c09fdc computes the inverse three-dimensional multi-level discrete wavelet transform (IDWT). This function reconstructs data from (possibly filtered or otherwise manipulated) wavelet transform coefficients calculated by c09fcc from an original input array. The initialization function c09acc must be called first to set up the IDWT options.

2 Specification

#include <nag.h>

void	c09fdc (Integer nwlinv, Integer lenc, const double c[], Integer m, Integer n, Integer fr, double b[], Integer ldb, Integer sdb, const Integer icomm[], NagError *fail)

The function may be called by the names: c09fdc, nag_wav_dim3_mxolap_multi_inv or nag_imldwt_3d.

3 Description

c09fdc performs the inverse operation of c09fcc. That is, given a set of wavelet coefficients, computed up to level

n_{fwd}

by c09fcc using a DWT as set up by the initialization function c09acc, on a real three-dimensional array,

A

, c09fdc 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 c09fcc. 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 c09fcc. The data will be reconstructed to level $(nwl - nwlinv)$ , where level $0$ is the original input dataset provided to c09fcc.

Constraint: $1 \leq nwlinv \leq nwl$ , where nwl is the value used in a preceding call to c09fcc.
2: $lenc$ – Integer Input: On entry: the dimension of the array c.

Constraint: $lenc \geq n_{ct}$ , where $n_{ct}$ is the total number of wavelet coefficients that correspond to a transform with nwlinv levels.
3: $c [lenc]$ – const double 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 function c09fcc.
Note that the coefficients in c may be extracted according to level and type into three-dimensional arrays using c09fyc, and inserted using c09fzc.
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 c09fcc, this must be the same as argument m used in a preceding call to c09fcc. For a partial reconstruction of $nwlinv < nwl$ levels, this must be equal to $dwtlvm [nwlinv]$ , as returned from c09fcc
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 c09fcc, this must be the same as argument n used in a preceding call to c09fcc. For a partial reconstruction of $nwlinv < nwl$ levels, this must be equal to $dwtlvn [nwlinv]$ , as returned from c09fcc.
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 c09fcc, this must be the same as argument fr used in a preceding call to c09fcc. For a partial reconstruction of $nwlinv < nwl$ levels, this must be equal to $dwtlvfr [nwlinv]$ , as returned from c09fcc.
7: $b [\dim]$ – double Output: Note: the dimension, dim, of the array b must be at least $ldb \times sdb \times fr$ .

On exit: the $m$ by $n$ by $fr$ reconstructed array, $B$ , with $B_{i j k}$ stored in $b [(k - 1) \times ldb \times sdb + (j - 1) \times ldb + i - 1]$ . The reconstruction is based on the input multi-level wavelet transform coefficients and the transform options supplied to the initialization function c09acc.
8: $ldb$ – Integer Input: On entry: the stride separating row elements of each of the sets of frame coefficients in the three-dimensional data stored in b.

Constraint: $ldb \geq m$ .
9: $sdb$ – Integer Input: On entry: the stride separating corresponding coefficients of consecutive frames in the three-dimensional data stored in b.

Constraint: $sdb \geq n$ .
10: $icomm [260]$ – const Integer Communication Array: On entry: contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization function c09acc.
11: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INITIALIZATION: Either the communication array icomm has been corrupted or there has not been a prior call to the initialization function c09acc.

The initialization function was called with $wtrans = Nag_SingleLevel$ .
NE_INT: 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.

On entry, $nwlinv = 〈value〉$ .
Constraint: $nwlinv \geq 1$ .
NE_INT_2: On entry, $ldb = 〈value〉$ and $m = 〈value〉$ .
Constraint: $ldb \geq m$ .

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 c09acc, lenc must be at least $n_{ct}$ , the value returned in nwct by the same call to c09acc.

On entry, $nwlinv = 〈value〉$ and $nwl = 〈value〉$ where nwl is as used in the computation of the wavelet coefficients by a call to c09fcc.
Constraint: $nwlinv \leq nwl$ as used in the call to c09fcc.

On entry, $sdb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $sdb \geq n$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL 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.