NAG CL Interface
c09ecc (dim2_multi_fwd)
1
Purpose
c09ecc computes the two-dimensional multi-level discrete wavelet transform (DWT). The initialization function
c09abc must be called first to set up the DWT options.
2
Specification
void |
c09ecc (Integer m,
Integer n,
const double a[],
Integer lda,
Integer lenc,
double c[],
Integer nwl,
Integer dwtlvm[],
Integer dwtlvn[],
Integer icomm[],
NagError *fail) |
|
The function may be called by the names: c09ecc, nag_wav_dim2_multi_fwd or nag_mldwt_2d.
3
Description
c09ecc computes the multi-level DWT of two-dimensional data. For a given wavelet and end extension method,
c09ecc will compute a multi-level transform of a matrix
, using a specified number,
, of levels. The number of levels specified,
, must be no more than the value
returned in
nwlmax by the initialization function
c09abc for the given problem. The transform is returned as a set of coefficients for the different levels (packed into a single array) and a representation of the multi-level structure.
The notation used here assigns level to the input matrix, . Level 1 consists of the first set of coefficients computed: the vertical (), horizontal () and diagonal () coefficients are stored at this level while the approximation () coefficients are used as the input to a repeat of the wavelet transform at the next level. This process is continued until, at level , all four types of coefficients are stored. The output array, , stores these sets of coefficients in reverse order, starting with followed by .
4
References
None.
5
Arguments
-
1:
– Integer
Input
-
On entry: number of rows, , of data matrix .
Constraint:
this must be the same as the value
m passed to the initialization function
c09abc.
-
2:
– Integer
Input
-
On entry: number of columns, , of data matrix .
Constraint:
this must be the same as the value
n passed to the initialization function
c09abc.
-
3:
– const double
Input
-
Note: the th element of the matrix is stored in .
On entry: the by data matrix .
-
4:
– Integer
Input
-
On entry: the stride separating matrix row elements in the array
a.
Constraint:
.
-
5:
– Integer
Input
-
On entry: the dimension of the array
c.
c must be large enough to contain,
, wavelet coefficients. The maximum value of
is returned in
nwct by the call to the initialization function
c09abc and corresponds to the DWT being continued for the maximum number of levels possible for the given data set. When the number of levels,
, is chosen to be less than the maximum,
, then
is correspondingly smaller and
lenc can be reduced by noting that the vertical, horizontal and diagonal coefficients are stored at every level and that in addition the approximation coefficients are stored for the final level only. The number of coefficients stored at each level is given by
for
in
c09abc and
for
,
or
, where the input data is of dimension
at that level and
is the filter length
nf provided by the call to
c09abc. At the final level the storage is
times this value to contain the set of approximation coefficients.
Constraint:
, where
is the total number of coefficients that correspond to a transform with
nwl levels.
-
6:
– double
Output
-
On exit: the coefficients of the discrete wavelet transform. If you need to access or modify the approximation coefficients or any specific set of detail coefficients then the use of
c09eyc or
c09ezc is recommended. For completeness the following description provides details of precisely how the coefficient are stored in
c but this information should only be required in rare cases.
Let
denote the number of coefficients (of each type) at level
, for
, such that
. Then, letting
and
, for
, the coefficients are stored in
c as follows:
- , for
- Contains the level approximation coefficients, .
- , for
- Contains the level vertical, horizontal and diagonal coefficients. These are:
- vertical coefficients if ;
- horizontal coefficients if ;
- diagonal coefficients if ,
for .
-
7:
– Integer
Input
-
On entry: the number of levels, , in the multi-level resolution to be performed.
Constraint:
, where
is the value returned in
nwlmax (the maximum number of levels) by the call to the initialization function
c09abc.
-
8:
– Integer
Output
-
On exit: the number of coefficients in the first dimension for each coefficient type at each level.
contains the number of coefficients in the first dimension (for each coefficient type computed) at the ()th level of resolution, for . Thus for the first levels of resolution, is the size of the first dimension of the matrices of vertical, horizontal and diagonal coefficients computed at this level; for the final level of resolution, is the size of the first dimension of the matrices of approximation, vertical, horizontal and diagonal coefficients computed.
-
9:
– Integer
Output
-
On exit: the number of coefficients in the second dimension for each coefficient type at each level.
contains the number of coefficients in the second dimension (for each coefficient type computed) at the ()th level of resolution, for . Thus for the first levels of resolution, is the size of the second dimension of the matrices of vertical, horizontal and diagonal coefficients computed at this level; for the final level of resolution, is the size of the second dimension of the matrices of approximation, vertical, horizontal and diagonal coefficients computed.
-
10:
– 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.
On exit: contains additional information on the computed transform.
-
11:
– 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 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
or
icomm has been corrupted.
- NE_INT
-
On entry,
.
Constraint:
, the value of
m on initialization (see
c09abc).
On entry,
.
Constraint:
, the value of
n on initialization (see
c09abc).
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, .
Constraint: , the total number of coefficents to be generated.
On entry,
and
in
c09abc.
Constraint:
in
c09abc.
- 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.
8
Parallelism and Performance
c09ecc is not threaded in any implementation.
The wavelet coefficients at each level can be extracted from the output array
c using the information contained in
dwtlvm and
dwtlvn on exit (see the descriptions of
c,
dwtlvm and
dwtlvn in
Section 5). For example, given an input data set,
, denoising can be carried out by applying a thresholding operation to the detail (vertical, horizontal and diagonal) coefficients at every level. The elements
to
, as described in
Section 5, contain the detail coefficients,
, for
and
, where
is the number of each type of coefficient at level
and
and
is the transformed noise term. If some threshold parameter
is chosen, a simple hard thresholding rule can be applied as
taking
to be an approximation to the required detail coefficient without noise,
. The resulting coefficients can then be used as input to
c09edc in order to reconstruct the denoised signal. See
Section 10 in
c09ezc for a simple example of denoising.
See the references given in the introduction to this chapter for a more complete account of wavelet denoising and other applications.
10
Example
This example performs a multi-level resolution transform of a dataset using the Daubechies wavelet (see
in
c09abc) using half-point symmetric end extensions, the maximum possible number of levels of resolution, where the number of coefficients in each level and the coefficients themselves are not changed. The original dataset is then reconstructed using
c09edc.
10.1
Program Text
10.2
Program Data
10.3
Program Results