of each type) produced by the wavelet transform at level

i

, for

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

. These values are returned in dwtlev. Setting

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

and

k_{j + 1} = k_{j} + q (n_{fwd} - j + 1)

, for

j = 1, 2, \dots, n_{fwd}

, the coefficients are stored as follows:

$c [i - 1]$ , for $i = 1, 2, \dots, k_{1}$: Contains the level $n_{fwd}$ approximation coefficients, $a_{n_{fwd}}$ .
$c [i - 1]$ , for $i = k_{1} + 1, \dots, k_{2}$: Contains the level $n_{fwd}$ detail coefficients $d_{n_{fwd}}$ .
$c [i - 1]$ , for $i = k_{j} + 1, \dots, k_{j + 1}$: Contains the level $n_{fwd} - j + 1$ detail coefficients, for $j = 2, 3, \dots, n_{fwd}$ .

5: nwl – IntegerInput

On entry: the number of levels,

n_{fwd}

, in the multi-level resolution to be performed.

Constraint:

1 \leq nwl \leq l_{\max}

, where

l_{\max}

is the value returned in nwlmax (the maximum number of levels) by the call to the initialization function nag_wfilt (c09aac).

6: dwtlev[ $nwl + 1$ ] – IntegerOutput

On exit: the number of transform coefficients at each level.

dwtlev [0]

and

dwtlev [1]

contain the number,

q (n_{fwd})

, of approximation and detail coefficients respectively, for the final level of resolution (these are equal);

dwtlev [i - 1]

contains the number of detail coefficients,

q (n_{fwd} - i + 2)

, for the (

n_{fwd} - i + 2

)th level, for

i = 3, 4, \dots, n_{fwd} + 1

7: icomm[ $100$ ] – IntegerCommunication Array

On entry: contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization function nag_wfilt (c09aac).

On exit: contains additional information on the computed transform.

8: fail – NagError *Input/Output

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

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_ARRAY_DIM_LEN: On entry, lenc is set too small: $lenc = ⟨value⟩$ .
Constraint: $lenc \geq ⟨value⟩$ .
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INITIALIZATION: Either the initialization function has not been called first or array icomm has been corrupted.
Either the initialization function was called with $wtrans = Nag_SingleLevel$ or array icomm has been corrupted.
On entry, n is inconsistent with the value passed to the initialization function: $n = ⟨value⟩$ , n should be $⟨value⟩$ .
On entry, nwl is larger than the maximum number of levels returned by the initialization function: $nwl = ⟨value⟩$ , maximum $= ⟨value⟩$ .
NE_INT: On entry, $nwl = ⟨value⟩$ .
Constraint: $nwl \geq 1$ .
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

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

Not applicable.

9 Further Comments

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

x

, denoising can be carried out by applying a thresholding operation to the detail coefficients at every level. The elements

c [i - 1]

, for

i = k_{1} + 1, \dots, k_{n_{fwd}} + 1

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

{\hat{d}}_{i j}

, for

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

and

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

, where

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

taking

{\bar{d}}_{i j}

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

d_{i j}

. The resulting coefficients can then be used as input to nag_imldwt (c09cdc) 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.

10 Example

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

wavnam = Nag_Daubechies4

in nag_wfilt (c09aac)) using zero end extensions, the number of levels of resolution, the number of coefficients in each level and the coefficients themselves are reused. The original dataset is then reconstructed using nag_imldwt (c09cdc).

NAG Library Function Documentnag_mldwt (c09ccc)

+− 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_mldwt (c09ccc)