NAG FL Interfacec09fzf (dim3_​coeff_​ins)

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1Purpose

c09fzf inserts a selected set of three-dimensional discrete wavelet transform (DWT) coefficients into the full set of coefficients stored in compact form, which may be later used as input to the reconstruction routines c09fbf or c09fdf.

2Specification

Fortran Interface
 Subroutine c09fzf ( ilev, lenc, c, d, ldd, sdd,
 Integer, Intent (In) :: ilev, cindex, lenc, ldd, sdd Integer, Intent (Inout) :: icomm(260), ifail Real (Kind=nag_wp), Intent (In) :: d(ldd,sdd,*) Real (Kind=nag_wp), Intent (Inout) :: c(lenc)
#include <nag.h>
 void c09fzf_ (const Integer *ilev, const Integer *cindex, const Integer *lenc, double c[], const double d[], const Integer *ldd, const Integer *sdd, Integer icomm[], Integer *ifail)
The routine may be called by the names c09fzf or nagf_wav_dim3_coeff_ins.

3Description

c09fzf inserts a selected set of three-dimensional DWT coefficients into the full set of coefficients stored in compact form in a one-dimensional array c. It is required that c09fzf is preceded by a call to the initialization routine c09acf and either the forwards transform routine c09faf or multi-level forwards transform routine c09fcf.
Given an initial three-dimensional data set $A$, a prior call to c09faf or c09fcf computes the approximation coefficients (at the highest requested level in the case of c09fcf) and, seven sets of detail coefficients (at all levels in the case of c09fcf) and stores these in compact form in a one-dimensional array c. c09fyf can then extract either the approximation coefficients or one of the sets of detail coefficients (at one of the levels following c09fcf) into a three-dimensional array, d. Following some calculation on this set of coefficients (for example, denoising), the updated coefficients in d are inserted back into the full set c using c09fzf. Several extractions and insertions may be performed. c09fbf or c09fdf can then be used to reconstruct a manipulated data set $\stackrel{~}{A}$. The dimensions of d depend on the level extracted and are available from either: the arrays dwtlvm, dwtlvn and dwtlvfr as returned by c09fcf if this was called first; or, otherwise from nwct, nwcn and nwcfr as returned by c09acf. See Section 2.1 in the C09 Chapter Introduction for a discussion of the three-dimensional DWT.

None.

