NAG Toolbox: nag_wav_3d_multi_fwd (c09fc)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

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_wav_3d_init (c09ac).

2:     $\mathrm{fr}$ – int64int32nag_int scalar

The number of two-dimensional frames.

Constraint: this must be the same as the value fr passed to the initialization function nag_wav_3d_init (c09ac).

3:     $\mathrm{a}\left(\mathit{lda},\mathit{sda},\mathbf{fr}\right)$ – double array

lda, the first dimension of the array, must satisfy the constraint $\mathit{lda}\ge \mathbf{m}$.

The $m$ by $n$ by $\mathit{fr}$ three-dimensional input data $A$, where with $A_{ijk}$ stored in $\mathbf{a}\left(i,j,k\right)$.

4:     $\mathrm{lenc}$ – int64int32nag_int scalar

The dimension of the array c.

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

5:     $\mathrm{nwl}$ – int64int32nag_int scalar

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_wav_3d_init (c09ac).

6:     $\mathrm{icomm}\left(260\right)$ – int64int32nag_int array

Contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization function nag_wav_3d_init (c09ac).

Optional Input Parameters

1:     $\mathrm{m}$ – int64int32nag_int scalar

Default: the first dimension of the array a.

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_wav_3d_init (c09ac).

Output Parameters

1:     $\mathrm{c}\left(\mathbf{lenc}\right)$ – double array

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 (c09fy) or nag_wav_3d_coeff_ins (c09fz) 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.

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 $q\left(i\right)=\mathbf{dwtlvm}\left(n_{\mathrm{fwd}}-i+1\right)\times \mathbf{dwtlvn}\left(n_{\mathrm{fwd}}-i+1\right)\times \mathbf{dwtlvfr}\left(n_{\mathrm{fwd}}-i+1\right)$. Then, letting $k_1=q\left(n_{\mathrm{fwd}}\right)$ and $k_{\mathit{j}+1}=k_{\mathit{j}}+q\left(n_{\mathrm{fwd}}-⌈\mathit{j}/7⌉+1\right)$, for $\mathit{j}=1,2,\dots ,7n_{\mathrm{fwd}}$, the coefficients are stored in c as follows:

$\mathbf{c}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,k_1$

Contains the level $n_{\mathrm{fwd}}$ approximation coefficients, $a_{n_{\mathrm{fwd}}}$. Note that for computational efficiency reasons these coefficients are stored as $\mathbf{dwtlvm}\left(1\right)\times \mathbf{dwtlvn}\left(1\right)\times \mathbf{dwtlvfr}\left(1\right)$ in c.

$\mathbf{c}\left(\mathit{i}\right)$, for $\mathit{i}=k_j+1,\dots ,k_{j+1}$

Contains the level $n_{\mathrm{fwd}}-⌈j/7⌉+1$ detail coefficients. These are:

• LLH coefficients if $j\mathrm{ mod }7=1$;
• LHL coefficients if $j\mathrm{ mod }7=2$;
• LHH coefficients if $j\mathrm{ mod }7=3$;
• HLL coefficients if $j\mathrm{ mod }7=4$;
• HLH coefficients if $j\mathrm{ mod }7=5$;
• HHL coefficients if $j\mathrm{ mod }7=6$;
• HHH coefficients if $j\mathrm{ mod }7=0$,

for $j=1,\dots ,7n_{\mathrm{fwd}}$. See Multiresolution and higher dimensional DWT in the C09 Chapter Introduction for a description of how these coefficients are produced.

Note that for computational efficiency reasons these coefficients are stored as $\mathbf{dwtlvfr}\left(⌈j/7⌉\right)\times \mathbf{dwtlvm}\left(⌈j/7⌉\right)\times \mathbf{dwtlvn}\left(⌈j/7⌉\right)$ in c.

2:     $\mathrm{dwtlvm}\left(\mathbf{nwl}\right)$ – int64int32nag_int array

The number of coefficients in the first dimension for each coefficient type at each level. $\mathbf{dwtlvm}\left(\mathit{i}\right)$ 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 $\mathit{i}=1,2,\dots ,n_{\mathrm{fwd}}$.

