c09fcc (dim3_multi_fwd) : NAG Library CL Interface, Mark 28

2 Specification

#include <nag.h>

void	c09fcc (Integer m, Integer n, Integer fr, const double a[], Integer lda, Integer sda, Integer lenc, double c[], Integer nwl, Integer dwtlvm[], Integer dwtlvn[], Integer dwtlvfr[], Integer icomm[], NagError *fail)

The function may be called by the names: c09fcc, nag_wav_dim3_multi_fwd or nag_mldwt_3d.

3 Description

c09fcc computes the multi-level DWT of three-dimensional data. For a given wavelet and end extension method, c09fcc will compute a multi-level transform of a three-dimensional array

A

, 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 c09acc 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 data,

A

. Level 1 consists of the first set of coefficients computed: the seven sets of detail coefficients are stored at this level while the approximation 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_{fwd}

, all eight types of coefficients are stored. All coefficients are packed into a single array.

5 Arguments

m

– Integer Input

On entry: the number of rows of each two-dimensional frame.

Constraint: this must be the same as the value m passed to the initialization function c09acc.

n

– Integer Input

On entry: the number of columns of each two-dimensional frame.

Constraint: this must be the same as the value n passed to the initialization function c09acc.

fr

– Integer Input

On entry: the number of two-dimensional frames.

Constraint: this must be the same as the value fr passed to the initialization function c09acc.

a [\dim]

– const double Input

Note: the dimension, dim, of the array a must be at least

lda \times sda \times fr

On entry: the

m \times n \times fr

three-dimensional input data

A

, where with

A_{i j k}

stored in

a [(k - 1) \times lda \times sda + (j - 1) \times lda + i - 1]

lda

– Integer Input

On entry: the stride separating row elements of each of the sets of frame coefficients in the three-dimensional data stored in a.

Constraint:

lda \geq m

sda

– Integer Input

On entry: the stride separating corresponding coefficients of consecutive frames in the three-dimensional data stored in a.

Constraint:

sda \geq n

lenc

– Integer Input

On entry: the dimension of the array c.

Constraint:

lenc \geq n_{ct}

, where

n_{ct}

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

c [lenc]

– double Output

On exit: the coefficients of the discrete wavelet transform. If you need to access or modify the approximation coefficients or any specific set of detail coefficients then the use of c09fyc or c09fzc is recommended. For completeness the following description provides details of precisely how the coefficients are stored in c but this information should only be required in rare cases.

Let

q (i)

denote the number of coefficients of each type at level

i

, for

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

, such that

q (i) = dwtlvm [n_{fwd} - i] \times dwtlvn [n_{fwd} - i] \times dwtlvfr [n_{fwd} - i]

. Then, letting

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

and

k_{j + 1} = k_{j} + q (n_{fwd} - ⌈ j / 7 ⌉ + 1)

, for

j = 1, 2, \dots, 7 n_{fwd}

, the coefficients are stored in c as follows:

$c [i - 1]$ , for $i = 1, 2, \dots, k_{1}$

Contains the level

n_{fwd}

approximation coefficients,

a_{n_{fwd}}

. Note that for computational efficiency reasons these coefficients are stored as

dwtlvm [0] \times dwtlvn [0] \times dwtlvfr [0]

in c.

$c [i - 1]$ , for $i = k_{j} + 1, \dots, k_{j + 1}$

Contains the level

n_{fwd} - ⌈ j / 7 ⌉ + 1

detail coefficients. These are:

LLH coefficients if $j mod 7 = 1$ ;
LHL coefficients if $j mod 7 = 2$ ;
LHH coefficients if $j mod 7 = 3$ ;
HLL coefficients if $j mod 7 = 4$ ;
HLH coefficients if $j mod 7 = 5$ ;
HHL coefficients if $j mod 7 = 6$ ;
HHH coefficients if $j mod 7 = 0$ ,

for

j = 1, \dots, 7 n_{fwd}

. See Section 2.1 in the C09 Chapter Introduction for a description of how these coefficients are produced.

Note that for computational efficiency reasons these coefficients are stored as

dwtlvfr [⌈ j / 7 ⌉ - 1] \times dwtlvm [⌈ j / 7 ⌉ - 1] \times dwtlvn [⌈ j / 7 ⌉ - 1]

in c.

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 c09acc.

10:

dwtlvm [nwl]

– Integer Output

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

dwtlvm [i - 1]

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

n_{fwd} - i + 1

)th level of resolution, for

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

11:

dwtlvn [nwl]

– Integer Output

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

dwtlvn [i - 1]

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

n_{fwd} - i + 1

)th level of resolution, for

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

12:

dwtlvfr [nwl]

– Integer Output

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

dwtlvfr [i - 1]

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

n_{fwd} - i + 1

)th level of resolution, for

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

13:

icomm [260]

– 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 c09acc.

On exit: contains additional information on the computed transform.

14:

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_BAD_PARAM

On entry, argument

⟨ value ⟩

had an illegal value.

NE_INITIALIZATION

Either the communication array icomm has been corrupted or there has not been a prior call to the initialization function c09acc.

The initialization function was called with

wtrans = Nag_SingleLevel

NE_INT

On entry,

fr = ⟨ value ⟩

.
Constraint:

fr = ⟨ value ⟩

, the value of fr on initialization (see c09acc).

On entry,

m = ⟨ value ⟩

.
Constraint:

m = ⟨ value ⟩

, the value of m on initialization (see c09acc).

On entry,

n = ⟨ value ⟩

.
Constraint:

n = ⟨ value ⟩

, the value of n on initialization (see c09acc).

On entry,

nwl = ⟨ value ⟩

.
Constraint:

nwl \geq 1

NE_INT_2

On entry,

lda = ⟨ value ⟩

and

m = ⟨ value ⟩

.
Constraint:

lda \geq m

On entry,

lenc = ⟨ value ⟩

.
Constraint:

lenc \geq ⟨ value ⟩

, the total number of coefficents to be generated.

On entry,

nwl = ⟨ value ⟩

and

nwlmax = ⟨ value ⟩

in c09acc.
Constraint:

nwl \leq nwlmax

in c09acc.

On entry,

sda = ⟨ value ⟩

and

n = ⟨ value ⟩

.
Constraint:

sda \geq n

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.

9 Further Comments

The example program shows how the wavelet coefficients at each level can be extracted from the output array c. Denoising can be carried out by applying a thresholding operation to the detail coefficients at every level. If

c_{i j}

is a detail coefficient then

{\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 c09fdc 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 computes the three-dimensional multi-level discrete wavelet decomposition for

7 \times 6 \times 5

input data using the biorthogonal wavelet of order

1.1

(set

wavnam = Nag_Biorthogonal1_1

in c09acc) with periodic end extension, prints a selected set of wavelet coefficients and then reconstructs and verifies that the reconstruction matches the original data.

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
c09fcc (dim3_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 Interfacec09fcc (dim3_​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
c09fcc (dim3_multi_fwd)