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

2 Specification

#include <nag.h>

void	c09ccc (Integer n, const double x[], Integer lenc, double c[], Integer nwl, Integer dwtlev[], Integer icomm[], NagError *fail)

The function may be called by the names: c09ccc, nag_wav_dim1_multi_fwd or nag_mldwt.

3 Description

c09ccc computes the multi-level DWT of one-dimensional data. For a given wavelet and end extension method, 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 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: $n$ – Integer Input

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 c09aac.

2: $x [n]$ – const double Input

On entry: x contains the one-dimensional input dataset

x_{i}

, for

i = 1, 2, \dots, n

3: $lenc$ – Integer Input

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 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 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 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: $c [lenc]$ – double Output

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: $nwl$ – Integer Input

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 c09aac.

6: $dwtlev [nwl + 1]$ – Integer Output

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: $icomm [100]$ – Integer Communication Array

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

On exit: contains additional information on the computed transform.

8: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL 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

c09ccc is not threaded in any implementation.

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 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 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 c09cdc.

c09cc: FL CL CPP AD

NAG CL Interfacec09ccc (dim1_​multi_​fwd)

▸▿ 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 CL Interface
c09ccc (dim1_multi_fwd)