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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_wav_2d_multi_inv (c09ed)

## Purpose

nag_wav_2d_multi_inv (c09ed) computes the inverse two-dimensional multi-level discrete wavelet transform (DWT). This function reconstructs data from (possibly filtered or otherwise manipulated) wavelet transform coefficients calculated by nag_wav_2d_multi_fwd (c09ec) from an original input matrix. The initialization function nag_wav_2d_init (c09ab) must be called first to set up the DWT options.

## Syntax

[b, ifail] = c09ed(nwl, c, m, n, icomm, 'lenc', lenc)
[b, ifail] = nag_wav_2d_multi_inv(nwl, c, m, n, icomm, 'lenc', lenc)

## Description

nag_wav_2d_multi_inv (c09ed) performs the inverse operation of nag_wav_2d_multi_fwd (c09ec). That is, given a set of wavelet coefficients, computed by nag_wav_2d_multi_fwd (c09ec) using a DWT as set up by the initialization function nag_wav_2d_init (c09ab), on a real matrix, A$A$, nag_wav_2d_multi_inv (c09ed) will reconstruct A$A$. The reconstructed matrix is referred to as B$B$ in the following since it will not be identical to A$A$ when the DWT coefficients have been filtered or otherwise manipulated prior to reconstruction. If the original input matrix is level 0$0$, 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_2d_multi_fwd (c09ec). This results in a partial reconstruction.

None.

## Parameters

### Compulsory Input Parameters

1:     nwl – int64int32nag_int scalar
The number, nl${n}_{l}$, of levels to be used in the inverse multi-level transform.
Constraint: 1nwlnfwd$1\le {\mathbf{nwl}}\le {n}_{\mathrm{fwd}}$, where nfwd${n}_{\mathrm{fwd}}$ is the value used in a preceding call to nag_wav_2d_multi_fwd (c09ec).
2:     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 coefficients that correspond to a transform with nfwd${n}_{\mathrm{fwd}}$ levels and is unchanged from the preceding call to nag_wav_2d_multi_fwd (c09ec).
The coefficients of a multi-level wavelet transform of the original matrix, A$A$, which may have been filtered or otherwise manipulated.
Let q(i)$q\left(\mathit{i}\right)$ be the number of coefficients (of each type) at level i$\mathit{i}$, for i = nfwd,nfwd1,,1$\mathit{i}={n}_{\mathrm{fwd}},{n}_{\mathrm{fwd}}-1,\dots ,1$. Then, setting k1 = q(nfwd)${k}_{1}=q\left({n}_{\mathrm{fwd}}\right)$ and kj + 1 = kj + q(nfwdj / 3 + 1)${k}_{j+1}={k}_{j}+q\left({n}_{\mathrm{fwd}}-⌈j/3⌉+1\right)$, for j = 1,2,,3nfwd$j=1,2,\dots ,3{n}_{\mathrm{fwd}}$, the coefficients are stored in c as follows:
c(i)${\mathbf{c}}\left(\mathit{i}\right)$, for i = 1,2,,k1$\mathit{i}=1,2,\dots ,{k}_{1}$
Contains the level nfwd${n}_{\mathrm{fwd}}$ approximation coefficients, anfwd${a}_{{n}_{\mathrm{fwd}}}$.
c(i)${\mathbf{c}}\left(\mathit{i}\right)$, for i = kj + 1,,kj + 1$\mathit{i}={k}_{j}+1,\dots ,{k}_{j+1}$
Contains the level nfwdj / 3 + 1${n}_{\mathrm{fwd}}-⌈j/3⌉+1$ vertical, horizontal and diagonal coefficients. These are:
• vertical coefficients if j  mod  3 = 1;
• horizontal coefficients if j  mod  3 = 2;
• diagonal coefficients if j  mod  3 = 0,
for j = 1,,3nfwd$j=1,\dots ,3{n}_{\mathrm{fwd}}$
3:     m – int64int32nag_int scalar
The number of elements, m$m$, in the first dimension of the reconstructed matrix B$B$. For a full reconstruction of nfwd${n}_{\mathrm{fwd}}$ levels this is the same as parameter m in nag_wav_2d_multi_fwd (c09ec). For a partial reconstruction of nl < nfwd${n}_{l}<{n}_{\mathrm{fwd}}$ levels this will be equal to dwtlvm(nl + 1)${\mathbf{dwtlvm}}\left({n}_{l}+1\right)$ as returned from nag_wav_2d_multi_fwd (c09ec).
4:     n – int64int32nag_int scalar
The number of elements, n$n$, in the second dimension of the reconstructed matrix B$B$. For a full reconstruction of nfwd${n}_{\mathrm{fwd}}$ levels this is the same as parameter n in nag_wav_2d_multi_fwd (c09ec). For a partial reconstruction of nl < nfwd${n}_{l}<{n}_{\mathrm{fwd}}$ levels this will be equal to dwtlvn(nl + 1)${\mathbf{dwtlvn}}\left({n}_{l}+1\right)$ as returned from nag_wav_2d_multi_fwd (c09ec).
5:     icomm(180$180$) – 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_2d_init (c09ab).

