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_3d_sngl_fwd (c09fa)

Purpose

nag_wav_3d_sngl_fwd (c09fa) computes the three-dimensional discrete wavelet transform (DWT) at a single level. The initialization function nag_wav_3d_init (c09ac) must be called first to set up the DWT options.

Syntax

[c, icomm, ifail] = c09fa(n, fr, a, lenc, icomm, 'm', m)
[c, icomm, ifail] = nag_wav_3d_sngl_fwd(n, fr, a, lenc, icomm, 'm', m)

Description

nag_wav_3d_sngl_fwd (c09fa) computes the three-dimensional DWT of a given input three-dimensional data array, considered as a number of two-dimensional frames, at a single level. For a chosen wavelet filter pair, the output coefficients are obtained by applying convolution and downsampling by two to the input, AA, first over columns, next over rows and finally across frames. The three-dimensional approximation coefficients are produced by the low pass filter over columns, rows and frames. In addition there are 7 sets of three-dimensional detail coefficients, each corresponding to a different order of low pass and high pass filters (see the C09 Chapter Introduction). All coefficients are packed into a single array. To reduce distortion effects at the ends of the data array, several end extension methods are commonly used. Those provided are: periodic or circular convolution end extension, half-point symmetric end extension, whole-point symmetric end extension and zero end extension. The total number, nctnct, of coefficients computed is returned by the initialization function nag_wav_3d_init (c09ac).

References

Daubechies I (1992) Ten Lectures on Wavelets SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1:     n – int64int32nag_int scalar
The second dimension of the input data: the number of columns of each two-dimensional frame.
Constraint: this must be the same as the value n passed to the initialization function nag_wav_3d_init (c09ac).
2:     fr – int64int32nag_int scalar
The third dimension of the input data: the number of two-dimensional frames.
Constraint: this must be the same as the value fr passed to the initialization function nag_wav_3d_init (c09ac).
3:     a(lda,sda,fr) – double array
lda, the first dimension of the array, must satisfy the constraint ldamldam.
The mm by nn by frfr input three-dimensional array AA.
4:     lenc – int64int32nag_int scalar
The dimension of the array c as declared in the (sub)program from which nag_wav_3d_sngl_fwd (c09fa) is called.
Constraint: lencnctlencnct, where nctnct is the total number of wavelet coefficients, as returned by nag_wav_3d_init (c09ac).
5:     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:     m – int64int32nag_int scalar
Default: The first dimension of the array a.
The first dimension of the input data: the number of rows of each two-dimensional frame.
Constraint: this must be the same as the value m passed to the initialization function nag_wav_3d_init (c09ac).

Input Parameters Omitted from the MATLAB Interface

lda sda

Output Parameters

1:     c(lenc) – double array
The coefficients of the discrete wavelet transform. The 88 sets of coefficients are stored in the following order: approximation coefficients (LLL) first, followed by 77 sets of detail coefficients: LLH, LHL, LHH, HLL, HLH, HHL, HHH, where L indicates the low pass filter, and H the high pass filter being applied to, respectively, the columns of length m, the rows of length n and then the frames of length fr. Note that for computational efficiency reasons each set of coefficients is stored in the order ncfr × ncm × ncnncfr×ncm×ncn (see output parameters nwcfr, nwct and nwcn in nag_wav_3d_init (c09ac)). See Section [Example] for details of how to access each set of coefficients in order to perform extraction from c following a call to this function, or insertion into c before a call to the three-dimensional inverse routine nag_wav_3d_sngl_inv (c09fb).
2:     icomm(260260) – int64int32nag_int array
Contains additional information on the computed transform.
3:     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: fr = frfr=fr, the value of fr on initialization (see nag_wav_3d_init (c09ac)).
Constraint: m = mm=m, the value of m on initialization (see nag_wav_3d_init (c09ac)).
Constraint: n = nn=n, the value of n on initialization (see nag_wav_3d_init (c09ac)).
  ifail = 2ifail=2
Constraint: ldamldam.
Constraint: sdansdan.
  ifail = 3ifail=3
Constraint: lencnctlencnct, where nctnct is the number of DWT coefficients returned by nag_wav_3d_init (c09ac) in parameter nwct.
  ifail = 6ifail=6
Either the initialization function nag_wav_3d_init (c09ac) has not been called first or the communication array icomm has been corrupted.
The initialization function was called with wtrans = 'M'wtrans='M'.
  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_sngl_fwd_example
m  = int64(5);
n  = int64(4);
fr = int64(3);
wavnam = 'Haar';
mode = 'half';
wtrans = 'Single Level';
a = zeros(m, n, fr);
a(:, :, 1) = [3, 2, 2, 2;
              2, 9, 1, 2;
              2, 5, 1, 2;
              1, 6, 2, 2;
              5, 3, 2, 2];
a(:, :, 2) = [2, 1, 5, 1;
              2, 9, 5, 2;
              2, 3, 2, 7;
              2, 1, 1, 2;
              2, 1, 2, 8];
a(:, :, 3) = [3, 1, 4, 1;
              1, 1, 2, 1;
              4, 1, 7, 2;
              3, 2, 1, 5;
              1, 1, 2, 2];


