NAG Toolbox: nag_wav_2d_multi_fwd (c09ec)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathit{lda},\mathbf{n}\right)$ – double array

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

The $m$ by $n$ data matrix $A$.

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

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_wav_2d_init (c09ab) 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 $\mathbf{mode}=\text{'P'}$ in nag_wav_2d_init (c09ab) and $3\times ⌊\left(\stackrel{-}{m}+n_f-1\right)/2⌋\times ⌊\left(\stackrel{-}{n}+n_f-1\right)/2⌋$ for $\mathbf{mode}=\text{'H'},\text{'W'},\text{'Z'}$, 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_wav_2d_init (c09ab). 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.

3:     $\mathrm{nwl}$ – 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_2d_init (c09ab).

4:     $\mathrm{icomm}\left(180\right)$ – 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_2d_init (c09ab).

Optional Input Parameters

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

Default: the first dimension of the array a.

Number of rows, $m$, of data matrix $A$.

Constraint: this must be the same as the value m passed to the initialization function nag_wav_2d_init (c09ab).

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

Default: the second dimension of the array a.

Number of columns, $n$, of data matrix $A$.

Constraint: this must be the same as the value n passed to the initialization function nag_wav_2d_init (c09ab).

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_2d_coeff_ext (c09ey) or nag_wav_2d_coeff_ins (c09ez) 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 $q\left(i\right)=\mathbf{dwtlvm}\left(n_{\mathrm{fwd}}-i+1\right)\times \mathbf{dwtlvn}\left(n_{\mathrm{fwd}}-i+1\right)$. Then, letting $k_1=q\left(n_{\mathrm{fwd}}\right)$ and $k_{\mathit{j}+1}=k_{\mathit{j}}+q\left(n_{\mathrm{fwd}}-⌈\mathit{j}/3⌉+1\right)$, for $\mathit{j}=1,2,\dots ,3n_{\mathrm{fwd}}$, the coefficients are stored in c as follows:

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

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

$\mathbf{c}\left(\mathit{i}\right)$, 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 $j=1,\dots ,3n_{\mathrm{fwd}}$

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 $\mathit{i}=1,2,\dots ,n_{\mathrm{fwd}}$. Thus for the first $n_{\mathrm{fwd}}-1$ levels of resolution, $\mathbf{dwtlvm}\left(n_{\mathrm{fwd}}-\mathit{i}+1\right)$ 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, $\mathbf{dwtlvm}\left(1\right)$ is the size of the first dimension of the matrices of approximation, vertical, horizontal and diagonal coefficients computed.

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 $\mathit{i}=1,2,\dots ,n_{\mathrm{fwd}}$. Thus for the first $n_{\mathrm{fwd}}-1$ levels of resolution, $\mathbf{dwtlvn}\left(n_{\mathrm{fwd}}-\mathit{i}+1\right)$ 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, $\mathbf{dwtlvn}\left(1\right)$ is the size of the second dimension of the matrices of approximation, vertical, horizontal and diagonal coefficients computed.

4:     $\mathrm{icomm}\left(180\right)$ – int64int32nag_int array

Contains additional information on the computed transform.

5:     $\mathrm{ifail}$ – 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: $\mathbf{m}=\_$, the value of m on initialization (see nag_wav_2d_init (c09ab)).

Constraint: $\mathbf{n}=\_$, the value of n on initialization (see nag_wav_2d_init (c09ab)).

   $\mathbf{ifail}=2$

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

   $\mathbf{ifail}=3$

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

   $\mathbf{ifail}=5$

Constraint: $\mathbf{nwl}\le \mathbf{nwlmax}$ in nag_wav_2d_init (c09ab).

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

   $\mathbf{ifail}=7$

Either the initialization function has not been called first or icomm has been corrupted.

Either the initialization function was called with $\mathbf{wtrans}=\text{'S'}$ or icomm has been corrupted.

   $\mathbf{ifail}=-99$

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: c09ec_example

function c09ec_example


fprintf('c09ec example results\n\n');

m = int64(7);
n = int64(8);
a = [3, 7, 9, 1, 9, 9, 1, 0;
     9, 9, 3, 3, 4, 1, 2, 4;
     7, 8, 1, 3, 8, 9, 3, 3;
     1, 1, 1, 1, 2, 8, 4, 0;
     1, 2, 4, 6, 5, 6, 5, 4;
     2, 2, 5, 7, 3, 6, 6, 8;
     7, 9, 3, 1, 3, 4, 7, 2];

fprintf('\nInput data a:\n');
disp(a);

wavnam = 'DB2';
mode = 'Half';
wtrans = 'Multilevel';
[nwl, nf, nwct, nwcn, icomm, ifail] = ...
  c09ab(...
        wavnam, wtrans, mode, m, n);

lenc = nwct;
% Perform Discrete Wavelet transform
[c, dwtlvm, dwtlvn, icomm, ifail] = c09ec(a, lenc, nwl, icomm);

fprintf('\nLength of wavelet filter : %d\n', nf);
fprintf('Number of Levels :         %d\n', nwl);
fprintf('Number of coefficients in first dimension for each level :\n');
disp(transpose(dwtlvm(1:nwl)));
fprintf('Number of coefficients in second dimension for each level :\n');
disp(transpose(dwtlvn(1:nwl)));