### 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_2d_multi_inv (c09ed) is called.
Constraint: lencnct${\mathbf{lenc}}\ge {n}_{\mathrm{ct}}$, where nct${n}_{\mathrm{ct}}$ is the total number of coefficients that correspond to a transform with nfwd${n}_{\mathrm{fwd}}$ levels and is unchanged from the preceding call to nag_wav_2d_multi_fwd (c09ec).

ldb

### Output Parameters

1:     b(ldb,n) – double array
ldbm$\mathit{ldb}\ge {\mathbf{m}}$.
The m$m$ by n$n$ reconstructed matrix, B$B$, based on the input multi-level wavelet transform coefficients and the transform options supplied to the initialization function nag_wav_2d_init (c09ab).
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: nwlnfwd${\mathbf{nwl}}\le {n}_{\mathrm{fwd}}$.
Constraint: nwl1${\mathbf{nwl}}\ge 1$.
ifail = 2${\mathbf{ifail}}=2$
Constraint: ldbm$\mathit{ldb}\ge {\mathbf{m}}$.
ifail = 3${\mathbf{ifail}}=3$
lenc is too small, the total number of coefficients generated by the preceding call to nag_wav_2d_multi_fwd (c09ec).
ifail = 4${\mathbf{ifail}}=4$
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${\mathbf{ifail}}=6$
Either the initialization function has not been called first or icomm has been corrupted.
Either the initialization function was called with wtrans = 'S'${\mathbf{wtrans}}=\text{'S'}$ or icomm has been corrupted.
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_2d_multi_inv_example
m = int64(7);
n = int64(8);
wavnam = 'DB2';
mode = 'Half';
wtrans = 'Multilevel';
a = [3, 7, 9, 1, 9, 9, 1, 0;
9, 9, 3, 3, 4, 1, 2, 4;
7, 8, 1, 3, 8, 9, 3, 3;
1, 1, 1, 1, 2, 8, 4, 0;
1, 2, 4, 6, 5, 6, 5, 4;
2, 2, 5, 7, 3, 6, 6, 8;
7, 9, 3, 1, 3, 4, 7, 2];

fprintf('\nInput data a:\n');
disp(a);
[nwl, nf, nwct, nwcn, icomm, ifail] = nag_wav_2d_init(wavnam, wtrans, mode, m, n);

lenc = nwct;
% Perform Discrete Wavelet transform
[c, dwtlvm, dwtlvn, icomm, ifail] = nag_wav_2d_multi_fwd(a, lenc, nwl, icomm);

fprintf('\nLength of wavelet filter : %d\n', nf);
fprintf('Number of Levels :         %d\n', nwl);
fprintf('Number of coefficients in first dimension for each level :\n');
disp(transpose(dwtlvm(1:double(nwl))));
fprintf('Number of coefficients in second dimension for each level :\n');
disp(transpose(dwtlvn(1:double(nwl))));

fprintf('\nTotal number of wavelet coefficients : %d\n', nwct);
fprintf('\nWavelet coefficients c :\n');
jstart = 1;
for ilevel = 1:double(nwl)
fprintf('-------------------------------------------------------\n');
fprintf('Level %d output is %d by %d\n', ...
nwl-ilevel+1, dwtlvm(ilevel), dwtlvn(ilevel));
fprintf('-------------------------------------------------------\n');

iskip = double(dwtlvm(ilevel));
i2 = iskip*double(dwtlvn(ilevel)) - 1;

for itype_coeffs = 1:4
switch itype_coeffs
case {1}
if (ilevel == nwl)
fprintf('Approximation coefficients:\n');
end
case {2}
fprintf('Vertical coefficients:\n');
case {3}
fprintf('Horizontal coefficients:\n');
case {4}
fprintf('Diagonal coefficients:\n');
end
if (itype_coeffs>1 || ilevel==1)
for i1 = jstart:jstart+iskip-1
fprintf('%8.4f',c(i1:iskip:i1+i2));
fprintf('\n');
end
jstart = jstart + i2 + 1;
end
end
fprintf('\n');
end

% Reconstruct original data
[b, ifail] = nag_wav_2d_multi_inv(nwl, c, m, n, icomm);
fprintf('Reconstruction       b:\n');
disp(b);
```
```

Input data a:
3     7     9     1     9     9     1     0
9     9     3     3     4     1     2     4
7     8     1     3     8     9     3     3
1     1     1     1     2     8     4     0
1     2     4     6     5     6     5     4
2     2     5     7     3     6     6     8
7     9     3     1     3     4     7     2