3:     $\mathrm{dwtlvn}\left(\mathbf{nwl}\right)$ – int64int32nag_int array

The number of coefficients in the second dimension for each coefficient type at each level. $\mathbf{dwtlvn}\left(\mathit{i}\right)$ 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 $\mathit{i}=1,2,\dots ,n_{\mathrm{fwd}}$.

4:     $\mathrm{dwtlvfr}\left(\mathbf{nwl}\right)$ – int64int32nag_int array

The number of coefficients in the third dimension for each coefficient type at each level. $\mathbf{dwtlvfr}\left(\mathit{i}\right)$ contains the number of coefficients in the third dimension (for each coefficient type computed) at the ($n_{\mathrm{fwd}}-\mathit{i}+1$)th level of resolution, for $\mathit{i}=1,2,\dots ,n_{\mathrm{fwd}}$.

5:     $\mathrm{icomm}\left(260\right)$ – int64int32nag_int array

Contains additional information on the computed transform.

6:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{ifail}=1$

Constraint: $\mathbf{fr}=\_$, the value of fr on initialization (see nag_wav_3d_init (c09ac)).

Constraint: $\mathbf{m}=\_$, the value of m on initialization (see nag_wav_3d_init (c09ac)).

Constraint: $\mathbf{n}=\_$, the value of n on initialization (see nag_wav_3d_init (c09ac)).

   $\mathbf{ifail}=2$

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

Constraint: $\mathit{sda}\ge \mathbf{n}$.

   $\mathbf{ifail}=3$

lenc is too small, the total number of coefficents to be generated.

   $\mathbf{ifail}=5$

Constraint: $\mathbf{nwl}\le \mathbf{nwlmax}$ in nag_wav_3d_init (c09ac).

Constraint: $\mathbf{nwl}\ge 1$.

   $\mathbf{ifail}=6$

Either the communication array icomm has been corrupted or there has not been a prior call to the initialization function nag_wav_3d_init (c09ac).

The initialization function was called with $\mathbf{wtrans}=\text{'S'}$.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: c09fc_example

function c09fc_example


fprintf('c09fc example results\n\n');

% Data
m  = int64(7);
n  = int64(6);
fr = int64(5);
a = zeros(m, n, fr);
a(:, :, 1) = [3, 2, 2, 2, 1, 1;
              2, 9, 1, 2, 1, 3;
              2, 5, 1, 2, 1, 1;
              1, 6, 2, 2, 7, 2;
              5, 3, 2, 2, 4, 7;
              2, 2, 1, 1, 2, 1;
              6, 2, 1, 3, 6, 9];
a(:, :, 2) = [2, 1, 5, 1, 2, 3;
              2, 9, 5, 2, 1, 2;
              2, 3, 2, 7, 1, 1;
              2, 1, 1, 2, 3, 1;
              2, 1, 2, 8, 3, 3;
              1, 4, 5, 1, 2, 7;
              8, 1, 3, 9, 1, 2];
a(:, :, 3) = [3, 1, 4, 1, 1, 1;
              1, 1, 2, 1, 2, 6;
              4, 1, 7, 2, 5, 6;
              3, 2, 1, 5, 9, 5;
              1, 1, 2, 2, 2, 1;
              2, 6, 3, 9, 5, 1;
              1, 1, 8, 2, 1, 3];
a(:, :, 4) = [5, 8, 1, 2, 2, 1;
              1, 2, 2, 9, 2, 9;
              2, 2, 2, 1, 1, 3;
              1, 1, 1, 5, 1, 2;
              3, 2, 8, 1, 9, 2;
              2, 1, 9, 1, 2, 2;
              3, 6, 5, 3, 2, 2];
a(:, :, 5) = [5, 2, 1, 2, 1, 1;
              3, 1, 9, 1, 2, 1;
              2, 3, 1, 1, 7, 2;
              7, 2, 2, 6, 1, 1;
              5, 1, 7, 2, 1, 1;
              2, 1, 3, 2, 2, 1;
              5, 3, 9, 1, 4, 1];

