c09 Chapter Contents
c09 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_wfilt_3d (c09acc)

## 1  Purpose

nag_wfilt_3d (c09acc) returns the details of the chosen three-dimensional discrete wavelet filter. For a chosen mother wavelet, discrete wavelet transform type (single-level or multi-level DWT) and end extension method, this function returns the maximum number of levels of resolution (appropriate to a multi-level transform), the filter length, the total number of coefficients and the number of wavelet coefficients in the second and third dimensions for the single-level case. This function must be called before any of the three-dimensional transform functions in this chapter.

## 2  Specification

 #include #include
 void nag_wfilt_3d (Nag_Wavelet wavnam, Nag_WaveletTransform wtrans, Nag_WaveletMode mode, Integer m, Integer n, Integer fr, Integer *nwlmax, Integer *nf, Integer *nwct, Integer *nwcn, Integer *nwcfr, Integer icomm[], NagError *fail)

## 3  Description

Three-dimensional discrete wavelet transforms (DWT) are characterised by the mother wavelet, the end extension method and whether multiresolution analysis is to be performed. For the selected combination of choices for these three characteristics, and for given dimensions ($m×n×\mathit{fr}$) of data array $A$, nag_wfilt_3d (c09acc) returns the dimension details for the transform determined by this combination. The dimension details are: ${l}_{\mathrm{max}}$, the maximum number of levels of resolution that would be computed were a multi-level DWT applied; ${n}_{f}$, the filter length; ${n}_{\mathrm{ct}}$ the total number of wavelet coefficients (over all levels in the multi-level DWT case); ${n}_{\mathrm{cn}}$, the number of coefficients in the second dimension for a single-level DWT; and ${n}_{\mathrm{cfr}}$, the number of coefficients in the third dimension for a single-level DWT. These values are also stored in the communication array icomm, as are the input choices, so that they may be conveniently communicated to the three-dimensional transform functions in this chapter.

None.