5Arguments

Note: the following notation is used in this section:
• ${n}_{\mathrm{cm}}$ is the number of wavelet coefficients in the first dimension. Following a call to c09faf (i.e., when ${\mathbf{ilev}}=0$) this is equal to ${\mathbf{nwct}}/\left(8×{\mathbf{nwcn}}×{\mathbf{nwcfr}}\right)$ as returned by c09acf. Following a call to c09fcf transforming nwl levels, and when inserting at level ${\mathbf{ilev}}>0$, this is equal to ${\mathbf{dwtlvm}}\left({\mathbf{nwl}}-{\mathbf{ilev}}+1\right)$.
• ${n}_{\mathrm{cn}}$ is the number of wavelet coefficients in the second dimension. Following a call to c09faf (i.e., when ${\mathbf{ilev}}=0$) this is equal to nwcn as returned by c09acf. Following a call to c09fcf transforming nwl levels, and when inserting at level ${\mathbf{ilev}}>0$, this is equal to ${\mathbf{dwtlvn}}\left({\mathbf{nwl}}-{\mathbf{ilev}}+1\right)$.
• ${n}_{\mathrm{cfr}}$ is the number of wavelet coefficients in the third dimension. Following a call to c09faf (i.e., when ${\mathbf{ilev}}=0$) this is equal to nwcfr as returned by c09acf. Following a call to c09fcf transforming nwl levels, and when inserting at level ${\mathbf{ilev}}>0$, this is equal to ${\mathbf{dwtlvfr}}\left({\mathbf{nwl}}-{\mathbf{ilev}}+1\right)$
1: $\mathbf{ilev}$Integer Input
On entry: the level at which coefficients are to be inserted.
If ${\mathbf{ilev}}=0$, it is assumed that the coefficient array c was produced by a preceding call to the single level routine c09faf.
If ${\mathbf{ilev}}>0$, it is assumed that the coefficient array c was produced by a preceding call to the multi-level routine c09fcf.
Constraints:
• ${\mathbf{ilev}}=0$ (following a call to c09faf);
• $0\le {\mathbf{ilev}}\le {\mathbf{nwl}}$, where nwl is as used in a preceding call to c09fcf;
• if ${\mathbf{cindex}}=0$, ${\mathbf{ilev}}={\mathbf{nwl}}$ (following a call to c09fcf).
2: $\mathbf{cindex}$Integer Input
On entry: identifies which coefficients to insert. The coefficients are identified as follows:
${\mathbf{cindex}}=0$
The approximation coefficients, produced by application of the low pass filter over columns, rows and frames of $A$ (LLL). After a call to the multi-level transform routine c09fcf (which implies that ${\mathbf{ilev}}>0$) the approximation coefficients are present only for ${\mathbf{ilev}}={\mathbf{nwl}}$, where nwl is the value used in a preceding call to c09fcf.
${\mathbf{cindex}}=1$
The detail coefficients produced by applying the low pass filter over columns and rows of $A$ and the high pass filter over frames (LLH).
${\mathbf{cindex}}=2$
The detail coefficients produced by applying the low pass filter over columns, high pass filter over rows and low pass filter over frames of $A$ (LHL).
${\mathbf{cindex}}=3$
The detail coefficients produced by applying the low pass filter over columns of $A$ and high pass filter over rows and frames (LHH).
${\mathbf{cindex}}=4$
The detail coefficients produced by applying the high pass filter over columns of $A$ and low pass filter over rows and frames (HLL).
${\mathbf{cindex}}=5$
The detail coefficients produced by applying the high pass filter over columns, low pass filter over rows and high pass filter over frames of $A$ (HLH).
${\mathbf{cindex}}=6$
The detail coefficients produced by applying the high pass filter over columns and rows of $A$ and the low pass filter over frames (HHL).
${\mathbf{cindex}}=7$
The detail coefficients produced by applying the high pass filter over columns, rows and frames of $A$ (HHH).
Constraints:
• if ${\mathbf{ilev}}=0$, $0\le {\mathbf{cindex}}\le 7$;
• if ${\mathbf{ilev}}={\mathbf{nwl}}$, following a call to c09fcf transforming nwl levels, $0\le {\mathbf{cindex}}\le 7$;
• otherwise $1\le {\mathbf{cindex}}\le 7$.
3: $\mathbf{lenc}$Integer Input
On entry: the dimension of the array c as declared in the (sub)program from which c09fzf is called.
Constraint: lenc must be unchanged from the value used in the preceding call to either c09faf or c09fcf.
4: $\mathbf{c}\left({\mathbf{lenc}}\right)$Real (Kind=nag_wp) array Input/Output
On entry: contains the DWT coefficients inserted by previous calls to c09fzf, or computed by a previous call to either c09faf or c09fcf.
On exit: contains the same DWT coefficients provided on entry except for those identified by ilev and cindex, which are updated with the values supplied in d, inserted into the correct locations as expected by one of the reconstruction routines c09fbf (if c09faf was called previously) or c09fdf (if c09fcf was called previously).
5: $\mathbf{d}\left({\mathbf{ldd}},{\mathbf{sdd}},*\right)$Real (Kind=nag_wp) array Input
Note: the last dimension of the array d must be at least ${n}_{\mathrm{cfr}}$.
On entry: the coefficients to be inserted.
If the DWT coefficients were computed by c09faf then
• if ${\mathbf{cindex}}=0$, the approximation coefficients must be stored in ${\mathbf{d}}\left(i,\mathit{j},\mathit{k}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{\mathrm{cm}}$, $\mathit{j}=1,2,\dots ,{n}_{\mathrm{cn}}$ and $\mathit{k}=1,2,\dots ,{n}_{\mathrm{cfr}}$;
