NAG Toolbox: nag_wav_3d_coeff_ins (c09fz)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

The level at which coefficients are to be inserted.

If $\mathbf{ilev}=0$, it is assumed that the coefficient array c was produced by a preceding call to the single level function nag_wav_3d_sngl_fwd (c09fa).

If $\mathbf{ilev}>0$, it is assumed that the coefficient array c was produced by a preceding call to the multi-level function nag_wav_3d_multi_fwd (c09fc).

Constraints:

• $\mathbf{ilev}=0$ (following a call to nag_wav_3d_sngl_fwd (c09fa));
• $0\le \mathbf{ilev}\le \mathbf{nwl}$, where nwl is as used in a preceding call to nag_wav_3d_multi_fwd (c09fc);
• if $\mathbf{cindex}=0$, $\mathbf{ilev}=\mathbf{nwl}$ (following a call to nag_wav_3d_multi_fwd (c09fc)).

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, rows and frames of $A$ (LLL). After a call to the multi-level transform function nag_wav_3d_multi_fwd (c09fc) (which implies that $\mathbf{ilev}>0$) the approximation coefficients are present only for $\mathbf{ilev}=\mathbf{nwl}$, where nwl is the value used in a preceding call to nag_wav_3d_multi_fwd (c09fc).

$\mathbf{cindex}=1$

The detail coefficients produced by applying the low pass filter over columns and rows of $A$ and the high pass filter over frames (LLH).

$\mathbf{cindex}=2$

The detail coefficients produced by applying the low pass filter over columns, high pass filter over rows and low pass filter over frames of $A$ (LHL).

$\mathbf{cindex}=3$

The detail coefficients produced by applying the low pass filter over columns of $A$ and high pass filter over rows and frames (LHH).

$\mathbf{cindex}=4$

The detail coefficients produced by applying the high pass filter over columns of $A$ and low pass filter over rows and frames (HLL).

$\mathbf{cindex}=5$

The detail coefficients produced by applying the high pass filter over columns, low pass filter over rows and high pass filter over frames of $A$ (HLH).

$\mathbf{cindex}=6$

The detail coefficients produced by applying the high pass filter over columns and rows of $A$ and the low pass filter over frames (HHL).

$\mathbf{cindex}=7$

The detail coefficients produced by applying the high pass filter over columns, rows and frames of $A$ (HHH).

Constraints:

• if $\mathbf{ilev}=0$, $0\le \mathbf{cindex}\le 7$;
• if $\mathbf{ilev}=\mathbf{nwl}$, following a call to nag_wav_3d_multi_fwd (c09fc) transforming nwl levels, $0\le \mathbf{cindex}\le 7$;
• otherwise $1\le \mathbf{cindex}\le 7$.

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

Contains the DWT coefficients inserted by previous calls to nag_wav_3d_coeff_ins (c09fz), or computed by a previous call to either nag_wav_3d_sngl_fwd (c09fa) or nag_wav_3d_multi_fwd (c09fc).

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

The last dimension of the array d must be at least $n_{\mathrm{cfr}}$

The coefficients to be inserted.

If the DWT coefficients were computed by nag_wav_3d_sngl_fwd (c09fa) then

• if $\mathbf{cindex}=0$, the approximation coefficients must be stored in $\mathbf{d}\left(i,\mathit{j},\mathit{k}\right)$, for $\mathit{i}=1,2,\dots ,n_{\mathrm{cm}}$, $\mathit{j}=1,2,\dots ,n_{\mathrm{cn}}$ and $\mathit{k}=1,2,\dots ,n_{\mathrm{cfr}}$;
• if $1\le \mathbf{cindex}\le 7$, the detail coefficients, as indicated by cindex, must be stored in $\mathbf{d}\left(i,\mathit{j},\mathit{k}\right)$, for $\mathit{i}=1,2,\dots ,n_{\mathrm{cm}}$, $\mathit{j}=1,2,\dots ,n_{\mathrm{cn}}$ and $\mathit{k}=1,2,\dots ,n_{\mathrm{cfr}}$.

