NAG Toolbox: nag_wav_2d_coeff_ins (c09ez)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

The level at which coefficients are to be inserted.

Constraints:

• $1\le \mathbf{ilev}\le \mathbf{nwl}$, where nwl is as used in a preceding call to nag_wav_2d_multi_fwd (c09ec);
• if $\mathbf{cindex}=0$, $\mathbf{ilev}=\mathbf{nwl}$.

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

Identifies which coefficients to insert. The coefficients are identified as follows:

$\mathbf{cindex}=0$

The approximation coefficients, produced by application of the low pass filter over columns and rows of the original matrix ($\mathrm{LL}$). The approximation coefficients are present only for $\mathbf{ilev}=\mathbf{nwl}$, where nwl is the value used in a preceding call to nag_wav_2d_multi_fwd (c09ec).

$\mathbf{cindex}=1$

The vertical detail coefficients produced by applying the low pass filter over columns of the original matrix and the high pass filter over rows ($\mathrm{LH}$).

$\mathbf{cindex}=2$

The horizontal detail coefficients produced by applying the high pass filter over columns of the original matrix and the low pass filter over rows ($\mathrm{HL}$).

$\mathbf{cindex}=3$

The diagonal detail coefficients produced by applying the high pass filter over columns and rows of the original matrix ($\mathrm{HH}$).

Constraint: $0\le \mathbf{cindex}\le 3$ when $\mathbf{ilev}=\mathbf{nwl}$ as used in nag_wav_2d_multi_fwd (c09ec), otherwise $1\le \mathbf{cindex}\le 3$.

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

Contains the DWT coefficients inserted by previous calls to nag_wav_2d_coeff_ins (c09ez), or computed by a previous call to nag_wav_2d_multi_fwd (c09ec).

4:     $\mathrm{d}\left(\mathit{ldd},:\right)$ – double array

The first dimension of the array d must be at least $n_{\mathrm{cm}}$.

The second dimension of the array d must be at least $\mathit{ncn}$.

The coefficients to be inserted.

If $\mathbf{ilev}=\mathbf{nwl}$ (as used in nag_wav_2d_multi_fwd (c09ec)) and $\mathbf{cindex}=0$, the $n_{\mathrm{cm}}$ by $n_{\mathrm{cn}}$ manipulated approximation coefficients $a_{\mathit{i}\mathit{j}}$ must be stored in $\mathbf{d}\left(\mathit{i},\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,n_{\mathrm{cm}}$ and $\mathit{i}=1,2,\dots ,n_{\mathrm{cn}}$.

Otherwise the $n_{\mathrm{cm}}$ by $n_{\mathrm{cn}}$ manipulated level ilev detail coefficients (of type specified by cindex) $d_{\mathit{i}\mathit{j}}$ must be stored in $\mathbf{d}\left(\mathit{i},\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,n_{\mathrm{cm}}$ and $\mathit{j}=1,2,\dots ,n_{\mathrm{cn}}$.

5:     $\mathrm{icomm}\left(180\right)$ – 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:     $\mathrm{lenc}$ – int64int32nag_int scalar

Default: the dimension of the array c.

The dimension of the array c.

Constraint: lenc must be unchanged from the value used in the preceding call to nag_wav_2d_multi_fwd (c09ec)..

Output Parameters

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

Contains the same DWT coefficients provided on entry except for those identified by ilev and cindex, which are updated with the values supplied in d, inserted into the correct locations as expected by the reconstruction function nag_wav_2d_multi_inv (c09ed).

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

Communication array, used to store information between calls to nag_wav_2d_coeff_ins (c09ez).

3:     $\mathrm{ifail}$ – 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: $\mathbf{ilev}\le \mathbf{nwl}$, where $\mathbf{nwl}$ is the number of levels used in the call to nag_wav_2d_multi_fwd (c09ec).

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

   $\mathbf{ifail}=2$

Constraint: $\mathbf{cindex}\le 3$.

Constraint: $\mathbf{cindex}\ge 0$.

   $\mathbf{ifail}=3$

Constraint: $\mathbf{lenc}\ge n_{\mathrm{ct}}$, where $n_{\mathrm{ct}}$ is the number of DWT coefficients computed in a previous call to nag_wav_2d_multi_fwd (c09ec).

   $\mathbf{ifail}=4$

Constraint: $\mathit{ldd}\ge n_{\mathrm{cm}}$, where $n_{\mathrm{cm}}$ is the number of DWT coefficients in the first dimension at the selected level ilev.