fprintf('\nTotal number of wavelet coefficients : %d\n', nwct);
fprintf('\nWavelet coefficients c :\n');

for ilevel = 1:nwl
  fprintf('-------------------------------------------------------\n');
  d1 = dwtlvm(ilevel);
  d2 = dwtlvn(ilevel);
  level = nwl-ilevel+1;
  fprintf('Level %d output is %d by %d\n', level, d1, d2);
  fprintf('-------------------------------------------------------\n');

  for itype_coeffs = int64(1:4)
    switch itype_coeffs
      case {1}
        if (ilevel == 1)
          fprintf('Approximation coefficients:\n');
        end
      case {2}
        fprintf('Vertical coefficients:\n');
      case {3}
        fprintf('Horizontal coefficients:\n');
      case {4}
        fprintf('Diagonal coefficients:\n');
    end
    if (itype_coeffs>1 || ilevel==1)
      % Use 2D extraction routine c09eyf
      [d, icomm, ifail] = c09ey(...
                                level, itype_coeffs-1, c, icomm);
      disp(d(1:d1,1:d2));
    end
  end
  fprintf('\n');
end

% Reconstruct original data
[b, ifail] = c09ed(nwl, c, m, n, icomm);
fprintf('Reconstruction       b:\n');
disp(b);

c09ec example results


Input data a:
     3     7     9     1     9     9     1     0
     9     9     3     3     4     1     2     4
     7     8     1     3     8     9     3     3
     1     1     1     1     2     8     4     0
     1     2     4     6     5     6     5     4
     2     2     5     7     3     6     6     8
     7     9     3     1     3     4     7     2


Length of wavelet filter : 4
Number of Levels :         2
Number of coefficients in first dimension for each level :
                    4                    5

Number of coefficients in second dimension for each level :
                    4                    5


Total number of wavelet coefficients : 139

Wavelet coefficients c :
-------------------------------------------------------
Level 2 output is 4 by 4
-------------------------------------------------------
Approximation coefficients:
   24.9724   25.6017   20.8900    7.9280
   27.6100   27.0955   18.7941    8.2804
   11.2663   11.0273   19.6410   18.6651
   27.6050   26.6443   14.5913   18.0835

Vertical coefficients:
   -2.5552   -6.1078   -4.0629    8.2136
   -1.6061   -7.2355   -3.3633    7.6075
   -0.2225   -1.6283   -0.5301    3.7415
   -0.9052   -6.5810    0.8023    1.8591

Horizontal coefficients:
   -3.8069   -3.0730    2.1121   -1.8525
   -2.7548   -4.5949   -0.8321   -4.8155
    4.8398    4.5104   -1.5308   -0.6456
   -6.4332   -4.5381    2.4753    6.8224

Diagonal coefficients:
   -0.8978   -0.2326   -1.2515    2.6346
    0.5708   -4.9783   -1.5309    6.4569
   -0.1854   -1.8430    0.2426   -0.0754
    0.0345    7.1864    1.5938   -5.9745


-------------------------------------------------------
Level 1 output is 5 by 5
-------------------------------------------------------
Vertical coefficients:
   -2.5981    4.6471    2.5392   -2.8415   -0.2165
   -1.3203   -0.0592    3.0490   -2.5837    1.0458
   -0.4330   -1.6405   -1.1752    0.2533   -2.3448
   -0.4118   -0.0682   -2.4608   -0.0167    0.4387
   -1.5368   -1.1450   -0.5547    4.5936   -3.6863

Horizontal coefficients:
   -4.3301   -1.8170    0.8023    5.7566   -2.8146
    4.3089    3.6908    0.8349    3.4653    1.7108
   -1.5311   -1.0736    1.5257    0.0212   -0.9608
    2.8873    3.1148   -1.9118   -0.4007   -1.5302
   -2.2377   -2.7611    2.4453   -0.3705    4.3448

Diagonal coefficients:
   -1.5000    4.4151   -0.0057   -0.8236   -1.1250
   -0.1953   -2.9530    1.8840   -1.7635    0.9877
   -0.4330    0.2745    1.1450    0.4632   -0.5547
   -0.3538   -0.3215    0.6462    1.3705   -1.2778
    0.7288    0.4587   -1.8873   -1.8828    2.4028


Reconstruction       b:
    3.0000    7.0000    9.0000    1.0000    9.0000    9.0000    1.0000    0.0000
    9.0000    9.0000    3.0000    3.0000    4.0000    1.0000    2.0000    4.0000
    7.0000    8.0000    1.0000    3.0000    8.0000    9.0000    3.0000    3.0000
    1.0000    1.0000    1.0000    1.0000    2.0000    8.0000    4.0000    0.0000
    1.0000    2.0000    4.0000    6.0000    5.0000    6.0000    5.0000    4.0000
    2.0000    2.0000    5.0000    7.0000    3.0000    6.0000    6.0000    8.0000
    7.0000    9.0000    3.0000    1.0000    3.0000    4.0000    7.0000    2.0000