% Query wavelet filter dimensions
[lmax, nf, nwct, nwcn, nwcfr, icomm, ifail] = ...
      nag_wav_3d_init(wavnam, wtrans, mode, m, n, fr);

nwcm = nwct/(8*nwcn*nwcfr);

% 3D DWT decomposition
[c, icomm, ifail] = nag_wav_3d_sngl_fwd(n, fr, a, nwct, icomm);

d = zeros(nwcm, nwcn, nwcfr);

for cindex = 0:7

  % Decide which coefficient type we are considering and advance the
  % pointer locc to the first element of that 3D array in C.
  switch (cindex)
    case {0}
      fprintf('Approximation coefficients (LLL)\n');
      locc = 1;
    case {1}
      fprintf('Detail coefficients (LLH)\n');
      % Advance pointer past approximation coefficients
      locc = nwcm*nwcn*nwcfr + 1;
    case {2}
      fprintf('Detail coefficients (LHL)\n');
      % Advance pointer past approximation coefficients and 1 set of
      % detail coefficients
      locc = 2*nwcm*nwcn*nwcfr + 1;
    case {3}
      fprintf('Detail coefficients (LHH)\n');
      % Advance pointer past approximation coefficients and 2 sets of
      % detail coefficients
      locc = 3*nwcm*nwcn*nwcfr + 1;
    case {4}
      fprintf('Detail coefficients (HLL)\n');
      % Advance pointer past approximation coefficients and 3 sets of
      % detail coefficients
      locc = 4*nwcm*nwcn*nwcfr + 1;
    case {5}
      fprintf('Detail coefficients (HLH)\n');
      % Advance pointer past approximation coefficients and 4 sets of
      % detail coefficients
      locc = 5*nwcm*nwcn*nwcfr + 1;
    case {6}
      fprintf('Detail coefficients (HHL)\n');
      % Advance pointer past approximation coefficients and 5 sets of
      % detail coefficients
      locc = 6*nwcm*nwcn*nwcfr + 1;
    case {7}
      fprintf('Detail coefficients (HHH)\n');
      % Advance pointer past approximation coefficients and 6 sets of
      % detail coefficients
      locc = 7*nwcm*nwcn*nwcfr + 1;
  end

  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

  for j = 1:nwcfr
    if (j==1)
      fprintf('Coefficients        Frame 1');
    else
      fprintf('          Frame %d', j);
    end
  end
  fprintf('\n');
  d2 = reshape(d, nwcm, nwcn*nwcfr);
  for i = 1:nwcm
    if i == 1
      fprintf('    %d         ', cindex);
    else
      fprintf('              ');
    end
    for j=1:nwcn*nwcfr
      fprintf('%8.4f ', d2(i,j));
    end
    fprintf('\n');
  end
end


% 3D DWT reconstruction
[b, ifail] = nag_wav_3d_sngl_inv(m, n, fr, c, icomm);

fprintf('\nOutput Data          b : \n');
% Convert to int16 to get more compact output
for i=1:nwcm
  fprintf('Frame %d :\n', i);
  disp(b(:, :, i));
end
 