   $\mathbf{ifail}=5$

Constraint: $\mathbf{cindex}>0$ when $\mathbf{ilev}<\mathbf{nwl}$ in the preceding call to nag_wav_2d_multi_fwd (c09ec).

   $\mathbf{ifail}=6$

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: c09ez_example

function c09ez_example


fprintf('c09ez example results\n\n');

% 2D Data
m = int64(7);
n = int64(6);
a = zeros(m,n);
a(1:2:m,:) = 0.01;
a(2:2:m,:) = 1;

% Add Noise
genid = int64(3);
subid = int64(0);
seed(1) = int64(642521);
[state, ifail] = g05kf(genid, subid, seed);
[state, x, ifail] = g05sk(m*n, 0, 1.0e-4, state);
an = a + reshape(x,[m,n]);

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

% Wavelet setup
wavnam = 'DB6';
mode = 'Period';
wtrans = 'Multilevel';
[nwl, nf, nwct, nwcn, icomm, ifail] = ...
  c09ab(...
        wavnam, wtrans, mode, m, n);

% Multi-level wavelet transform on noisy data
lenc = nwct;
% Perform Discrete Wavelet transform
[c, dwtlvm, dwtlvn, icomm, ifail] = ...
  c09ec(...
        an, lenc, nwl, icomm);

% Reconstruct without thresholding of detail coefficients
[b, ifail] = c09ed(nwl, c, m, n, icomm);
 
% Mean square error of noisy reconstruction
mse = (norm(reshape(a-b,[m*n,1]))^2)/(double(m*n));
fprintf('Without denoising Mean Square Error is %9.6f\n',mse);

% De-noise by applying hard threshold to detail coefficients
thresh = 0.01*sqrt(2*log(double(m*n)));
nt = 0;
nnt = 0;
for ilev = 1:nwl
  level = int64(nwl - ilev + 1);

  for detail = int64(1:3)
    % Extract the selected set of coefficients. 
    [d, icomm, ifail] = c09ey(...
                               level, detail, c, icomm);
  
    % Threshold
    d1 = dwtlvm(ilev);
    d2 = dwtlvn(ilev);
    for i = 1:d1
      for j = 1:d2
        if abs(d(i,j))<thresh
           d(i,j) = 0;
           nt = nt + 1;
        end
        nnt = nnt + 1;
      end
    end
    % Insert de-noised coefficients back into c
    [c, icomm, ifail] = c09ez(...
                               level, detail, c, d, icomm);
  end
end

fprintf('\nNumber of coefficients denoised is %3d out of %3d\n',nt,nnt);

% Reconstruct data after threholding
[b, ifail] = c09ed(nwl, c, m, n, icomm);

% Mean square error of de-noised reconstruction
mse = (norm(reshape(a-b,[m*n,1]))^2)/(double(m*n));
fprintf('With denoising Mean Square Error is %9.6f\n\n',mse);

disp('Reconstruction of denoised input: ');
disp(b);

c09ez example results


Input data a:
    0.0100    0.0100    0.0100    0.0100    0.0100    0.0100
    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000
    0.0100    0.0100    0.0100    0.0100    0.0100    0.0100
    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000
    0.0100    0.0100    0.0100    0.0100    0.0100    0.0100
    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000
    0.0100    0.0100    0.0100    0.0100    0.0100    0.0100


Noisy data an:
    0.0135    0.0170   -0.0049   -0.0009    0.0002    0.0123
    1.0015    0.9896    0.9983    1.0044    1.0097    0.9847
   -0.0017    0.0107    0.0194   -0.0084    0.0114   -0.0006
    0.9899    1.0038    1.0005    0.9921    0.9923    0.9982
   -0.0093    0.0149    0.0094    0.0160    0.0058    0.0257
    0.9842    1.0278    0.9991    0.9956    1.0113    0.9911
    0.0139   -0.0011    0.0180    0.0187    0.0106    0.0118

Without denoising Mean Square Error is  0.000098

Number of coefficients denoised is  32 out of  48
With denoising Mean Square Error is  0.000018

Reconstruction of denoised input: 
    0.0127    0.0094    0.0030    0.0007    0.0009    0.0065
    0.9913    0.9940    1.0000    1.0027    1.0032    0.9976
    0.0084    0.0086    0.0072    0.0048    0.0028    0.0050
    1.0009    0.9998    0.9966    0.9942    0.9930    0.9965
    0.0061    0.0070    0.0103    0.0134    0.0154    0.0114
    1.0034    1.0036    1.0028    1.0011    0.9996    1.0011
    0.0135    0.0113    0.0093    0.0114    0.0147    0.0148