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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_wav_3d_mxolap_multi_inv (c09fd)

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(nwl, c, m, n, fr, icomm, 'lenc', lenc)
[b, ifail] = nag_wav_3d_mxolap_multi_inv(nwl, 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 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, AA, nag_wav_3d_mxolap_multi_inv (c09fd) will reconstruct AA. The reconstructed array is referred to as BB in the following since it will not be identical to AA when the DWT coefficients have been filtered or otherwise manipulated prior to reconstruction. If the original input array is level 00, 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

None.

Parameters

Compulsory Input Parameters

1:     nwl – int64int32nag_int scalar
The number, nlnl, of levels to be used in the inverse multi-level transform.
Constraint: 1nwlnfwd1nwlnfwd, where nfwdnfwd is the value used in a preceding call to nag_wav_3d_multi_fwd (c09fc).
2:     c(lenc) – double array
lenc, the dimension of the array, must satisfy the constraint lencnctlencnct, where nctnct is the total number of wavelet coefficients that correspond to a transform with nwl levels.
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).
3:     m – int64int32nag_int scalar
The number of elements, mm, in the first dimension of the reconstructed array BB. For a full reconstruction of nfwdnfwd levels this must be the same as parameter m used in a preceding call to nag_wav_3d_multi_fwd (c09fc). For reconstruction of nl < nfwdnl<nfwd levels this must be equal to dwtlvm(nl + 1)dwtlvmnl+1 as returned from nag_wav_3d_multi_fwd (c09fc).
4:     n – int64int32nag_int scalar
The number of elements, nn, in the second dimension of the reconstructed array BB. For a full reconstruction of nfwdnfwd levels this must be the same as parameter n used in a preceding call to nag_wav_3d_multi_fwd (c09fc). For a partial reconstruction of nl < nfwdnl<nfwd levels this must be equal to dwtlvn(nl + 1)dwtlvnnl+1 as returned from nag_wav_3d_multi_fwd (c09fc).
5:     fr – int64int32nag_int scalar
The number of elements, frfr, in the third dimension of the reconstructed array BB. For a full reconstruction of nfwdnfwd levels this must be the same as parameter fr used in a preceding call to nag_wav_3d_multi_fwd (c09fc). For a partial reconstruction of nl < nfwdnl<nfwd levels this must be equal to dwtlvfr(nl + 1)dwtlvfrnl+1 as returned from nag_wav_3d_multi_fwd (c09fc).
6:     icomm(260260) – 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 as declared in the (sub)program from which nag_wav_3d_mxolap_multi_inv (c09fd) is called.
Constraint: lencnctlencnct, where nctnct is the total number of wavelet coefficients that correspond to a transform with nwl levels.

Input Parameters Omitted from the MATLAB Interface

ldb sdb

Output Parameters

1:     b(ldb,sdb,fr) – double array
ldbmldbm.
sdbnsdbn.
The mm by nn by frfr reconstructed array, BB, 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 = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
Constraint: nwlnfwdnwlnfwd.
Constraint: nwl1nwl1.
  ifail = 2ifail=2
Constraint: ldbmldbm.
Constraint: sdbnsdbn.
  ifail = 3ifail=3
lenc is too small, the number of wavelet coefficients required for a transform operating on nwl levels. If nwl = lmaxnwl=lmax, the maximum number of levels as returned in nwl by the initial call to nag_wav_3d_init (c09ac), then lenc must be at least nctnct, the value returned in nwct by the same call to nag_wav_3d_init (c09ac).
  ifail = 4ifail=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 = 6ifail=6
Either the initialization function has not been called first or the communication array icomm has been corrupted.
The initialization function was called with wtrans = 'S'wtrans='S'.
  ifail = 999ifail=-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

function nag_wav_3d_mxolap_multi_inv
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] = ...
      nag_wav_3d_init(wavnam, wtrans, mode, m, n, fr);

% Perform Discrete Wavelet transform
[c, dwtlvm, dwtlvn, dwtlvfr, icomm, ifail] = ...
      nag_wav_3d_multi_fwd(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] = nag_wav_3d_mxolap_multi_inv(nwl, c, m, n, fr, icomm);

% Check reconstruction matches original
eps = 10*double(m*n*fr)*nag_machine_precision;
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
 
 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.

function c09fd_example
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
 
 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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