# NAG Library Function Document

## 1Purpose

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.

## 2Specification

 #include #include
 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)

## 3Description

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, ${n}_{\mathrm{ct}}$, of coefficients computed is returned by the initialization function nag_wfilt_3d (c09acc).

## 4References

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

## 5Arguments

1:    $\mathbf{m}$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:    $\mathbf{n}$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:    $\mathbf{fr}$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:    $\mathbf{a}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array a must be at least ${\mathbf{lda}}×{\mathbf{sda}}×{\mathbf{fr}}$.
On entry: the $m$ by $n$ by $\mathit{fr}$ three-dimensional input data $A$, where ${A}_{ijk}$ is stored in ${\mathbf{a}}\left[\left(k-1\right)×{\mathbf{lda}}×{\mathbf{sda}}+\left(j-1\right)×{\mathbf{lda}}+i-1\right]$.
5:    $\mathbf{lda}$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: ${\mathbf{lda}}\ge {\mathbf{m}}$.
6:    $\mathbf{sda}$IntegerInput
On entry: the stride separating corresponding coefficients of consecutive frames in the three-dimensional data stored in a.
Constraint: ${\mathbf{sda}}\ge {\mathbf{n}}$.
7:    $\mathbf{lenc}$IntegerInput
On entry: the dimension of the array c.
Constraint: ${\mathbf{lenc}}\ge {n}_{\mathrm{ct}}$, where ${n}_{\mathrm{ct}}$ is the total number of wavelet coefficients, as returned by nag_wfilt_3d (c09acc).
8:    $\mathbf{c}\left[{\mathbf{lenc}}\right]$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 ${n}_{\mathrm{cfr}}×{n}_{\mathrm{cm}}×{n}_{\mathrm{cn}}$ (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:    $\mathbf{icomm}\left[260\right]$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:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ 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 ${\mathbf{wtrans}}=\mathrm{Nag_MultiLevel}$.
NE_INT
On entry, ${\mathbf{fr}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{fr}}=〈\mathit{\text{value}}〉$, the value of fr on initialization (see nag_wfilt_3d (c09acc)).
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}=〈\mathit{\text{value}}〉$, the value of m on initialization (see nag_wfilt_3d (c09acc)).
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}=〈\mathit{\text{value}}〉$, the value of n on initialization (see nag_wfilt_3d (c09acc)).
NE_INT_2
On entry, ${\mathbf{lda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lda}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{lenc}}=〈\mathit{\text{value}}〉$ and ${n}_{\mathrm{ct}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lenc}}\ge {n}_{\mathrm{ct}}$, where ${n}_{\mathrm{ct}}$ is the number of DWT coefficients returned by nag_wfilt_3d (c09acc) in argument nwct.
On entry, ${\mathbf{sda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{sda}}\ge {\mathbf{n}}$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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.

## 8Parallelism and Performance

nag_dwt_3d (c09fac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

None.

## 10Example

This example computes the three-dimensional discrete wavelet decomposition for $5×4×3$ input data using the Haar wavelet, ${\mathbf{wavnam}}=\mathrm{Nag_Haar}$, 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.

### 10.1Program Text

Program Text (c09face.c)

### 10.2Program Data

Program Data (c09face.d)

### 10.3Program Results

Program Results (c09face.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017