NAG Library Function Document

nag_dwt_2d (c09eac)

+− 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_2d (c09eac) computes the two-dimensional discrete wavelet transform (DWT) at a single level. The initialization function nag_wfilt_2d (c09abc) must be called first to set up the DWT options.

2 Specification

#include <nag.h>

#include <nagc09.h>

void	nag_dwt_2d (Integer m, Integer n, const double a[], Integer lda, double ca[], Integer ldca, double ch[], Integer ldch, double cv[], Integer ldcv, double cd[], Integer ldcd, Integer icomm[], NagError *fail)

3 Description

nag_dwt_2d (c09eac) computes the two-dimensional DWT of a given input data array, considered as a matrix

A

, 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,

A

, first over columns and then to the result over rows. The matrix of approximation (or smooth) coefficients,

C_{a}

, is produced by the low pass filter over columns and rows; the matrix of horizontal coefficients,

C_{h}

, is produced by the high pass filter over columns and the low pass filter over rows; the matrix of vertical coefficients,

C_{v}

, is produced by the low pass filter over columns and the high pass filter over rows; and the matrix of diagonal coefficients,

C_{d}

, is produced by the high pass filter over columns and rows. 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 for

C_{a}

C_{h}

C_{v}

, and

C_{d}

together and the number of columns of each coefficients matrix,

n_{cn}

, are returned by the initialization function nag_wfilt_2d (c09abc). These values can be used to calculate the number of rows of each coefficients matrix,

n_{cm}

, using the formula

n_{cm} = n_{ct} / (4 n_{cn})

4 References

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

5 Arguments

1: m – IntegerInput: On entry: number of rows, $m$ , of data matrix $A$ .
Constraint: this must be the same as the value m passed to the initialization function nag_wfilt_2d (c09abc).
2: n – IntegerInput: On entry: number of columns, $n$ , of data matrix $A$ .
Constraint: this must be the same as the value n passed to the initialization function nag_wfilt_2d (c09abc).
3: a[ $lda \times n$ ] – const doubleInput: Note: the $(i, j)$ th element of the matrix $A$ is stored in $a [(j - 1) \times lda + i - 1]$ .
On entry: the $m$ by $n$ data matrix $A$ .
4: lda – IntegerInput: On entry: the stride separating matrix row elements in the array a.

Constraint: $lda \geq m$ .
5: ca[ $\dim$ ] – doubleOutput: Note: the dimension, dim, of the array ca must be at least $ldca \times n_{cn}$ where $n_{cn}$ is the argument nwcn returned by function nag_wfilt_2d (c09abc).

The $(i, j)$ th element of the matrix is stored in $ca [(j - 1) \times ldca + i - 1]$ .
On exit: contains the $n_{cm}$ by $n_{cn}$ matrix of approximation coefficients, $C_{a}$ .
6: ldca – IntegerInput: On entry: the stride separating matrix row elements in the array ca.

Constraint: $ldca \geq n_{cm}$ where $n_{cm} = n_{ct} / (4 n_{cn})$ and $n_{cn}$ , $n_{ct}$ are returned by the initialization function nag_wfilt_2d (c09abc).
7: ch[ $\dim$ ] – doubleOutput: Note: the dimension, dim, of the array ch must be at least $ldch \times n_{cn}$ where $n_{cn}$ is the argument nwcn returned by function nag_wfilt_2d (c09abc).

The $(i, j)$ th element of the matrix is stored in $ch [(j - 1) \times ldch + i - 1]$ .
On exit: contains the $n_{cm}$ by $n_{cn}$ matrix of horizontal coefficients, $C_{h}$ .
8: ldch – IntegerInput: On entry: the stride separating matrix row elements in the array ch.

Constraint: $ldch \geq n_{cm}$ where $n_{cm} = n_{ct} / (4 n_{cn})$ and $n_{cn}$ , $n_{ct}$ are returned by the initialization function nag_wfilt_2d (c09abc).
9: cv[ $\dim$ ] – doubleOutput: Note: the dimension, dim, of the array cv must be at least $ldcv \times n_{cn}$ where $n_{cn}$ is the argument nwcn returned by function nag_wfilt_2d (c09abc).

The $(i, j)$ th element of the matrix is stored in $cv [(j - 1) \times ldcv + i - 1]$ .
On exit: contains the $n_{cm}$ by $n_{cn}$ matrix of vertical coefficients, $C_{v}$ .
10: ldcv – IntegerInput: On entry: the stride separating matrix row elements in the array cv.

Constraint: $ldcv \geq n_{cm}$ where $n_{cm} = n_{ct} / (4 n_{cn})$ and $n_{cn}$ , $n_{ct}$ are returned by the initialization function nag_wfilt_2d (c09abc).
11: cd[ $\dim$ ] – doubleOutput: Note: the dimension, dim, of the array cd must be at least $ldcd \times n_{cn}$ where $n_{cn}$ is the argument nwcn returned by function nag_wfilt_2d (c09abc).

The $(i, j)$ th element of the matrix is stored in $cd [(j - 1) \times ldcd + i - 1]$ .
On exit: contains the $n_{cm}$ by $n_{cn}$ matrix of diagonal coefficients, $C_{d}$ .
12: ldcd – IntegerInput: On entry: the stride separating matrix row elements in the array cd.

Constraint: $ldcd \geq n_{cm}$ where $n_{cm} = n_{ct} / (4 n_{cn})$ and $n_{cn}$ , $n_{ct}$ are returned by the initialization function nag_wfilt_2d (c09abc).
13: icomm[ $180$ ] – 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_2d (c09abc).
14: 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 initialization function has not been called first or icomm has been corrupted.
Either the initialization function was called with $wtrans = Nag_MultiLevel$ or icomm has been corrupted.
NE_INT: On entry, $ldca = ⟨value⟩$ .
Constraint: $ldca \geq ⟨value⟩$ , the number of wavelet coefficients in the first dimension.
On entry, $ldcd = ⟨value⟩$ .
Constraint: $ldcd \geq ⟨value⟩$ , the number of wavelet coefficients in the first dimension.
On entry, $ldch = ⟨value⟩$ .
Constraint: $ldch \geq ⟨value⟩$ , the number of wavelet coefficients in the first dimension.
On entry, $ldcv = ⟨value⟩$ .
Constraint: $ldcv \geq ⟨value⟩$ , the number of wavelet coefficients in the first dimension.
On entry, $m = ⟨value⟩$ .
Constraint: $m = ⟨value⟩$ , the value of m on initialization (see nag_wfilt_2d (c09abc)).
On entry, $n = ⟨value⟩$ .
Constraint: $n = ⟨value⟩$ , the value of n on initialization (see nag_wfilt_2d (c09abc)).
NE_INT_2: On entry, $lda = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $lda \geq m$ .
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_2d (c09eac) 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 two-dimensional discrete wavelet decomposition for a

6 \times 6

input matrix using the Daubechies wavelet,

wavnam = Nag_Daubechies4

, with half point symmetric end extension.

NAG Library Function Documentnag_dwt_2d (c09eac)