Length of wavelet filter : 4
Number of Levels :         2
Number of coefficients in first dimension for each level :
4                    5

Number of coefficients in second dimension for each level :
4                    5

Total number of wavelet coefficients : 139

Wavelet coefficients c :
-------------------------------------------------------
Level 2 output is 4 by 4
-------------------------------------------------------
24.9724 25.6017 20.8900  7.9280
27.6100 27.0955 18.7941  8.2804
11.2663 11.0273 19.6410 18.6651
27.6050 26.6443 14.5913 18.0835
Vertical coefficients:
-2.5552 -6.1078 -4.0629  8.2136
-1.6061 -7.2355 -3.3633  7.6075
-0.2225 -1.6283 -0.5301  3.7415
-0.9052 -6.5810  0.8023  1.8591
Horizontal coefficients:
-3.8069 -3.0730  2.1121 -1.8525
-2.7548 -4.5949 -0.8321 -4.8155
4.8398  4.5104 -1.5308 -0.6456
-6.4332 -4.5381  2.4753  6.8224
Diagonal coefficients:
-0.8978 -0.2326 -1.2515  2.6346
0.5708 -4.9783 -1.5309  6.4569
-0.1854 -1.8430  0.2426 -0.0754
0.0345  7.1864  1.5938 -5.9745

-------------------------------------------------------
Level 1 output is 5 by 5
-------------------------------------------------------
Approximation coefficients:
Vertical coefficients:
-2.5981  4.6471  2.5392 -2.8415 -0.2165
-1.3203 -0.0592  3.0490 -2.5837  1.0458
-0.4330 -1.6405 -1.1752  0.2533 -2.3448
-0.4118 -0.0682 -2.4608 -0.0167  0.4387
-1.5368 -1.1450 -0.5547  4.5936 -3.6863
Horizontal coefficients:
-4.3301 -1.8170  0.8023  5.7566 -2.8146
4.3089  3.6908  0.8349  3.4653  1.7108
-1.5311 -1.0736  1.5257  0.0212 -0.9608
2.8873  3.1148 -1.9118 -0.4007 -1.5302
-2.2377 -2.7611  2.4453 -0.3705  4.3448
Diagonal coefficients:
-1.5000  4.4151 -0.0057 -0.8236 -1.1250
-0.1953 -2.9530  1.8840 -1.7635  0.9877
-0.4330  0.2745  1.1450  0.4632 -0.5547
-0.3538 -0.3215  0.6462  1.3705 -1.2778
0.7288  0.4587 -1.8873 -1.8828  2.4028

Reconstruction       b:
3.0000    7.0000    9.0000    1.0000    9.0000    9.0000    1.0000    0.0000
9.0000    9.0000    3.0000    3.0000    4.0000    1.0000    2.0000    4.0000
7.0000    8.0000    1.0000    3.0000    8.0000    9.0000    3.0000    3.0000
1.0000    1.0000    1.0000    1.0000    2.0000    8.0000    4.0000    0.0000
1.0000    2.0000    4.0000    6.0000    5.0000    6.0000    5.0000    4.0000
2.0000    2.0000    5.0000    7.0000    3.0000    6.0000    6.0000    8.0000
7.0000    9.0000    3.0000    1.0000    3.0000    4.0000    7.0000    2.0000

```
```function c09ed_example
m = int64(7);
n = int64(8);
wavnam = 'DB2';
mode = 'Half';
wtrans = 'Multilevel';
a = [3, 7, 9, 1, 9, 9, 1, 0;
9, 9, 3, 3, 4, 1, 2, 4;
7, 8, 1, 3, 8, 9, 3, 3;
1, 1, 1, 1, 2, 8, 4, 0;
1, 2, 4, 6, 5, 6, 5, 4;
2, 2, 5, 7, 3, 6, 6, 8;
7, 9, 3, 1, 3, 4, 7, 2];

fprintf('\nInput data a:\n');
disp(a);
[nwl, nf, nwct, nwcn, icomm, ifail] = c09ab(wavnam, wtrans, mode, m, n);

lenc = nwct;
% Perform Discrete Wavelet transform
[c, dwtlvm, dwtlvn, icomm, ifail] = c09ec(a, lenc, nwl, icomm);

fprintf('\nLength of wavelet filter : %d\n', nf);
fprintf('Number of Levels :         %d\n', nwl);
fprintf('Number of coefficients in first dimension for each level :\n');
disp(transpose(dwtlvm(1:double(nwl))));
fprintf('Number of coefficients in second dimension for each level :\n');
disp(transpose(dwtlvn(1:double(nwl))));

