c09fdf computes the inverse three-dimensional multi-level discrete wavelet transform (IDWT). This routine reconstructs data from (possibly filtered or otherwise manipulated) wavelet transform coefficients calculated by c09fcf from an original input array. The initialization routine c09acf must be called first to set up the IDWT options.

2 Specification

Fortran Interface

Subroutine c09fdf (

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

Integer, Intent (In)	::	nwlinv, lenc, m, n, fr, ldb, sdb, icomm(260)
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	c(lenc)
Real (Kind=nag_wp), Intent (Inout)	::	b(ldb,sdb,fr)

C Header Interface

#include <nag.h>

void	c09fdf_ (const Integer nwlinv, const Integer lenc, const double c[], const Integer m, const Integer n, const Integer fr, double b[], const Integer ldb, const Integer sdb, const Integer icomm[], Integer ifail)

The routine may be called by the names c09fdf or nagf_wav_dim3_mxolap_multi_inv.

3 Description

c09fdf performs the inverse operation of c09fcf. That is, given a set of wavelet coefficients, computed up to level

n_{fwd}

by c09fcf using a DWT as set up by the initialization routine c09acf, on a real three-dimensional array,

A

, c09fdf will reconstruct

A

. The reconstructed array 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 array 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 c09fcf. This results in a partial reconstruction.

4 References

Wang Y, Che X and Ma S (2012) Nonlinear filtering based on 3D wavelet transform for MRI denoising URASIP Journal on Advances in Signal Processing 2012:40

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 c09fcf. The data will be reconstructed to level $(nwl - nwlinv)$ , where level $0$ is the original input dataset provided to c09fcf.

Constraint: $1 \leq nwlinv \leq nwl$ , where nwl is the value used in a preceding call to c09fcf.
2: $lenc$ – Integer Input: On entry: the dimension of the array c as declared in the (sub)program from which c09fdf is called.

Constraint: $lenc \geq n_{ct}$ , where $n_{ct}$ is the total number of wavelet coefficients that correspond to a transform with nwlinv levels.
3: $c (lenc)$ – Real (Kind=nag_wp) array Input: On entry: the coefficients of the multi-level discrete wavelet transform. This will normally be the result of some transformation on the coefficients computed by routine c09fcf.
Note that the coefficients in c may be extracted according to level and type into three-dimensional arrays using c09fyf, and inserted using c09fzf.
4: $m$ – Integer Input: On entry: the number of elements, $m$ , in the first dimension of the reconstructed array $B$ . For a full reconstruction of nwl levels, where nwl is as supplied to c09fcf, this must be the same as argument m used in a preceding call to c09fcf. For a partial reconstruction of $nwlinv < nwl$ levels, this must be equal to $dwtlvm (nwlinv + 1)$ , as returned from c09fcf
5: $n$ – Integer Input: On entry: the number of elements, $n$ , in the second dimension of the reconstructed array $B$ . For a full reconstruction of nwl, levels, where nwl is as supplied to c09fcf, this must be the same as argument n used in a preceding call to c09fcf. For a partial reconstruction of $nwlinv < nwl$ levels, this must be equal to $dwtlvn (nwlinv + 1)$ , as returned from c09fcf.
6: $fr$ – Integer Input: On entry: the number of elements, $fr$ , in the third dimension of the reconstructed array $B$ . For a full reconstruction of nwl levels, where nwl is as supplied to c09fcf, this must be the same as argument fr used in a preceding call to c09fcf. For a partial reconstruction of $nwlinv < nwl$ levels, this must be equal to $dwtlvfr (nwlinv + 1)$ , as returned from c09fcf.
7: $b (ldb, sdb, fr)$ – Real (Kind=nag_wp) array Output: On exit: the $m \times n \times fr$ reconstructed array, $B$ , with $B_{i j k}$ stored in $b (i, j, k)$ . The reconstruction is based on the input multi-level wavelet transform coefficients and the transform options supplied to the initialization routine c09acf.
8: $ldb$ – Integer Input: On entry: the first dimension of the array b as declared in the (sub)program from which c09fdf is called.

Constraint: $ldb \geq m$ .
9: $sdb$ – Integer Input: On entry: the second dimension of the array b as declared in the (sub)program from which c09fdf is called.

Constraint: $sdb \geq n$ .
10: $icomm (260)$ – 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 c09acf.
11: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ 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 $−1$ or $1$ 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 $nwl = ⟨ value ⟩$ where nwl is as used in the computation of the wavelet coefficients by a call to c09fcf.
Constraint: $nwlinv \leq nwl$ as used in the call to c09fcf.

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

On entry, $sdb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $sdb \geq n$ .

$ifail = 3$: On entry, $lenc = ⟨ value ⟩$ .
Constraint: $lenc \geq ⟨ value ⟩$ , the number of wavelet coefficients required for a transform operating on nwlinv levels. If $nwlinv = nwlmax$ , the maximum number of levels as returned by the initial call to c09acf, lenc must be at least $n_{ct}$ , the value returned in nwct by the same call to c09acf.

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

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 communication array icomm has been corrupted or there has not been a prior call to the initialization routine c09acf.

The initialization routine was called with $wtrans ='S'$ .

$ifail = - 99$: 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.