Approximation coefficients (LLL)
Coefficients        Frame 1          Frame 2
    0          10.6066   7.0711   4.2426   5.6569 
                7.7782   6.7175   7.0711  10.6066 
                7.7782   9.8995   2.8284   5.6569 
Detail coefficients (LLH)
Coefficients        Frame 1          Frame 2
    1           0.7071  -2.1213   0.0000   0.0000 
                2.1213  -1.7678   0.0000   0.0000 
                3.5355  -4.2426   0.0000   0.0000 
Detail coefficients (LHL)
Coefficients        Frame 1          Frame 2
    2          -4.2426   2.1213   1.4142   2.8284 
               -2.8284  -2.4749   2.8284   0.7071 
                2.1213  -4.2426   0.0000   0.0000 
Detail coefficients (LHH)
Coefficients        Frame 1          Frame 2
    3           0.0000  -2.8284   0.0000   0.0000 
               -2.8284   1.7678   0.0000   0.0000 
                0.7071   4.2426   0.0000   0.0000 
Detail coefficients (HLL)
Coefficients        Frame 1          Frame 2
    4          -4.9497   0.0000   1.4142   1.4142 
                0.7071   1.7678  -0.0000   2.1213 
                0.0000   0.0000   0.0000   0.0000 
Detail coefficients (HLH)
Coefficients        Frame 1          Frame 2
    5           0.7071   0.7071   0.0000   0.0000 
               -0.7071  -2.4749   0.0000   0.0000 
                0.0000   0.0000   0.0000   0.0000 
Detail coefficients (HHL)
Coefficients        Frame 1          Frame 2
    6           5.6569   0.7071   1.4142   1.4142 
                0.0000  -1.7678   1.4142   6.3640 
                0.0000   0.0000   0.0000   0.0000 
Detail coefficients (HHH)
Coefficients        Frame 1          Frame 2
    7           0.0000   0.0000   0.0000   0.0000 
                1.4142   1.0607   0.0000   0.0000 
                0.0000   0.0000   0.0000   0.0000 

Output Data          b : 
Frame 1 :
    3.0000    2.0000    2.0000    2.0000
    2.0000    9.0000    1.0000    2.0000
    2.0000    5.0000    1.0000    2.0000
    1.0000    6.0000    2.0000    2.0000
    5.0000    3.0000    2.0000    2.0000

Frame 2 :
    2.0000    1.0000    5.0000    1.0000
    2.0000    9.0000    5.0000    2.0000
    2.0000    3.0000    2.0000    7.0000
    2.0000    1.0000    1.0000    2.0000
    2.0000    1.0000    2.0000    8.0000

Frame 3 :
    3.0000    1.0000    4.0000    1.0000
    1.0000    1.0000    2.0000    1.0000
    4.0000    1.0000    7.0000    2.0000
    3.0000    2.0000    1.0000    5.0000
    1.0000    1.0000    2.0000    2.0000


function c09fa_example
m  = int64(5);
n  = int64(4);
fr = int64(3);
wavnam = 'Haar';
mode = 'half';
wtrans = 'Single Level';
a = zeros(m, n, fr);
a(:, :, 1) = [3, 2, 2, 2;
              2, 9, 1, 2;
              2, 5, 1, 2;
              1, 6, 2, 2;
              5, 3, 2, 2];
a(:, :, 2) = [2, 1, 5, 1;
              2, 9, 5, 2;
              2, 3, 2, 7;
              2, 1, 1, 2;
              2, 1, 2, 8];
a(:, :, 3) = [3, 1, 4, 1;
              1, 1, 2, 1;
              4, 1, 7, 2;
              3, 2, 1, 5;
              1, 1, 2, 2];


% Query wavelet filter dimensions
[lmax, nf, nwct, nwcn, nwcfr, icomm, ifail] = ...
      c09ac(wavnam, wtrans, mode, m, n, fr);

nwcm = nwct/(8*nwcn*nwcfr);

% 3D DWT decomposition
[c, icomm, ifail] = c09fa(n, fr, a, nwct, icomm);

d = zeros(nwcm, nwcn, nwcfr);

for cindex = 0:7

  % Decide which coefficient type we are considering and advance the
  % pointer locc to the first element of that 3D array in C.
  switch (cindex)
    case {0}
      fprintf('Approximation coefficients (LLL)\n');
      locc = 1;
    case {1}
      fprintf('Detail coefficients (LLH)\n');
      % Advance pointer past approximation coefficients
      locc = nwcm*nwcn*nwcfr + 1;
    case {2}
      fprintf('Detail coefficients (LHL)\n');
      % Advance pointer past approximation coefficients and 1 set of
      % detail coefficients
      locc = 2*nwcm*nwcn*nwcfr + 1;
    case {3}
      fprintf('Detail coefficients (LHH)\n');
      % Advance pointer past approximation coefficients and 2 sets of
      % detail coefficients
      locc = 3*nwcm*nwcn*nwcfr + 1;
    case {4}
      fprintf('Detail coefficients (HLL)\n');
      % Advance pointer past approximation coefficients and 3 sets of
      % detail coefficients
      locc = 4*nwcm*nwcn*nwcfr + 1;
    case {5}
      fprintf('Detail coefficients (HLH)\n');
      % Advance pointer past approximation coefficients and 4 sets of
      % detail coefficients
      locc = 5*nwcm*nwcn*nwcfr + 1;
    case {6}
      fprintf('Detail coefficients (HHL)\n');
      % Advance pointer past approximation coefficients and 5 sets of
      % detail coefficients
      locc = 6*nwcm*nwcn*nwcfr + 1;
    case {7}
      fprintf('Detail coefficients (HHH)\n');
      % Advance pointer past approximation coefficients and 6 sets of
      % detail coefficients
      locc = 7*nwcm*nwcn*nwcfr + 1;
  end