% Query wavelet filter dimensions
wavnam = 'Bior1.1';
mode   = 'period';
wtrans = 'Multilevel';
[nwl, nf, nwct, nwcn, nwcfr, icomm, ifail] = ...
  c09ac(...
         wavnam, wtrans, mode, m, n, fr);

% Perform Discrete Wavelet transform
[c, dwtlvm, dwtlvn, dwtlvfr, icomm, ifail] = ...
  c09fc(...
         n, fr, a, nwct, nwl, icomm);

fprintf(' Number of Levels : %d\n\n', nwl);
fprintf(' Number of coefficients in 1st dimension for each level:\n');
fprintf(' %8d', dwtlvm(1:nwl));
fprintf('\n');
fprintf(' Number of coefficients in 2nd dimension for each level:\n');
fprintf(' %8d', dwtlvn(1:nwl));
fprintf('\n');
fprintf(' Number of coefficients in 3rd dimension for each level:\n');
fprintf(' %8d', dwtlvfr(1:nwl));
fprintf('\n');

% Print the first level HLL detail coefficients
want_level = int64(1);
want_coeffs = int64(4);

cpass = char('LLH','LHL','LHH','HLL','HLH','HHL','HHH');

if (want_coeffs==0) 
  title = 'Approximation coefficients (LLL)'
else
  title = sprintf('Detail coefficients (%s)',cpass(want_coeffs,:));
end

% Extract coefficients
[d, icomm, ifail] = c09fy(...
                           want_level, want_coeffs, c, icomm);
fprintf('\n%s\n Level %2d, Coefficients %2d :\n',title, want_level, ...
        want_coeffs );

% Matrix printing arguments
matrix = 'General'; diag   = 'Non-unit'; fmt = 'F9.4';
labrow = 'Integer'; labcol = labrow;
rlabs  = {' '};     clabs  = rlabs;
ncols  = int64(80); indent = int64(0);

nk = dwtlvfr(nwl-want_level+1);
for k = 1:nk
  title = sprintf('Frame: %3d',k);
  [ifail] =  x04cb(...
                    matrix, diag, d(:,:,k), fmt, title, labrow, ...
                   rlabs, labcol, clabs, ncols, indent);
end
 
% Reconstruct original data
[b, ifail] = c09fd(nwl, c, m, n, fr, icomm);

% Check reconstruction matches original
eps = 10*double(m*n*fr)*x02aj;
err = a-b;
frob = 0;
for i=1:fr
  fnew = sqrt(sum(sum(err(:,:,i).^2)));
  frob = max(frob,fnew);
end

if frob < eps
  fprintf('\nSuccess: the reconstruction matches the original.\n');
else
  fprintf('\nFail: Frobenius norm of b-a is too large.\n');
end

c09fc example results

 Number of Levels : 2

 Number of coefficients in 1st dimension for each level:
        2        4
 Number of coefficients in 2nd dimension for each level:
        2        3
 Number of coefficients in 3rd dimension for each level:
        2        3

Detail coefficients (HLL)
 Level  1, Coefficients  4 :
 Frame:   1
           1        2        3
 1   -4.9497   0.0000   0.0000
 2    0.7071   1.7678  -3.1820
 3    0.7071   2.1213   1.7678
 4    0.0000   0.0000   0.0000
 Frame:   2
           1        2        3
 1    4.2426  -2.1213  -4.9497
 2    0.7071  -0.0000  -0.7071
 3   -1.4142  -3.1820   1.4142
 4    0.0000   0.0000   0.0000
 Frame:   3
           1        2        3
 1    2.1213  -4.9497  -0.7071
 2   -2.8284  -4.2426   4.9497
 3    2.1213   2.8284  -0.7071
 4    0.0000   0.0000   0.0000

Success: the reconstruction matches the original.