naginterfaces.library.wav.dim2_​multi_​fwd

naginterfaces.library.wav.dim2_multi_fwd(a, nwl, comm)

dim2_multi_fwd computes the two-dimensional multi-level discrete wavelet transform (DWT). The initialization function dim2_init() must be called first to set up the DWT options.

For full information please refer to the NAG Library document for c09ec

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/c09/c09ecf.html

Parameters

afloat, array-like, shape $(m,n)$

The $m\times n$ data matrix $A$.

nwlint

The number of levels, $n_{fwd}$, in the multi-level resolution to be performed.

commdict, communication object, modified in place

Communication structure.

This argument must have been initialized by a prior call to dim2_init().

Returns

cfloat, ndarray, shape $\left(\text{lenc}\right)$

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 dim2_coeff_ext() or dim2_coeff_ins() 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 $q\left(\text{i}\right)$ denote the number of coefficients (of each type) at level $\text{i}$, for $\text{i}=1,2,\dots ,n_{fwd}$, such that $q\left(i\right)=dwtlvm[n_{fwd}-i+1-1]\times dwtlvn[n_{fwd}-i+1-1]$.

Then, letting $k_1=q\left(n_{fwd}\right)$ and $k_{\text{j}+1}=k_{\text{j}}+q(n_{fwd}-\lceil \text{j}/3\rceil +1)$, for $\text{j}=1,2,\dots ,3n_{fwd}$, the coefficients are stored in $c$ as follows:

$c[\text{i}-1]$, for $\text{i}=1,2,\dots ,k_1$

Contains the level $n_{fwd}$ approximation coefficients, $a_{n_{fwd}}$.

$c[\text{i}-1]$, for $\text{i}=k_j+1,\dots ,k_{j+1}$

Contains the level $n_{fwd}-\lceil j/3\rceil +1$ vertical, horizontal and diagonal coefficients. These are:

vertical coefficients if $mod(j,3)=1$;

horizontal coefficients if $mod(j,3)=2$;

diagonal coefficients if $mod(j,3)=0$,

for $j=1,\dots ,3n_{fwd}$.

dwtlvmint, ndarray, shape $\left(nwl\right)$

The number of coefficients in the first dimension for each coefficient type at each level. $dwtlvm[\text{i}-1]$ contains the number of coefficients in the first dimension (for each coefficient type computed) at the ($n_{fwd}-\text{i}+1$)th level of resolution, for $\text{i}=1,2,\dots ,n_{fwd}$. Thus for the first $n_{fwd}-1$ levels of resolution, $dwtlvm[n_{fwd}-\text{i}+1-1]$ 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, $dwtlvm\left[0\right]$ is the size of the first dimension of the matrices of approximation, vertical, horizontal and diagonal coefficients computed.

dwtlvnint, ndarray, shape $\left(nwl\right)$

The number of coefficients in the second dimension for each coefficient type at each level. $dwtlvn[\text{i}-1]$ contains the number of coefficients in the second dimension (for each coefficient type computed) at the ($n_{fwd}-\text{i}+1$)th level of resolution, for $\text{i}=1,2,\dots ,n_{fwd}$. Thus for the first $n_{fwd}-1$ levels of resolution, $dwtlvn[n_{fwd}-\text{i}+1-1]$ 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, $dwtlvn\left[0\right]$ is the size of the second dimension of the matrices of approximation, vertical, horizontal and diagonal coefficients computed.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n=⟨value⟩$, the value of $\text{n}$ on initialization (see dim2_init()).

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m=⟨value⟩$, the value of $\text{m}$ on initialization (see dim2_init()).

(errno $5$)

On entry, $nwl=⟨value⟩$ and $\text{nwlmax}=⟨value⟩$ in dim2_init().

Constraint: $nwl\le \text{nwlmax}$ in dim2_init().

(errno $5$)

On entry, $nwl=⟨value⟩$.

Constraint: $nwl\ge 1$.

(errno $7$)

Either the initialization function has not been called first or $comm$[‘icomm’] has been corrupted.

(errno $7$)

Either the initialization function was called with $wtrans=\text{‘S'}$ or $comm$[‘icomm’] has been corrupted.

Notes

dim2_multi_fwd computes the multi-level DWT of two-dimensional data. For a given wavelet and end extension method, dim2_multi_fwd will compute a multi-level transform of a matrix $A$, using a specified number, $n_{fwd}$, of levels. The number of levels specified, $n_{fwd}$, must be no more than the value $l_{max}$ returned in $\text{nwlmax}$ by the initialization function dim2_init() 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 $0$ to the input matrix, $A$. Level 1 consists of the first set of coefficients computed: the vertical ($v_1$), horizontal ($h_1$) and diagonal ($d_1$) coefficients are stored at this level while the approximation ($a_1$) coefficients are used as the input to a repeat of the wavelet transform at the next level. This process is continued until, at level $n_{fwd}$, all four types of coefficients are stored. The output array, $C$, stores these sets of coefficients in reverse order, starting with $a_{n_{fwd}}$ followed by $v_{n_{fwd}},h_{n_{fwd}},d_{n_{fwd}},v_{n_{fwd}-1},h_{n_{fwd}-1},d_{n_{fwd}-1},\dots ,v_1,h_1,d_1$.