  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

  for j = 1:nwcfr
    if (j==1)
      fprintf('Coefficients        Frame 1');
    else
      fprintf('          Frame %d', j);
    end
  end
  fprintf('\n');
  d2 = reshape(d, nwcm, nwcn*nwcfr);
  for i = 1:nwcm
    if i == 1
      fprintf('    %d         ', cindex);
    else
      fprintf('              ');
    end
    for j=1:nwcn*nwcfr
      fprintf('%8.4f ', d2(i,j));
    end
    fprintf('\n');
  end
end


% 3D DWT reconstruction
[b, ifail] = c09fb(m, n, fr, c, icomm);

fprintf('\nOutput Data          b : \n');
% Convert to int16 to get more compact output
for i=1:nwcm
  fprintf('Frame %d :\n', i);
  disp(b(:, :, i));
end
 
Approximation coefficients (LLL)
Coefficients        Frame 1          Frame 2
    0          10.6066   7.0711   4.2426   5.6569 
                7.7782   6.7175   7.0711  10.6066 
                7.7782   9.8995   2.8284   5.6569 
Detail coefficients (LLH)
Coefficients        Frame 1          Frame 2
    1           0.7071  -2.1213   0.0000   0.0000 
                2.1213  -1.7678   0.0000   0.0000 
                3.5355  -4.2426   0.0000   0.0000 
Detail coefficients (LHL)
Coefficients        Frame 1          Frame 2
    2          -4.2426   2.1213   1.4142   2.8284 
               -2.8284  -2.4749   2.8284   0.7071 
                2.1213  -4.2426   0.0000   0.0000 
Detail coefficients (LHH)
Coefficients        Frame 1          Frame 2
    3           0.0000  -2.8284   0.0000   0.0000 
               -2.8284   1.7678   0.0000   0.0000 
                0.7071   4.2426   0.0000   0.0000 
Detail coefficients (HLL)
Coefficients        Frame 1          Frame 2
    4          -4.9497   0.0000   1.4142   1.4142 
                0.7071   1.7678  -0.0000   2.1213 
                0.0000   0.0000   0.0000   0.0000 
Detail coefficients (HLH)
Coefficients        Frame 1          Frame 2
    5           0.7071   0.7071   0.0000   0.0000 
               -0.7071  -2.4749   0.0000   0.0000 
                0.0000   0.0000   0.0000   0.0000 
Detail coefficients (HHL)
Coefficients        Frame 1          Frame 2
    6           5.6569   0.7071   1.4142   1.4142 
                0.0000  -1.7678   1.4142   6.3640 
                0.0000   0.0000   0.0000   0.0000 
Detail coefficients (HHH)
Coefficients        Frame 1          Frame 2
    7           0.0000   0.0000   0.0000   0.0000 
                1.4142   1.0607   0.0000   0.0000 
                0.0000   0.0000   0.0000   0.0000 

Output Data          b : 
Frame 1 :
    3.0000    2.0000    2.0000    2.0000
    2.0000    9.0000    1.0000    2.0000
    2.0000    5.0000    1.0000    2.0000
    1.0000    6.0000    2.0000    2.0000
    5.0000    3.0000    2.0000    2.0000

Frame 2 :
    2.0000    1.0000    5.0000    1.0000
    2.0000    9.0000    5.0000    2.0000
    2.0000    3.0000    2.0000    7.0000
    2.0000    1.0000    1.0000    2.0000
    2.0000    1.0000    2.0000    8.0000

Frame 3 :
    3.0000    1.0000    4.0000    1.0000
    1.0000    1.0000    2.0000    1.0000
    4.0000    1.0000    7.0000    2.0000
    3.0000    2.0000    1.0000    5.0000
    1.0000    1.0000    2.0000    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. 2009–2013