# NAG CL Interfacec09edc (dim2_​multi_​inv)

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## 1Purpose

c09edc 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 c09ecc from an original input matrix. The initialization function c09abc must be called first to set up the DWT options.

## 2Specification

 #include
 void c09edc (Integer nwlinv, Integer lenc, const double c[], Integer m, Integer n, double b[], Integer ldb, const Integer icomm[], NagError *fail)
The function may be called by the names: c09edc, nag_wav_dim2_multi_inv or nag_imldwt_2d.

## 3Description

c09edc performs the inverse operation of c09ecc. That is, given a set of wavelet coefficients, computed up to level ${n}_{\mathrm{fwd}}$ by c09ecc using a DWT as set up by the initialization function c09abc, on a real matrix, $A$, c09edc 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 c09ecc. This results in a partial reconstruction.
None.

## 5Arguments

1: $\mathbf{nwlinv}$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 ${n}_{\mathrm{fwd}}$, which has the value of argument nwl as used in the computation of the wavelet coefficients using c09ecc. The data will be reconstructed to level $\left({\mathbf{nwl}}-{\mathbf{nwlinv}}\right)$, where level $0$ is the original input dataset provided to c09ecc.
Constraint: $1\le {\mathbf{nwlinv}}\le {\mathbf{nwl}}$, where nwl is the value used in a preceding call to c09ecc.
2: $\mathbf{lenc}$Integer Input
On entry: the dimension of the array c.
Constraint: ${\mathbf{lenc}}\ge {n}_{\mathrm{ct}}$, where ${n}_{\mathrm{ct}}$ is the total number of coefficients that correspond to a transform with nwlinv levels and is unchanged from the preceding call to c09ecc.
3: $\mathbf{c}\left[{\mathbf{lenc}}\right]$const double 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 $q\left(\mathit{i}\right)$ be the number of coefficients (of each type) at level $\mathit{i}$, for $\mathit{i}={n}_{\mathrm{fwd}},{n}_{\mathrm{fwd}}-1,\dots ,1$. Then, setting ${k}_{1}=q\left({n}_{\mathrm{fwd}}\right)$ and ${k}_{j+1}={k}_{j}+q\left({n}_{\mathrm{fwd}}-⌈j/3⌉+1\right)$, for $j=1,2,\dots ,3{n}_{\mathrm{fwd}}$, the coefficients are stored in c as follows:
${\mathbf{c}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{k}_{1}$
Contains the level ${n}_{\mathrm{fwd}}$ approximation coefficients, ${a}_{{n}_{\mathrm{fwd}}}$.
${\mathbf{c}}\left[\mathit{i}-1\right]$, for $\mathit{i}={k}_{j}+1,\dots ,{k}_{j+1}$
Contains the level ${n}_{\mathrm{fwd}}-⌈j/3⌉+1$ vertical, horizontal and diagonal coefficients. These are:
• vertical coefficients if ;
• horizontal coefficients if ;
• diagonal coefficients if ,
for $j=1,\dots ,3{n}_{\mathrm{fwd}}$.
Note that the coefficients in c may be extracted according to level and type into two-dimensional arrays using c09eyc, and inserted using c09ezc.
4: $\mathbf{m}$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 c09ecc, this must be the same as argument m used in the call to c09ecc. For a partial reconstruction of ${\mathbf{nwlinv}}<{\mathbf{nwl}}$ levels, this must be equal to ${\mathbf{dwtlvm}}\left[{\mathbf{nwlinv}}\right]$, as returned from c09ecc.
5: $\mathbf{n}$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 c09fcc, this must be the same as argument n used in the call to c09ecc. For a partial reconstruction of ${\mathbf{nwlinv}}<{\mathbf{nwl}}$, this must be equal to ${\mathbf{dwtlvn}}\left[{\mathbf{nwlinv}}\right]$, as returned from c09ecc.
6: $\mathbf{b}\left[{\mathbf{ldb}}×{\mathbf{n}}\right]$double Output
Note: the $\left(i,j\right)$th element of the matrix $B$ is stored in ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{ldb}}+i-1\right]$.
On exit: the $m×n$ reconstructed matrix, $B$, based on the input multi-level wavelet transform coefficients and the transform options supplied to the initialization function c09abc.
7: $\mathbf{ldb}$Integer Input
On entry: the stride separating matrix row elements in the array b.
Constraint: ${\mathbf{ldb}}\ge {\mathbf{m}}$.
8: $\mathbf{icomm}\left[180\right]$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 c09abc.
9: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error 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.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INITIALIZATION
Either the initialization function has not been called first or icomm has been corrupted.
Either the initialization function was called with ${\mathbf{wtrans}}=\mathrm{Nag_SingleLevel}$ or icomm has been corrupted.
NE_INT
On entry, ${\mathbf{lenc}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lenc}}\ge ⟨\mathit{\text{value}}⟩$, the total number of coefficients generated by the preceding call to c09ecc.
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge ⟨\mathit{\text{value}}⟩$, the number of coefficients in the first dimension at the required level of reconstruction.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge ⟨\mathit{\text{value}}⟩$, the number of coefficients in the second dimension at the required level of reconstruction.
On entry, ${\mathbf{nwlinv}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nwlinv}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{ldb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldb}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{nwlinv}}=⟨\mathit{\text{value}}⟩$ and ${n}_{\mathrm{fwd}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nwlinv}}\le {n}_{\mathrm{fwd}}$.
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.

## 7Accuracy

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.

## 8Parallelism and Performance

c09edc is not threaded in any implementation.