If the DWT coefficients were computed by nag_wav_3d_multi_fwd (c09fc) then

• if $\mathbf{cindex}=0$ and $\mathbf{ilev}=\mathbf{nwl}$, the approximation coefficients must be stored in $\mathbf{d}\left(i,\mathit{j},\mathit{k}\right)$, for $\mathit{i}=1,2,\dots ,n_{\mathrm{cm}}$, $\mathit{j}=1,2,\dots ,n_{\mathrm{cn}}$ and $\mathit{k}=1,2,\dots ,n_{\mathrm{cfr}}$;
• if $1\le \mathbf{cindex}\le 7$, the detail coefficients, as indicated by cindex, for level ilev must be stored in $\mathbf{d}\left(i,\mathit{j},\mathit{k}\right)$, for $\mathit{i}=1,2,\dots ,n_{\mathrm{cm}}$, $\mathit{j}=1,2,\dots ,n_{\mathrm{cn}}$ and $\mathit{k}=1,2,\dots ,n_{\mathrm{cfr}}$.

5:     $\mathrm{icomm}\left(260\right)$ – 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:     $\mathrm{lenc}$ – 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 either nag_wav_3d_sngl_fwd (c09fa) or nag_wav_3d_multi_fwd (c09fc)..

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 one of the reconstruction functions nag_wav_3d_sngl_inv (c09fb) (if nag_wav_3d_sngl_fwd (c09fa) was called previously) or nag_wav_3d_mxolap_multi_inv (c09fd) (if nag_wav_3d_multi_fwd (c09fc) was called previously).

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

Communication array, used to store information between calls to nag_wav_3d_coeff_ins (c09fz).

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: $\mathbf{ilev}=0$ following a call to the single level function nag_wav_3d_sngl_fwd (c09fa).

Constraint: $\mathbf{ilev}>0$ following a call to the multi-level function nag_wav_3d_multi_fwd (c09fc).

Constraint: $\mathbf{ilev}\le \mathbf{nwl}$, where $\mathbf{nwl}$ is the number of levels used in the call to nag_wav_3d_multi_fwd (c09fc).

   $\mathbf{ifail}=2$

Constraint: $\mathbf{cindex}\le 7$.

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_3d_sngl_fwd (c09fa).

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_3d_multi_fwd (c09fc).

   $\mathbf{ifail}=4$

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.

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

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

Constraint: $\mathit{sdd}\ge n_{\mathrm{cn}}$, where $n_{\mathrm{cn}}$ is the number of DWT coefficients in the second dimension following the single level transform.

   $\mathbf{ifail}=5$

Constraint: $\mathbf{cindex}>0$ when $\mathbf{ilev}<\mathbf{nwl}$ in the preceding call to nag_wav_3d_multi_fwd (c09fc).

   $\mathbf{ifail}=6$

Either the initialization function has not been called first 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: c09fz_example

function c09fz_example


fprintf('c09fz example results\n\n');

% 3D Data
m = int64(4);
n = int64(4);
f = int64(4);
a = zeros(m,n,f);
a(1:2:m,:,1:2:f) = 0.01;
a(2:2:m,:,2:2:f) = 0.01;
a(2:2:m,:,1:2:f) = 1;
a(1:2:m,:,2:2:f) = 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*f, 0, 1.0e-4, state);
an = a + reshape(x,[m,n,f]);

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

% Wavelet setup
wavnam = 'Haar';
mode = 'Period';
wtrans = 'Multilevel';
[nwl, nf, nwct, nwcn, nwcf, icomm, ifail] = ...
c09ac(...
      wavnam, wtrans, mode, m, n, f);

% Multi-level wavelet transform on noisy data
[c, dwtlvm, dwtlvn, dwtlvfr, icomm, ifail] = ...
c09fc(...
       n, f, an, nwct, nwl, icomm);

% Reconstruct without thresholding of detail coefficients
[b, ifail] = c09fd(nwl, c, m, n, f, icomm);
 
