NAG Library Function Document

nag_mldwt_2d (c09ecc)

1  Purpose

2  Specification

3  Description

4  References

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[$\mathbf{lda}\times \mathbf{n}$] – const doubleInput

Note: the $\left(i,j\right)$th element of the matrix $A$ is stored in $\mathbf{a}\left[\left(j-1\right)\times \mathbf{lda}+i-1\right]$.

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: $\mathbf{lda}\ge \mathbf{m}$.

5:     lenc – IntegerInput

On entry: the dimension of the array c. c must be large enough to contain, $n_{\mathrm{ct}}$, wavelet coefficients. The maximum value of $n_{\mathrm{ct}}$ is returned in nwct by the call to the initialization function nag_wfilt_2d (c09abc) and corresponds to the DWT being continued for the maximum number of levels possible for the given data set. When the number of levels, $n_{\mathrm{fwd}}$, is chosen to be less than the maximum, $l_{\max }$, then $n_{\mathrm{ct}}$ is correspondingly smaller and lenc can be reduced by noting that the vertical, horizontal and diagonal coefficients are stored at every level and that in addition the approximation coefficients are stored for the final level only. The number of coefficients stored at each level is given by $3\times ⌈\stackrel{-}{m}/2⌉\times ⌈\stackrel{-}{n}/2⌉$ for $\mathbf{mode}=\mathrm{Nag\_Periodic}$ in nag_wfilt_2d (c09abc) and $3\times ⌊\left(\stackrel{-}{m}+n_f-1\right)/2⌋\times ⌊\left(\stackrel{-}{n}+n_f-1\right)/2⌋$ for $\mathbf{mode}=\mathrm{Nag\_HalfPointSymmetric}$, $\mathrm{Nag\_WholePointSymmetric}$ or $\mathrm{Nag\_ZeroPadded}$, where the input data is of dimension $\stackrel{-}{m}\times \stackrel{-}{n}$ at that level and $n_f$ is the filter length nf provided by the call to nag_wfilt_2d (c09abc). At the final level the storage is $4/3$ times this value to contain the set of approximation coefficients.

Constraint: $\mathbf{lenc}\ge n_{\mathrm{ct}}$, where $n_{\mathrm{ct}}$ is the total number of coefficients that correspond to a transform with nwl levels.

6:     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_2d_coeff_ext (c09eyc) or nag_wav_2d_coeff_ins (c09ezc) is recommended. For completeness the following description provides details of precisely how the coefficient are stored in c but this information should only be required in rare cases.

Let $q\left(\mathit{i}\right)$ denote the number of coefficients (of each type) at level $\mathit{i}$, for $\mathit{i}=1,2,\dots ,n_{\mathrm{fwd}}$, such that $q\left(i\right)=\mathbf{dwtlvm}\left[n_{\mathrm{fwd}}-i\right]\times \mathbf{dwtlvn}\left[n_{\mathrm{fwd}}-i\right]$. Then, letting $k_1=q\left(n_{\mathrm{fwd}}\right)$ and $k_{\mathit{j}+1}=k_{\mathit{j}}+q\left(n_{\mathrm{fwd}}-⌈\mathit{j}/3⌉+1\right)$, for $\mathit{j}=1,2,\dots ,3n_{\mathrm{fwd}}$, the coefficients are stored in c as follows:

$\mathbf{c}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,k_1$

Contains the level $n_{\mathrm{fwd}}$ approximation coefficients, $a_{n_{\mathrm{fwd}}}$.

$\mathbf{c}\left[\mathit{i}-1\right]$, for $\mathit{i}=k_j+1,\dots ,k_{j+1}$

Contains the level $n_{\mathrm{fwd}}-⌈j/3⌉+1$ vertical, horizontal and diagonal coefficients. These are:

• vertical coefficients if $j\mathrm{ mod }3=1$;
• horizontal coefficients if $j\mathrm{ mod }3=2$;
• diagonal coefficients if $j\mathrm{ mod }3=0$,

for $j=1,\dots ,3n_{\mathrm{fwd}}$.

7:     nwl – IntegerInput

On entry: the number of levels, $n_{\mathrm{fwd}}$, in the multi-level resolution to be performed.

Constraint: $1\le \mathbf{nwl}\le l_{\max }$, where $l_{\max }$ is the value returned in nwlmax (the maximum number of levels) by the call to the initialization function nag_wfilt_2d (c09abc).

8:     dwtlvm[nwl] – IntegerOutput

On exit: the number of coefficients in the first dimension for each coefficient type at each level. $\mathbf{dwtlvm}\left[\mathit{i}-1\right]$ contains the number of coefficients in the first dimension (for each coefficient type computed) at the ($n_{\mathrm{fwd}}-\mathit{i}+1$)th level of resolution, for $\mathit{i}=1,2,\dots ,n_{\mathrm{fwd}}$. Thus for the first $n_{\mathrm{fwd}}-1$ levels of resolution, $\mathbf{dwtlvm}\left[n_{\mathrm{fwd}}-\mathit{i}\right]$ is the size of the first dimension of the matrices of vertical, horizontal and diagonal coefficients computed at this level; for the final level of resolution, $\mathbf{dwtlvm}\left[0\right]$ is the size of the first dimension of the matrices of approximation, vertical, horizontal and diagonal coefficients computed.

9:     dwtlvn[nwl] – IntegerOutput

On exit: the number of coefficients in the second dimension for each coefficient type at each level. $\mathbf{dwtlvn}\left[\mathit{i}-1\right]$ contains the number of coefficients in the second dimension (for each coefficient type computed) at the ($n_{\mathrm{fwd}}-\mathit{i}+1$)th level of resolution, for $\mathit{i}=1,2,\dots ,n_{\mathrm{fwd}}$. Thus for the first $n_{\mathrm{fwd}}-1$ levels of resolution, $\mathbf{dwtlvn}\left[n_{\mathrm{fwd}}-\mathit{i}\right]$ is the size of the second dimension of the matrices of vertical, horizontal and diagonal coefficients computed at this level; for the final level of resolution, $\mathbf{dwtlvn}\left[0\right]$ is the size of the second dimension of the matrices of approximation, vertical, horizontal and diagonal coefficients computed.

10:   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).

On exit: contains additional information on the computed transform.

11:   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 $⟨\mathit{\text{value}}⟩$ 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 $\mathbf{wtrans}=\mathrm{Nag\_SingleLevel}$ or icomm has been corrupted.

NE_INT

On entry, $\mathbf{m}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{m}=⟨\mathit{\text{value}}⟩$, the value of m on initialization (see nag_wfilt_2d (c09abc)).

On entry, $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{n}=⟨\mathit{\text{value}}⟩$, the value of n on initialization (see nag_wfilt_2d (c09abc)).

On entry, $\mathbf{nwl}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nwl}\ge 1$.

NE_INT_2

On entry, $\mathbf{lda}=⟨\mathit{\text{value}}⟩$ and $\mathbf{m}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{lda}\ge \mathbf{m}$.

On entry, $\mathbf{lenc}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{lenc}\ge ⟨\mathit{\text{value}}⟩$, the total number of coefficents to be generated.

On entry, $\mathbf{nwl}=⟨\mathit{\text{value}}⟩$ and $\mathbf{nwlmax}=⟨\mathit{\text{value}}⟩$ in nag_wfilt_2d (c09abc).
Constraint: $\mathbf{nwl}\le \mathbf{nwlmax}$ in nag_wfilt_2d (c09abc).

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

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (c09ecce.c)

10.2  Program Data

Program Data (c09ecce.d)

10.3  Program Results

Program Results (c09ecce.r)