c09ezc inserts a selected set of two-dimensional discrete wavelet transform (DWT) coefficients into the full set of coefficients stored in compact form, which may be later used as input to the multi-level reconstruction function c09edc.

2 Specification

#include <nag.h>

void	c09ezc (Integer ilev, Integer cindex, Integer lenc, double c[], const double d[], Integer pdd, Integer icomm[], NagError *fail)

The function may be called by the names: c09ezc, nag_wav_dim2_coeff_ins or nag_wav_2d_coeff_ins.

3 Description

c09ezc inserts a selected set of two-dimensional DWT coefficients into the full set of coefficients stored in compact form in a one-dimensional array c. It is required that c09ezc is preceded by a call to the initialization function c09abc and the forward multi-level transform function c09ecc.

Given an initial two-dimensional data set

A

, a prior call to c09ecc computes the approximation coefficients (at the highest requested level) and three sets of detail coefficients at all levels and stores these in compact form in a one-dimensional array c. c09eyc can then extract either the approximation coefficients or one of the sets of detail coefficients at one of the levels as two-dimensional data into the array, d. Following some calculation on this set of coefficients (for example, denoising), the updated coefficients in d are inserted back into the full set c using c09ezc. Several extractions and insertions may be performed at different levels. c09edc can then be used to reconstruct a manipulated data set

\tilde{A}

. The dimensions of the two-dimensional data stored in d depend on the level extracted and are available from the arrays dwtlvm and dwtlvn as returned by c09ecc which contain the first and second dimensions respectively. See Section 2.1 in the C09 Chapter Introduction for a discussion of the multi-level two-dimensional DWT.

4 References

None.

5 Arguments

Note: the following notation is used in this section:

$n_{cm}$ is the number of wavelet coefficients in the first dimension, which, at level ilev, is equal to $dwtlvm [nwl - ilev]$ as returned by a call to c09ecc transforming nwl levels.
$n_{cn}$ is the number of wavelet coefficients in the second dimension, which, at level ilev, is equal to $dwtlvn [nwl - ilev]$ as returned by a call to c09ecc transforming nwl levels

1: $ilev$ – Integer Input

On entry: the level at which coefficients are to be inserted.

Constraints:

$1 \leq ilev \leq nwl$ , where nwl is as used in a preceding call to c09ecc;
if $cindex = 0$ , $ilev = nwl$ .

2: $cindex$ – Integer Input

On entry: identifies which coefficients to insert. The coefficients are identified as follows:

$cindex = 0$: The approximation coefficients, produced by application of the low pass filter over columns and rows of the original matrix (LL). The approximation coefficients are present only for $ilev = nwl$ , where nwl is the value used in a preceding call to c09ecc.
$cindex = 1$: The vertical detail coefficients produced by applying the low pass filter over columns of the original matrix and the high pass filter over rows (LH).
$cindex = 2$: The horizontal detail coefficients produced by applying the high pass filter over columns of the original matrix and the low pass filter over rows (HL).
$cindex = 3$: The diagonal detail coefficients produced by applying the high pass filter over columns and rows of the original matrix (HH).

Constraint:

0 \leq cindex \leq 3

when

ilev = nwl

as used in c09ecc, otherwise

1 \leq cindex \leq 3

3: $lenc$ – Integer Input

On entry: the dimension of the array c.

Constraint: lenc must be unchanged from the value used in the preceding call to c09ecc.

4: $c [lenc]$ – double Input/Output

On entry: contains the DWT coefficients inserted by previous calls to c09ezc, or computed by a previous call to c09ecc.

On exit: contains the same DWT coefficients provided on entry except for those identified by ilev and cindex, which are updated with the values supplied in d, inserted into the correct locations as expected by the reconstruction function c09edc.

5: $d [\dim]$ – const double Input

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

pdd \times n_{cn}

On entry: the coefficients to be inserted.

ilev = nwl

(as used in c09ecc) and

cindex = 0

, the

n_{cm} \times n_{cn}

manipulated approximation coefficients

a_{i j}

must be stored in

d [(j - 1) \times pdd + i - 1]

, for

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

and

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

Otherwise the

n_{cm} \times n_{cn}

manipulated level ilev detail coefficients (of type specified by cindex)

d_{i j}

must be stored in

d [(j - 1) \times pdd + i - 1]

, for

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

and

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

6: $pdd$ – Integer Input

On entry: the stride separating row elements in the two-dimensional data stored in the array d.

Constraint:

pdd \geq n_{cm}

7: $icomm [180]$ – 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 c09abc.

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_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INITIALIZATION: Either the initialization function has not been called first or icomm has been corrupted.

Either the initialization function was called with $wtrans = Nag_SingleLevel$ or icomm has been corrupted.
NE_INT: On entry, $cindex = ⟨ value ⟩$ .
Constraint: $cindex \leq 3$ .

On entry, $cindex = ⟨ value ⟩$ .
Constraint: $cindex \geq 0$ .

On entry, $ilev = ⟨ value ⟩$ .
Constraint: $ilev \geq 1$ .
NE_INT_2: On entry, $ilev = ⟨ value ⟩$ and $nwl = ⟨ value ⟩$ .
Constraint: $ilev \leq nwl$ , where $nwl$ is the number of levels used in the call to c09ecc.

On entry, $lenc = ⟨ value ⟩$ and $n_{ct} = ⟨ value ⟩$ .
Constraint: $lenc \geq n_{ct}$ , where $n_{ct}$ is the number of DWT coefficients computed in a previous call to c09ecc.

On entry, $pdd = ⟨ value ⟩$ and $n_{cm} = ⟨ value ⟩$ .
Constraint: $pdd \geq n_{cm}$ , where $n_{cm}$ is the number of DWT coefficients in the first dimension at the selected level ilev.
NE_INT_3: On entry, $ilev = ⟨ value ⟩$ and $nwl = ⟨ value ⟩$ , but $cindex = 0$ .
Constraint: $cindex > 0$ when $ilev < nwl$ in the preceding call to c09ecc.
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

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

c09ezc is not threaded in any implementation.

9 Further Comments

None.

10 Example

The following example demonstrates using the coefficient extraction and insertion functions in order to apply denoising using a thresholding operation. The original input data, which is horizontally striped, has artificial noise introduced to it, taken from a normal random number distribution. Reconstruction then takes place on both the noisy data and denoised data. The Mean Square Errors (MSE) of the two reconstructions are printed along with the reconstruction of the denoised data. The MSE of the denoised reconstruction is less than that of the noisy reconstruction.

c09ez: FL CL CPP AD PY MB

NAG CL Interfacec09ezc (dim2_​coeff_​ins)

▸▿ 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
c09ezc (dim2_coeff_ins)