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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_wav_3d_mxolap_multi_inv (c09fd)

▸▿ 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_3d_mxolap_multi_inv (c09fd) computes the inverse three-dimensional multi-level discrete wavelet transform (IDWT). This function reconstructs data from (possibly filtered or otherwise manipulated) wavelet transform coefficients calculated by nag_wav_3d_multi_fwd (c09fc) from an original input array. The initialization function nag_wav_3d_init (c09ac) must be called first to set up the IDWT options.

Syntax

[b, ifail] = c09fd(nwlinv, c, m, n, fr, icomm, 'lenc', lenc)
[b, ifail] = nag_wav_3d_mxolap_multi_inv(nwlinv, c, m, n, fr, icomm, 'lenc', lenc)

Description

nag_wav_3d_mxolap_multi_inv (c09fd) performs the inverse operation of nag_wav_3d_multi_fwd (c09fc). That is, given a set of wavelet coefficients, computed up to level nfwd by nag_wav_3d_multi_fwd (c09fc) using a DWT as set up by the initialization function nag_wav_3d_init (c09ac), on a real three-dimensional array, A, nag_wav_3d_mxolap_multi_inv (c09fd) will reconstruct A. The reconstructed array 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 array 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_3d_multi_fwd (c09fc). This results in a partial reconstruction.

References

Wang Y, Che X and Ma S (2012) Nonlinear filtering based on 3D wavelet transform for MRI denoising URASIP Journal on Advances in Signal Processing 2012:40

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_3d_multi_fwd (c09fc). The data will be reconstructed to level nwl-nwlinv, where level 0 is the original input dataset provided to nag_wav_3d_multi_fwd (c09fc).
Constraint: 1≤nwlinv≤nwl, where nwl is the value used in a preceding call to nag_wav_3d_multi_fwd (c09fc).
2:     clenc – double array
The coefficients of the multi-level discrete wavelet transform. This will normally be the result of some transformation on the coefficients computed by function nag_wav_3d_multi_fwd (c09fc).
Note that the coefficients in c may be extracted according to level and type into three-dimensional arrays using nag_wav_3d_coeff_ext (c09fy), and inserted using nag_wav_3d_coeff_ins (c09fz).
3:     m – int64int32nag_int scalar
The number of elements, m, in the first dimension of the reconstructed array 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 m used in a preceding call to nag_wav_3d_multi_fwd (c09fc). For a partial reconstruction of nwlinv<nwl levels, this must be equal to dwtlvmnwlinv+1, as returned from nag_wav_3d_multi_fwd (c09fc)
4:     n – int64int32nag_int scalar
The number of elements, n, in the second dimension of the reconstructed array 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 a preceding call to nag_wav_3d_multi_fwd (c09fc). For a partial reconstruction of nwlinv<nwl levels, this must be equal to dwtlvnnwlinv+1, as returned from nag_wav_3d_multi_fwd (c09fc).
5:     fr – int64int32nag_int scalar
The number of elements, fr, in the third dimension of the reconstructed array 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 fr used in a preceding call to nag_wav_3d_multi_fwd (c09fc). For a partial reconstruction of nwlinv<nwl levels, this must be equal to dwtlvfrnwlinv+1, as returned from nag_wav_3d_multi_fwd (c09fc).
6:     icomm260 – 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:     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 wavelet coefficients that correspond to a transform with nwlinv levels.

Output Parameters

1:     bldbsdbfr – double array
sdb=n.
The m by n by fr reconstructed array, B, with Bijk stored in bijk. The reconstruction is based on the input multi-level wavelet transform coefficients and the transform options supplied to the initialization function nag_wav_3d_init (c09ac).
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≤nwl as used in the call to nag_wav_3d_multi_fwd (c09fc).
Constraint: nwlinv≥1.
   ifail=2
Constraint: ldb≥m.
Constraint: sdb≥n.
   ifail=3
lenc is too small, the number of wavelet coefficients required for a transform operating on nwlinv levels. If nwlinv=nwlmax, the maximum number of levels as returned by the initial call to nag_wav_3d_init (c09ac), then lenc must be at least nct, the value returned in nwct by the same call to nag_wav_3d_init (c09ac).
   ifail=4
fr is too small, the number of coefficients in the third dimension at the required level of reconstruction.
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 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 wtrans='S'.
   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_3d_multi_fwd (c09fc).

