c09edf computes the inverse two-dimensional multi-level discrete wavelet transform (DWT). This routine reconstructs data from (possibly filtered or otherwise manipulated) wavelet transform coefficients calculated by c09ecf from an original input matrix. The initialization routine c09abf must be called first to set up the DWT options.

2 Specification

Fortran Interface

Subroutine c09edf (

nwlinv, lenc, c, m, n, b, ldb, icomm, ifail)

Integer, Intent (In)	::	nwlinv, lenc, m, n, ldb, icomm(180)
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	c(lenc)
Real (Kind=nag_wp), Intent (Inout)	::	b(ldb,n)

C Header Interface

#include <nag.h>

void	c09edf_ (const Integer nwlinv, const Integer lenc, const double c[], const Integer m, const Integer n, double b[], const Integer ldb, const Integer icomm[], Integer ifail)

The routine may be called by the names c09edf or nagf_wav_dim2_multi_inv.

3 Description

c09edf performs the inverse operation of c09ecf. That is, given a set of wavelet coefficients, computed up to level

n_{fwd}

by c09ecf using a DWT as set up by the initialization routine c09abf, on a real matrix,

A

, c09edf will reconstruct

A

. The reconstructed matrix 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 matrix 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 c09ecf. This results in a partial reconstruction.

4 References

None.

5 Arguments

1: $nwlinv$ – Integer Input

On entry: the number of levels to be used in the inverse multi-level transform. The number of levels must be less than or equal to

n_{fwd}

, which has the value of argument nwl as used in the computation of the wavelet coefficients using c09ecf. The data will be reconstructed to level

(nwl - nwlinv)

, where level

0

is the original input dataset provided to c09ecf.

Constraint:

1 \leq nwlinv \leq nwl

, where nwl is the value used in a preceding call to c09ecf.

2: $lenc$ – Integer Input

On entry: the dimension of the array c as declared in the (sub)program from which c09edf is called.

Constraint:

lenc \geq n_{ct}

, where

n_{ct}

is the total number of coefficients that correspond to a transform with nwlinv levels and is unchanged from the preceding call to c09ecf.

3: $c (lenc)$ – Real (Kind=nag_wp) array Input

On entry: the coefficients of a multi-level wavelet transform of the original matrix,

A

, which may have been filtered or otherwise manipulated.

Let

q (i)

be the number of coefficients (of each type) at level

i

, for

i = n_{fwd}, n_{fwd} - 1, \dots, 1

. Then, setting

k_{1} = q (n_{fwd})

and

k_{j + 1} = k_{j} + q (n_{fwd} - ⌈ j / 3 ⌉ + 1)

, for

j = 1, 2, \dots, 3 n_{fwd}

, the coefficients are stored in c as follows:

$c (i)$ , for $i = 1, 2, \dots, k_{1}$

Contains the level

n_{fwd}

approximation coefficients,

a_{n_{fwd}}

$c (i)$ , for $i = k_{j} + 1, \dots, k_{j + 1}$

Contains the level

n_{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, \dots, 3 n_{fwd}

Note that the coefficients in c may be extracted according to level and type into two-dimensional arrays using c09eyf, and inserted using c09ezf.

4: $m$ – Integer Input

On entry: the number of elements,

m

, in the first dimension of the reconstructed matrix

B

. For a full reconstruction of nwl levels, where nwl is as supplied to c09ecf, this must be the same as argument m used in the call to c09ecf. For a partial reconstruction of

nwlinv < nwl

levels, this must be equal to

dwtlvm (nwlinv + 1)

, as returned from c09ecf.

5: $n$ – Integer Input

On entry: the number of elements,

n

, in the second dimension of the reconstructed matrix

B

. For a full reconstruction of nwl levels, where nwl is as supplied to c09fcf, this must be the same as argument n used in the call to c09ecf. For a partial reconstruction of

nwlinv < nwl

, this must be equal to

dwtlvn (nwlinv + 1)

, as returned from c09ecf.

6: $b (ldb, n)$ – Real (Kind=nag_wp) array Output

On exit: the

m \times n

reconstructed matrix,

B

, based on the input multi-level wavelet transform coefficients and the transform options supplied to the initialization routine c09abf.

7: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which c09edf is called.

Constraint:

ldb \geq m

8: $icomm (180)$ – Integer array Communication Array

On entry: contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization routine c09abf.

9: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $nwlinv = ⟨ value ⟩$ .
Constraint: $nwlinv \geq 1$ .

On entry, $nwlinv = ⟨ value ⟩$ and $n_{fwd} = ⟨ value ⟩$ .
Constraint: $nwlinv \leq n_{fwd}$ .

$ifail = 2$: On entry, $ldb = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $ldb \geq m$ .

$ifail = 3$: On entry, $lenc = ⟨ value ⟩$ .
Constraint: $lenc \geq ⟨ value ⟩$ , the total number of coefficients generated by the preceding call to c09ecf.

$ifail = 4$: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq ⟨ value ⟩$ , the number of coefficients in the first dimension at the required level of reconstruction.

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq ⟨ value ⟩$ , the number of coefficients in the second dimension at the required level of reconstruction.

$ifail = 6$: Either the initialization routine has not been called first or icomm has been corrupted.

Either the initialization routine was called with $wtrans ='S'$ or icomm has been corrupted.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 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.