## 5  Arguments

1:     wavnamNag_WaveletInput
On entry: the name of the mother wavelet. See the c09 Chapter Introduction for details.
${\mathbf{wavnam}}=\mathrm{Nag_Haar}$
Haar wavelet.
${\mathbf{wavnam}}=\mathrm{Nag_Daubechies}\mathbit{n}$, where $\mathbit{n}=2,3,\dots ,10$
Daubechies wavelet with $\mathbit{n}$ vanishing moments ($2\mathbit{n}$ coefficients). For example, ${\mathbf{wavnam}}=\mathrm{Nag_Daubechies4}$ is the name for the Daubechies wavelet with $4$ vanishing moments ($8$ coefficients).
${\mathbf{wavnam}}=\mathrm{Nag_Biorthogonal}\mathbit{x}_\mathbit{y}$, where $\mathbit{x}_\mathbit{y}$ can be one of 1_1, 1_3, 1_5, 2_2, 2_4, 2_6, 2_8, 3_1, 3_3, 3_5 or 3_7
Biorthogonal wavelet of order $\mathbit{x}$.$\mathbit{y}$. For example ${\mathbf{wavnam}}=\mathrm{Nag_Biorthogonal1_1}$ is the name for the Biorthogonal wavelet of order $1.1$.
Constraint: ${\mathbf{wavnam}}=\mathrm{Nag_Haar}$, $\mathrm{Nag_Daubechies2}$, $\mathrm{Nag_Daubechies3}$, $\mathrm{Nag_Daubechies4}$, $\mathrm{Nag_Daubechies5}$, $\mathrm{Nag_Daubechies6}$, $\mathrm{Nag_Daubechies7}$, $\mathrm{Nag_Daubechies8}$, $\mathrm{Nag_Daubechies9}$, $\mathrm{Nag_Daubechies10}$, $\mathrm{Nag_Biorthogonal1_1}$, $\mathrm{Nag_Biorthogonal1_3}$, $\mathrm{Nag_Biorthogonal1_5}$, $\mathrm{Nag_Biorthogonal2_2}$, $\mathrm{Nag_Biorthogonal2_4}$, $\mathrm{Nag_Biorthogonal2_6}$, $\mathrm{Nag_Biorthogonal2_8}$, $\mathrm{Nag_Biorthogonal3_1}$, $\mathrm{Nag_Biorthogonal3_3}$, $\mathrm{Nag_Biorthogonal3_5}$ or $\mathrm{Nag_Biorthogonal3_7}$.
2:     wtransNag_WaveletTransformInput
On entry: the type of discrete wavelet transform that is to be applied.
${\mathbf{wtrans}}=\mathrm{Nag_SingleLevel}$
Single-level decomposition or reconstruction by discrete wavelet transform.
${\mathbf{wtrans}}=\mathrm{Nag_MultiLevel}$
Multiresolution, by a multi-level DWT or its inverse.
Constraint: ${\mathbf{wtrans}}=\mathrm{Nag_SingleLevel}$ or $\mathrm{Nag_MultiLevel}$.
3:     modeNag_WaveletModeInput
On entry: the end extension method.
${\mathbf{mode}}=\mathrm{Nag_Periodic}$
Periodic end extension.
${\mathbf{mode}}=\mathrm{Nag_HalfPointSymmetric}$
Half-point symmetric end extension.
${\mathbf{mode}}=\mathrm{Nag_WholePointSymmetric}$
Whole-point symmetric end extension.
${\mathbf{mode}}=\mathrm{Nag_ZeroPadded}$
Zero end extension.
Constraint: ${\mathbf{mode}}=\mathrm{Nag_Periodic}$, $\mathrm{Nag_HalfPointSymmetric}$, $\mathrm{Nag_WholePointSymmetric}$ or $\mathrm{Nag_ZeroPadded}$.
4:     mIntegerInput
On entry: the number of elements, $m$, in the first dimension (number of rows of each two-dimensional frame) of the input data, $A$.
Constraint: ${\mathbf{m}}\ge 2$.
5:     nIntegerInput
On entry: the number of elements, $n$, in the second dimension (number of columns of each two-dimensional frame) of the input data, $A$.
Constraint: ${\mathbf{n}}\ge 2$.
6:     frIntegerInput
On entry: the number of elements, $\mathit{fr}$, in the third dimension (number of frames) of the input data, $A$.
Constraint: ${\mathbf{fr}}\ge 2$.
7:     nwlmaxInteger *Output
On exit: the maximum number of levels of resolution, ${l}_{\mathrm{max}}$, that can be computed if a multi-level discrete wavelet transform is applied (${\mathbf{wtrans}}=\mathrm{Nag_MultiLevel}$). It is such that ${2}^{{l}_{\mathrm{max}}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n,\mathit{fr}\right)<{2}^{{l}_{\mathrm{max}}+1}$, for ${l}_{\mathrm{max}}$ an integer.
If ${\mathbf{wtrans}}=\mathrm{Nag_SingleLevel}$, nwlmax is not set.
8:     nfInteger *Output
On exit: the filter length, ${n}_{f}$, for the supplied mother wavelet. This is used to determine the number of coefficients to be generated by the chosen transform.
9:     nwctInteger *Output
On exit: the total number of wavelet coefficients, ${n}_{\mathrm{ct}}$, that will be generated. When ${\mathbf{wtrans}}=\mathrm{Nag_SingleLevel}$ the number of rows required (i.e., the first dimension of each two-dimensional frame) in each of the output coefficient arrays can be calculated as ${n}_{\mathrm{cm}}={n}_{\mathrm{ct}}/\left(8×{n}_{\mathrm{cn}}×{n}_{\mathrm{cfr}}\right)$. When ${\mathbf{wtrans}}=\mathrm{Nag_MultiLevel}$ the length of the array used to store all of the coefficient matrices must be at least ${n}_{\mathrm{ct}}$.
10:   nwcnInteger *Output
On exit: for a single-level transform (${\mathbf{wtrans}}=\mathrm{Nag_SingleLevel}$), the number of coefficients that would be generated in the second dimension, ${n}_{\mathrm{cn}}$, for each coefficient type. For a multi-level transform (${\mathbf{wtrans}}=\mathrm{Nag_MultiLevel}$) this is set to $1$.
11:   nwcfrInteger *Output
On exit: for a single-level transform (${\mathbf{wtrans}}=\mathrm{Nag_SingleLevel}$), the number of coefficients that would be generated in the third dimension, ${n}_{\mathrm{cfr}}$, for each coefficient type. For a multi-level transform (${\mathbf{wtrans}}=\mathrm{Nag_MultiLevel}$) this is set to $1$.
12:   icomm[$260$]IntegerCommunication Array
On exit: contains details of the wavelet transform and the problem dimension which is to be communicated to the two-dimensional discrete transform functions in this chapter.
13:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{fr}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{fr}}\ge 2$.
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 2$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 2$.
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.

Not applicable.

Not applicable.

None.

## 10  Example

This example computes the three-dimensional multi-level resolution for $8×8×8$ input data by a discrete wavelet transform using the Daubechies wavelet with four vanishing moments (see ${\mathbf{wavnam}}=\mathrm{Nag_Daubechies4}$ in nag_wfilt_3d (c09acc)) and zero end extension. The number of levels of transformation actually performed is one less than the maximum possible. This number of levels, the length of the wavelet filter, the total number of coefficients and the number of coefficients in each dimension for each level are printed along with the approximation coefficients before a reconstruction is performed. This example also demonstrates in general how to access any set of coefficients at any level following a multi-level transform.

### 10.1  Program Text

Program Text (c09acce.c)

### 10.2  Program Data

Program Data (c09acce.d)

### 10.3  Program Results

Program Results (c09acce.r)