fprintf('\nTotal number of wavelet coefficients : %d\n', nwct);
fprintf('\nWavelet coefficients c :\n');
jstart = 1;
for ilevel = 1:double(nwl)
fprintf('-------------------------------------------------------\n');
fprintf('Level %d output is %d by %d\n', ...
nwl-ilevel+1, dwtlvm(ilevel), dwtlvn(ilevel));
fprintf('-------------------------------------------------------\n');

iskip = double(dwtlvm(ilevel));
i2 = iskip*double(dwtlvn(ilevel)) - 1;

for itype_coeffs = 1:4
switch itype_coeffs
case {1}
if (ilevel == nwl)
fprintf('Approximation coefficients:\n');
end
case {2}
fprintf('Vertical coefficients:\n');
case {3}
fprintf('Horizontal coefficients:\n');
case {4}
fprintf('Diagonal coefficients:\n');
end
if (itype_coeffs>1 || ilevel==1)
for i1 = jstart:jstart+iskip-1
fprintf('%8.4f',c(i1:iskip:i1+i2));
fprintf('\n');
end
jstart = jstart + i2 + 1;
end
end
fprintf('\n');
end

% Reconstruct original data
[b, ifail] = c09ed(nwl, c, m, n, icomm);
fprintf('Reconstruction       b:\n');
disp(b);
```
```

Input data a:
3     7     9     1     9     9     1     0
9     9     3     3     4     1     2     4
7     8     1     3     8     9     3     3
1     1     1     1     2     8     4     0
1     2     4     6     5     6     5     4
2     2     5     7     3     6     6     8
7     9     3     1     3     4     7     2

Length of wavelet filter : 4
Number of Levels :         2
Number of coefficients in first dimension for each level :
4                    5

Number of coefficients in second dimension for each level :
4                    5

Total number of wavelet coefficients : 139

Wavelet coefficients c :
-------------------------------------------------------
Level 2 output is 4 by 4
-------------------------------------------------------
24.9724 25.6017 20.8900  7.9280
27.6100 27.0955 18.7941  8.2804
11.2663 11.0273 19.6410 18.6651
27.6050 26.6443 14.5913 18.0835
Vertical coefficients:
-2.5552 -6.1078 -4.0629  8.2136
-1.6061 -7.2355 -3.3633  7.6075
-0.2225 -1.6283 -0.5301  3.7415
-0.9052 -6.5810  0.8023  1.8591
Horizontal coefficients:
-3.8069 -3.0730  2.1121 -1.8525
-2.7548 -4.5949 -0.8321 -4.8155
4.8398  4.5104 -1.5308 -0.6456
-6.4332 -4.5381  2.4753  6.8224
Diagonal coefficients:
-0.8978 -0.2326 -1.2515  2.6346
0.5708 -4.9783 -1.5309  6.4569
-0.1854 -1.8430  0.2426 -0.0754
0.0345  7.1864  1.5938 -5.9745

-------------------------------------------------------
Level 1 output is 5 by 5
-------------------------------------------------------
Approximation coefficients:
Vertical coefficients:
-2.5981  4.6471  2.5392 -2.8415 -0.2165
-1.3203 -0.0592  3.0490 -2.5837  1.0458
-0.4330 -1.6405 -1.1752  0.2533 -2.3448
-0.4118 -0.0682 -2.4608 -0.0167  0.4387
-1.5368 -1.1450 -0.5547  4.5936 -3.6863
Horizontal coefficients:
-4.3301 -1.8170  0.8023  5.7566 -2.8146
4.3089  3.6908  0.8349  3.4653  1.7108
-1.5311 -1.0736  1.5257  0.0212 -0.9608
2.8873  3.1148 -1.9118 -0.4007 -1.5302
-2.2377 -2.7611  2.4453 -0.3705  4.3448
Diagonal coefficients:
-1.5000  4.4151 -0.0057 -0.8236 -1.1250
-0.1953 -2.9530  1.8840 -1.7635  0.9877
-0.4330  0.2745  1.1450  0.4632 -0.5547
-0.3538 -0.3215  0.6462  1.3705 -1.2778
0.7288  0.4587 -1.8873 -1.8828  2.4028

Reconstruction       b:
3.0000    7.0000    9.0000    1.0000    9.0000    9.0000    1.0000    0.0000
9.0000    9.0000    3.0000    3.0000    4.0000    1.0000    2.0000    4.0000
7.0000    8.0000    1.0000    3.0000    8.0000    9.0000    3.0000    3.0000
1.0000    1.0000    1.0000    1.0000    2.0000    8.0000    4.0000    0.0000
1.0000    2.0000    4.0000    6.0000    5.0000    6.0000    5.0000    4.0000
2.0000    2.0000    5.0000    7.0000    3.0000    6.0000    6.0000    8.0000
7.0000    9.0000    3.0000    1.0000    3.0000    4.0000    7.0000    2.0000

```