% Mean square error of noisy reconstruction
mse = (norm(reshape(a-b,[m*n*f,1]))^2)/(double(m*n*f));
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*f)));
nt = 0;
nnt = 0;
for ilev = 1:nwl
  level = int64(nwl - ilev + 1);

  for detail = int64(1:7)
    % Extract the selected set of coefficients. 
    [d, icomm, ifail] = c09fy(...
                              level, detail, c, icomm);

    % Threshold
    d1 = dwtlvm(ilev);
    d2 = dwtlvn(ilev);
    d3 = dwtlvfr(ilev);
    for i = 1:d1
      for j = 1:d2
        for k = 1:d3
          if abs(d(i,j,k))<thresh
            d(i,j,k) = 0;
            nt = nt + 1;
          end
          nnt = nnt + 1;
        end
      end
    end
    % Insert de-noised coefficients back into c
    [c, icomm, ifail] = c09fz(...
                               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] = c09fd(nwl, c, m, n, f, icomm);

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

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

c09fz example results


Input data a:

(:,:,1) =

    0.0100    0.0100    0.0100    0.0100
    1.0000    1.0000    1.0000    1.0000
    0.0100    0.0100    0.0100    0.0100
    1.0000    1.0000    1.0000    1.0000


(:,:,2) =

    1.0000    1.0000    1.0000    1.0000
    0.0100    0.0100    0.0100    0.0100
    1.0000    1.0000    1.0000    1.0000
    0.0100    0.0100    0.0100    0.0100


(:,:,3) =

    0.0100    0.0100    0.0100    0.0100
    1.0000    1.0000    1.0000    1.0000
    0.0100    0.0100    0.0100    0.0100
    1.0000    1.0000    1.0000    1.0000


(:,:,4) =

    1.0000    1.0000    1.0000    1.0000
    0.0100    0.0100    0.0100    0.0100
    1.0000    1.0000    1.0000    1.0000
    0.0100    0.0100    0.0100    0.0100


Noisy data an:

(:,:,1) =

    0.0135   -0.0093   -0.0004    0.0378
    1.0015    0.9842    1.0007    0.9889
   -0.0017    0.0139    0.0138   -0.0049
    0.9899    1.0070    1.0049    0.9983


(:,:,2) =

    1.0094    1.0080    0.9921    0.9902
    0.0105   -0.0009    0.0160    0.0197
    0.9994    1.0044    0.9956    1.0014
    0.0091   -0.0084    0.0187    0.0023


(:,:,3) =

    0.0058   -0.0053    0.0011    0.0159
    1.0113    0.9894    1.0018    0.9992
    0.0106    0.0082    0.0093    0.0153
    1.0023    1.0157    1.0084    0.9834


(:,:,4) =

    0.9969    1.0010    0.9904    0.9968
    0.0227    0.0022    0.0062    0.0214
    0.9948    0.9981    0.9951    0.9968
    0.0121    0.0103    0.0114    0.0206

Without denoising Mean Square Error is  0.000081

Number of coefficients denoised is  55 out of  63
With denoising Mean Square Error is  0.000015

Reconstruction of denoised input: 

(:,:,1) =

    0.0053    0.0053    0.0166    0.0166
    1.0026    1.0026    0.9913    0.9913
    0.0055    0.0055    0.0077    0.0077
    1.0025    1.0025    1.0003    1.0003


(:,:,2) =

    1.0026    1.0026    0.9913    0.9913
    0.0053    0.0053    0.0166    0.0166
    1.0025    1.0025    1.0003    1.0003
    0.0055    0.0055    0.0077    0.0077


(:,:,3) =

    0.0073    0.0073    0.0110    0.0110
    1.0006    1.0006    0.9969    0.9969
    0.0078    0.0078    0.0131    0.0131
    1.0002    1.0002    0.9949    0.9949


(:,:,4) =

    1.0006    1.0006    0.9969    0.9969
    0.0073    0.0073    0.0110    0.0110
    1.0002    1.0002    0.9949    0.9949
    0.0078    0.0078    0.0131    0.0131