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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_wav_3d_sngl_inv (c09fb)

Purpose

nag_wav_3d_sngl_inv (c09fb) computes the three-dimensional inverse discrete wavelet transform (IDWT) at a single level. The initialization function nag_wav_3d_init (c09ac) must be called first to set up the DWT options.

Syntax

[b, ifail] = c09fb(m, n, fr, c, icomm, 'lenc', lenc)
[b, ifail] = nag_wav_3d_sngl_inv(m, n, fr, c, icomm, 'lenc', lenc)

Description

nag_wav_3d_sngl_inv (c09fb) performs the inverse operation of function nag_wav_3d_sngl_fwd (c09fa). That is, given sets of wavelet coefficients computed by function nag_wav_3d_sngl_fwd (c09fa) using a DWT as set up by the initialization function nag_wav_3d_init (c09ac), on a real data array, B$B$, nag_wav_3d_sngl_inv (c09fb) will reconstruct B$B$.

None.

Parameters

Compulsory Input Parameters

1:     m – int64int32nag_int scalar
The first dimension of the output 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).
2:     n – int64int32nag_int scalar
The second dimension of the output 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).
3:     fr – int64int32nag_int scalar
The third dimension of the output data: the number two-dimensional frames.
Constraint: this must be the same as the value fr passed to the initialization function nag_wav_3d_init (c09ac).
4:     c(lenc) – double array
lenc, the dimension of the array, must satisfy the constraint lencnct${\mathbf{lenc}}\ge {n}_{\mathrm{ct}}$, where nct${n}_{\mathrm{ct}}$ is the total number of wavelet coefficients, as returned by nag_wav_3d_init (c09ac).
The coefficients of the discrete wavelet transform. This will normally be the result of some transformation on the coefficients computed by function nag_wav_3d_sngl_fwd (c09fa).
5:     icomm(260$260$) – 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_sngl_inv (c09fb) is called.
Constraint: lencnct${\mathbf{lenc}}\ge {n}_{\mathrm{ct}}$, where nct${n}_{\mathrm{ct}}$ is the total number of wavelet coefficients, as returned by nag_wav_3d_init (c09ac).

ldb sdb

Output Parameters

1:     b(ldb,sdb,fr) – double array
ldbm$\mathit{ldb}\ge {\mathbf{m}}$.
sdbn$\mathit{sdb}\ge {\mathbf{n}}$.
The m$m$ by n$n$ by fr$\mathit{fr}$ reconstructed array, B$B$, based on the input wavelet coefficients and the transform options supplied to the initialization function nag_wav_3d_init (c09ac).
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
Constraint: fr = fr${\mathbf{fr}}=\mathrm{fr}$, the value of fr on initialization (see nag_wav_3d_init (c09ac)).
Constraint: m = m${\mathbf{m}}=m$, the value of m on initialization (see nag_wav_3d_init (c09ac)).
Constraint: n = n${\mathbf{n}}=n$, the value of n on initialization (see nag_wav_3d_init (c09ac)).
ifail = 2${\mathbf{ifail}}=2$
Constraint: ldbm$\mathit{ldb}\ge {\mathbf{m}}$.
Constraint: sdbn$\mathit{sdb}\ge {\mathbf{n}}$.
ifail = 3${\mathbf{ifail}}=3$
Constraint: lencnct${\mathbf{lenc}}\ge {n}_{\mathrm{ct}}$, where nct${n}_{\mathrm{ct}}$ is the number of DWT coefficients returned by nag_wav_3d_init (c09ac) in parameter nwct.
ifail = 6${\mathbf{ifail}}=6$
Either the initialization function has not been called first or the communication array icomm has been corrupted.
The initialization function was called with wtrans = 'M'${\mathbf{wtrans}}=\text{'M'}$.
ifail = 999${\mathbf{ifail}}=-999$
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.

None.

Example

```function nag_wav_3d_sngl_inv_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 c09fb_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

```