hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int
PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_wav_2d_multi_inv (c09ed)

▸▿ Contents

    1  Purpose
    2  Syntax
    3  Description
    4  References
▸▿ 5  Parameters
    6  Error Indicators and Warnings
    7  Accuracy
    8  Further Comments
    9  Example

Purpose

nag_wav_2d_multi_inv (c09ed) computes the inverse two-dimensional multi-level discrete wavelet transform (DWT). This function reconstructs data from (possibly filtered or otherwise manipulated) wavelet transform coefficients calculated by nag_wav_2d_multi_fwd (c09ec) from an original input matrix. The initialization function nag_wav_2d_init (c09ab) must be called first to set up the DWT options.

Syntax

[b, ifail] = c09ed(nwlinv, c, m, n, icomm, 'lenc', lenc)
[b, ifail] = nag_wav_2d_multi_inv(nwlinv, c, m, n, icomm, 'lenc', lenc)

Description

nag_wav_2d_multi_inv (c09ed) performs the inverse operation of nag_wav_2d_multi_fwd (c09ec). That is, given a set of wavelet coefficients, computed up to level nfwd by nag_wav_2d_multi_fwd (c09ec) using a DWT as set up by the initialization function nag_wav_2d_init (c09ab), on a real matrix, A, nag_wav_2d_multi_inv (c09ed) will reconstruct A. The reconstructed matrix is referred to as B in the following since it will not be identical to A when the DWT coefficients have been filtered or otherwise manipulated prior to reconstruction. If the original input matrix is level 0, then it is possible to terminate reconstruction at a higher level by specifying fewer than the number of levels used in the call to nag_wav_2d_multi_fwd (c09ec). This results in a partial reconstruction.

References

None.

Parameters

Compulsory Input Parameters

1:     nwlinv – int64int32nag_int scalar
The number of levels to be used in the inverse multi-level transform. The number of levels must be less than or equal to nfwd, which has the value of argument nwl as used in the computation of the wavelet coefficients using nag_wav_2d_multi_fwd (c09ec). The data will be reconstructed to level nwl-nwlinv, where level 0 is the original input dataset provided to nag_wav_2d_multi_fwd (c09ec).
Constraint: 1≤nwlinv≤nwl, where nwl is the value used in a preceding call to nag_wav_2d_multi_fwd (c09ec).
2:     clenc – double array
The coefficients of a multi-level wavelet transform of the original matrix, A, which may have been filtered or otherwise manipulated.
Let qi be the number of coefficients (of each type) at level i, for i=nfwd,nfwd-1,…,1. Then, setting k1=qnfwd and kj+1=kj+qnfwd-j/3+1, for j=1,2,…,3nfwd, the coefficients are stored in c as follows:
ci, for i=1,2,…,k1
Contains the level nfwd approximation coefficients, anfwd.
ci, for i=kj+1,…,kj+1
Contains the level nfwd-j/3+1 vertical, horizontal and diagonal coefficients. These are:
  • vertical coefficients if j mod 3=1;
  • horizontal coefficients if j mod 3=2;
  • diagonal coefficients if j mod 3=0,
for j=1,…,3⁢nfwd
Note that the coefficients in c may be extracted according to level and type into two-dimensional arrays using nag_wav_2d_coeff_ext (c09ey), and inserted using nag_wav_2d_coeff_ins (c09ez).
3:     m – int64int32nag_int scalar
The number of elements, m, in the first dimension of the reconstructed matrix B. For a full reconstruction of nwl levels, where nwl is as supplied to nag_wav_2d_multi_fwd (c09ec), this must be the same as argument m used in the call to nag_wav_2d_multi_fwd (c09ec). For a partial reconstruction of nwlinv<nwl levels, this must be equal to dwtlvmnwlinv+1, as returned from nag_wav_2d_multi_fwd (c09ec).
4:     n – int64int32nag_int scalar
The number of elements, n, in the second dimension of the reconstructed matrix B. For a full reconstruction of nwl levels, where nwl is as supplied to nag_wav_3d_multi_fwd (c09fc), this must be the same as argument n used in the call to nag_wav_2d_multi_fwd (c09ec). For a partial reconstruction of nwlinv<nwl, this must be equal to dwtlvnnwlinv+1, as returned from nag_wav_2d_multi_fwd (c09ec).
5:     icomm180 – 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:     lenc – int64int32nag_int scalar
Default: the dimension of the array c.
The dimension of the array c.
Constraint: lenc≥nct, where nct is the total number of coefficients that correspond to a transform with nwlinv levels and is unchanged from the preceding call to nag_wav_2d_multi_fwd (c09ec).

Output Parameters

1:     bldbn – double array
The m by n reconstructed matrix, B, based on the input multi-level wavelet transform coefficients and the transform options supplied to the initialization function nag_wav_2d_init (c09ab).
2:     ifail – int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=1
Constraint: nwlinv≤nfwd.
Constraint: nwlinv≥1.
   ifail=2
Constraint: ldb≥m.
   ifail=3
lenc is too small, the total number of coefficients generated by the preceding call to nag_wav_2d_multi_fwd (c09ec).
   ifail=4
m is too small, the number of coefficients in the first dimension at the required level of reconstruction.
n is too small, the number of coefficients in the second dimension at the required level of reconstruction.
   ifail=6
Either the initialization function has not been called first or icomm has been corrupted.
Either the initialization function was called with wtrans='S' or icomm has been corrupted.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

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.

Further Comments

None.

Example

See Example in nag_wav_2d_multi_fwd (c09ec).

Open in the MATLAB editor: c09ed_example

function c09ed_example


fprintf('c09ed example results\n\n');

m = int64(7);
n = int64(8);
wavnam = 'DB2';
mode = 'Half';
wtrans = 'Multilevel';
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);
[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:double(nwl))));
fprintf('Number of coefficients in second dimension for each level :\n');
disp(transpose(dwtlvn(1:double(nwl))));

fprintf('\nTotal number of wavelet coefficients : %d\n', nwct);
fprintf('\nWavelet coefficients c :\n');
jstart = 1;
for ilevel = 1:double(nwl)
  fprintf('-------------------------------------------------------\n');
  fprintf('Level %d output is %d by %d\n', ...
     nwl-ilevel+1, dwtlvm(ilevel), dwtlvn(ilevel));
  fprintf('-------------------------------------------------------\n');

  iskip = double(dwtlvm(ilevel));
  i2 = iskip*double(dwtlvn(ilevel)) - 1;

  for itype_coeffs = 1:4
    switch itype_coeffs
      case {1}
        if (ilevel == nwl)
          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)
      for i1 = jstart:jstart+iskip-1
        fprintf('%8.4f',c(i1:iskip:i1+i2));
        fprintf('\n');
      end
      jstart = jstart + i2 + 1;
    end
  end
  fprintf('\n');
end

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



c09ed 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
-------------------------------------------------------
 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
-------------------------------------------------------
Approximation coefficients:
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
PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox
© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015