Open in the MATLAB editor: c09fd_example

function c09fd_example


fprintf('c09fd example results\n\n');

m  = int64(7);
n  = int64(6);
fr = int64(5);
wavnam = 'Bior1.1';
mode = 'period';
wtrans = 'Multilevel';
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
[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 coefficients
want_level = 1;

% Select the approximation coefficients.
want_coeffs = 4;

% Identify each set of coefficients in c
for ilevel = nwl:-1:1

  if ilevel ~= want_level
    continue
  end

  nwcm = dwtlvm(nwl-ilevel+1);
  nwcn = dwtlvn(nwl-ilevel+1);
  nwcfr = dwtlvfr(nwl-ilevel+1);

  fprintf('\n--------------------------------\n');
  fprintf(' Level %d output is %d by %d by %d.\n', ilevel, nwcm, nwcn, nwcfr);
  fprintf('--------------------------------\n\n');

  for itype_coeffs = 0:7

    if itype_coeffs ~= want_coeffs
      continue
    end

    % Unless we're looking at the deepest level of nesting, which contains
    % approximation coefficients, advance the pointer on past the preceding
    % levels
    if ilevel == nwl
      locc = 0;
    else
      locc = 8*dwtlvm(1)*dwtlvn(1)*dwtlvfr(1);
      for i = ilevel + 1 : nwl - 1
        locc = locc + 7*dwtlvm(nwl-i+1)*dwtlvn(nwl-i+1)*dwtlvfr(nwl-i+1);
      end
    end

    % Now decide which coefficient type we are considering
    switch (itype_coeffs)
      case {0}
        if (ilevel==nwl)
          fprintf('Approximation coefficients (LLL)\n');
          locc = locc + 1;
        end
      case {1}
        fprintf('Detail coefficients (LLH)\n');
        if (ilevel==nwl)
          % Advance pointer past approximation coefficients
          locc = locc + nwcm*nwcn*nwcfr + 1;
        else
          locc = locc + 1;
        end
      case {2}
        fprintf('Detail coefficients (LHL)\n');
        if (ilevel==nwl)
          % Advance pointer past approximation coefficients and 1 set of
          % detail coefficients
          locc = locc + 2*nwcm*nwcn*nwcfr + 1;
        else
          % Advance pointer past 1 set of detail coefficients
          locc = locc + nwcm*nwcn*nwcfr + 1;
        end
      case {3}
        fprintf('Detail coefficients (LHH)\n');
        if (ilevel==nwl)
          % Advance pointer past approximation coefficients and 2 sets of
          % detail coefficients
          locc = locc + 3*nwcm*nwcn*nwcfr + 1;
        else
          % Advance pointer past 2 sets of detail coefficients
          locc = locc + 2*nwcm*nwcn*nwcfr + 1;
        end
      case {4}
        fprintf('Detail coefficients (HLL)\n');
        if (ilevel==nwl)
          % Advance pointer past approximation coefficients and 3 sets of
          % detail coefficients
          locc = locc + 4*nwcm*nwcn*nwcfr + 1;
        else
          % Advance pointer past 3 sets of detail coefficients
          locc = locc + 3*nwcm*nwcn*nwcfr + 1;
        end
      case {5}
        fprintf('Detail coefficients (HLH)\n');
        if (ilevel==nwl)
          % Advance pointer past approximation coefficients and 4 sets of
          % detail coefficients
          locc = locc + 5*nwcm*nwcn*nwcfr + 1;
        else
          % Advance pointer past 4 sets of detail coefficients
          locc = locc + 4*nwcm*nwcn*nwcfr + 1;
        end
      case {6}
        fprintf('Detail coefficients (HHL)\n');
        if (ilevel==nwl)
          % Advance pointer past approximation coefficients and 5 sets of
          % detail coefficients
          locc = locc + 6*nwcm*nwcn*nwcfr + 1;
        else
          % Advance pointer past 4 sets of detail coefficients
          locc = locc + 5*nwcm*nwcn*nwcfr + 1;
        end
      case {7}
        fprintf('Detail coefficients (HHH)\n');
        if (ilevel==nwl)
          % Advance pointer past approximation coefficients and 6 sets of
          % detail coefficients
          locc = locc + 7*nwcm*nwcn*nwcfr + 1;
        else
          % Advance pointer past 5 sets of detail coefficients
          locc = locc + 6*nwcm*nwcn*nwcfr + 1;
        end
      end

  if itype_coeffs > 0 || ilevel == nwl

    if (itype_coeffs==0)
      % For a multi level transform approx coeffs stored as
      % nwcm x nwcn x nwcfr
      i1 = locc;
      for k = 1:nwcfr
        for j = 1:nwcn
          for i = 1:nwcm
            d(i,j,k) = c(i1);
            i1 = i1 + 1;
          end
        end
      end
    else
      % ... but detail coefficients are stored as ncwfr x nwcm x nwcn
      for k = 1:nwcfr
        for j = 1:nwcn
          for i = 1:nwcm
            i1 = locc - 1 + (j-1)*nwcfr*nwcm + (i-1)*nwcfr + k;
            d(i,j,k) = c(i1);
          end
        end
      end
    end

    % Print out the selected set of coefficients
    fprintf('Level %d, Coefficients %d:\n', ilevel, itype_coeffs);
    for k = 1:nwcfr
      fprintf('Frame %d:\n', k);
      for i = 1:nwcm
        for j=1:nwcn
          fprintf('%8.4f ', d(i, j, k));
        end
        fprintf('\n');
      end
    end

  end

  end
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



c09fd 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

--------------------------------
 Level 1 output is 4 by 3 by 3.
--------------------------------

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

Success: the reconstruction matches the original.
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