NAG Library Function Document

nag_dwt_3d (c09fac)

+− 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_dwt_3d (c09fac) computes the three-dimensional discrete wavelet transform (DWT) at a single level. The initialization function nag_wfilt_3d (c09acc) must be called first to set up the DWT options.

2 Specification

#include <nag.h>

#include <nagc09.h>

void	nag_dwt_3d (Integer m, Integer n, Integer fr, const double a[], Integer lda, Integer sda, Integer lenc, double c[], Integer icomm[], NagError *fail)

3 Description

nag_dwt_3d (c09fac) computes the three-dimensional DWT of some given three-dimensional input data, considered as a number of two-dimensional frames, at a single level. For a chosen wavelet filter pair, the output coefficients are obtained by applying convolution and downsampling by two to the input data,

A

, first over columns, next over rows and finally across frames. The three-dimensional approximation coefficients are produced by the low pass filter over columns, rows and frames. In addition there are 7 sets of three-dimensional detail coefficients, each corresponding to a different order of low pass and high pass filters (see the c09 Chapter Introduction). All coefficients are packed into a single array. To reduce distortion effects at the ends of the data array, several end extension methods are commonly used. Those provided are: periodic or circular convolution end extension, half-point symmetric end extension, whole-point symmetric end extension and zero end extension. The total number,

n_{ct}

, of coefficients computed is returned by the initialization function nag_wfilt_3d (c09acc).

4 References

Daubechies I (1992) Ten Lectures on Wavelets SIAM, Philadelphia

5 Arguments

1: m – IntegerInput: 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 nag_wfilt_3d (c09acc).
2: n – IntegerInput: 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 nag_wfilt_3d (c09acc).
3: fr – IntegerInput: On entry: the number of two-dimensional frames.
Constraint: this must be the same as the value fr passed to the initialization function nag_wfilt_3d (c09acc).
4: a[ $\dim$ ] – const doubleInput: Note: the dimension, dim, of the array a must be at least $lda \times sda \times fr$ .

On entry: the $m$ by $n$ by $fr$ three-dimensional input data $A$ , where $A_{i j k}$ is stored in $a [(k - 1) \times lda \times sda + (j - 1) \times lda + i - 1]$ .
5: lda – IntegerInput: On entry: the stride separating row elements of each of the sets of frame coefficients in the three-dimensional data stored in a.

Constraint: $lda \geq m$ .
6: sda – IntegerInput: On entry: the stride separating corresponding coefficients of consecutive frames in the three-dimensional data stored in a.

Constraint: $sda \geq n$ .
7: lenc – IntegerInput: On entry: the dimension of the array c.
Constraint: $lenc \geq n_{ct}$ , where $n_{ct}$ is the total number of wavelet coefficients, as returned by nag_wfilt_3d (c09acc).
8: c[lenc] – doubleOutput: 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 nag_wav_3d_coeff_ext (c09fyc) or nag_wav_3d_coeff_ins (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.
The $8$ sets of coefficients are stored in the following order: approximation coefficients (LLL) first, followed by $7$ sets of detail coefficients: LLH, LHL, LHH, HLL, HLH, HHL, HHH, where L indicates the low pass filter, and H the high pass filter being applied to, respectively, the columns of length m, the rows of length n and then the frames of length fr. Note that for computational efficiency reasons each set of coefficients is stored in the order $n_{cfr} \times n_{cm} \times n_{cn}$ (see output arguments nwcfr, nwct and nwcn in nag_wfilt_3d (c09acc)). See Section 10 for details of how to access each set of coefficients in order to perform extraction from c following a call to this function, or insertion into c before a call to the three-dimensional inverse function nag_idwt_3d (c09fbc).
9: icomm[ $260$ ] – 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_3d (c09acc).

On exit: contains additional information on the computed transform.
10: 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_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 nag_wfilt_3d (c09acc).
The initialization function was called with $wtrans = Nag_MultiLevel$ .
NE_INT: On entry, $fr = ⟨value⟩$ .
Constraint: $fr = ⟨value⟩$ , the value of fr on initialization (see nag_wfilt_3d (c09acc)).
On entry, $m = ⟨value⟩$ .
Constraint: $m = ⟨value⟩$ , the value of m on initialization (see nag_wfilt_3d (c09acc)).
On entry, $n = ⟨value⟩$ .
Constraint: $n = ⟨value⟩$ , the value of n on initialization (see nag_wfilt_3d (c09acc)).
NE_INT_2: On entry, $lda = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $lda \geq m$ .
On entry, $lenc = ⟨value⟩$ and $n_{ct} = ⟨value⟩$ .
Constraint: $lenc \geq n_{ct}$ , where $n_{ct}$ is the number of DWT coefficients returned by nag_wfilt_3d (c09acc) in argument nwct.
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.

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

nag_dwt_3d (c09fac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

This example computes the three-dimensional discrete wavelet decomposition for

5 \times 4 \times 3

input data using the Haar wavelet,

wavnam = Nag_Haar

, with half point end extension, prints the wavelet coefficients and then reconstructs the original data using nag_idwt_3d (c09fbc). This example also demonstrates in general how to access any set of coefficients following a single level transform.

NAG Library Function Documentnag_dwt_3d (c09fac)