• if $1\le {\mathbf{cindex}}\le 7$, the detail coefficients, as indicated by cindex, must be stored in ${\mathbf{d}}\left(i,\mathit{j},\mathit{k}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{\mathrm{cm}}$, $\mathit{j}=1,2,\dots ,{n}_{\mathrm{cn}}$ and $\mathit{k}=1,2,\dots ,{n}_{\mathrm{cfr}}$.
If the DWT coefficients were computed by c09fcf then
• if ${\mathbf{cindex}}=0$ and ${\mathbf{ilev}}={\mathbf{nwl}}$, the approximation coefficients must be stored in ${\mathbf{d}}\left(i,\mathit{j},\mathit{k}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{\mathrm{cm}}$, $\mathit{j}=1,2,\dots ,{n}_{\mathrm{cn}}$ and $\mathit{k}=1,2,\dots ,{n}_{\mathrm{cfr}}$;
• if $1\le {\mathbf{cindex}}\le 7$, the detail coefficients, as indicated by cindex, for level ilev must be stored in ${\mathbf{d}}\left(i,\mathit{j},\mathit{k}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{\mathrm{cm}}$, $\mathit{j}=1,2,\dots ,{n}_{\mathrm{cn}}$ and $\mathit{k}=1,2,\dots ,{n}_{\mathrm{cfr}}$.
6: $\mathbf{ldd}$Integer Input
On entry: the first dimension of the array d as declared in the (sub)program from which c09fzf is called.
Constraint: ${\mathbf{ldd}}>{n}_{\mathrm{cm}}$.
7: $\mathbf{sdd}$Integer Input
On entry: the second dimension of the array d as declared in the (sub)program from which c09fzf is called.
Constraint: ${\mathbf{sdd}}>{n}_{\mathrm{cn}}$.
8: $\mathbf{icomm}\left(260\right)$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.
9: $\mathbf{ifail}$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 $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ilev}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ilev}}=0$ following a call to the single level routine c09faf.
On entry, ${\mathbf{ilev}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ilev}}>0$ following a call to the multi-level routine c09fcf.
On entry, ${\mathbf{ilev}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nwl}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ilev}}\le {\mathbf{nwl}}$, where ${\mathbf{nwl}}$ is the number of levels used in the call to c09fcf.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{cindex}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{cindex}}\le 7$.
On entry, ${\mathbf{cindex}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{cindex}}\ge 0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{lenc}}=⟨\mathit{\text{value}}⟩$ and ${n}_{\mathrm{ct}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lenc}}\ge {n}_{\mathrm{ct}}$, where ${n}_{\mathrm{ct}}$ is the number of DWT coefficients computed in a previous call to c09faf.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{ldd}}=⟨\mathit{\text{value}}⟩$ and ${n}_{\mathrm{cm}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldd}}\ge {n}_{\mathrm{cm}}$, where ${n}_{\mathrm{cm}}$ is the number of DWT coefficients in the first dimension at the selected level ilev.
On entry, ${\mathbf{ldd}}=⟨\mathit{\text{value}}⟩$ and ${n}_{\mathrm{cm}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldd}}\ge {n}_{\mathrm{cm}}$, where ${n}_{\mathrm{cm}}$ is the number of DWT coefficients in the first dimension following the single level transform.
On entry, ${\mathbf{sdd}}=⟨\mathit{\text{value}}⟩$ and ${n}_{\mathrm{cn}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{sdd}}\ge {n}_{\mathrm{cn}}$, where ${n}_{\mathrm{cn}}$ is the number of DWT coefficients in the second dimension at the selected level ilev.
On entry, ${\mathbf{sdd}}=⟨\mathit{\text{value}}⟩$ and ${n}_{\mathrm{cn}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{sdd}}\ge {n}_{\mathrm{cn}}$, where ${n}_{\mathrm{cn}}$ is the number of DWT coefficients in the second dimension following the single level transform.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{ilev}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nwl}}=⟨\mathit{\text{value}}⟩$, but ${\mathbf{cindex}}=0$.
Constraint: ${\mathbf{cindex}}>0$ when ${\mathbf{ilev}}<{\mathbf{nwl}}$ in the preceding call to c09fcf.
${\mathbf{ifail}}=6$
Either the initialization routine has not been called first or icomm has been corrupted.
${\mathbf{ifail}}=-99$
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.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

Not applicable.

8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
c09fzf is not threaded in any implementation.

None.

10Example

The following example demonstrates using the coefficient extraction and insertion routines in order to apply denoising using a thresholding operation. The original input data has artificial noise introduced to it, taken from a normal random number distribution. Reconstruction then takes place on both the noisy data and denoised data. The Mean Square Errors (MSE) of the two reconstructions are printed along with the reconstruction of the denoised data. The MSE of the denoised reconstruction is less than that of the noisy reconstruction.

10.1Program Text

Program Text (c09fzfe.f90)

10.2Program Data

Program Data (c09fzfe.d)

10.3Program Results

Program Results (c09fzfe.r)