naginterfaces.library.wav.dim3_​multi_​fwd

naginterfaces.library.wav.dim3_multi_fwd(a, nwl, comm)

dim3_multi_fwd computes the three-dimensional multi-level discrete wavelet transform (DWT). The initialization function dim3_init() must be called first to set up the DWT options.

For full information please refer to the NAG Library document for c09fc

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/c09/c09fcf.html

Parameters

afloat, array-like, shape $(m,n,\text{fr})$

The $m\times n\times \text{fr}$ three-dimensional input data $A$, where with $A_{ijk}$ stored in $a[i-1,j-1,k-1]$.

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 dim3_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 dim3_coeff_ext() or dim3_coeff_ins() is recommended. For completeness the following description provides details of precisely how the coefficients 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 $q\left(i\right)=dwtlvm[n_{fwd}-i]\times dwtlvn[n_{fwd}-i]\times dwtlvfr[n_{fwd}-i]$.

Then, letting $k_1=q\left(n_{fwd}\right)$ and $k_{\text{j}+1}=k_{\text{j}}+q(n_{fwd}-\lceil \text{j}/7\rceil +1)$, for $\text{j}=1,2,\dots ,7n_{fwd}$, 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, $a_{n_{fwd}}$. Note that for computational efficiency reasons these coefficients are stored as $dwtlvm\left[0\right]\times dwtlvn\left[0\right]\times dwtlvfr\left[0\right]$ in $c$.

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

Contains the level $n_{fwd}-\lceil j/7\rceil +1$ detail coefficients. These are:

LLH coefficients if $mod(j,7)=1$;

LHL coefficients if $mod(j,7)=2$;

LHH coefficients if $mod(j,7)=3$;

HLL coefficients if $mod(j,7)=4$;

HLH coefficients if $mod(j,7)=5$;

HHL coefficients if $mod(j,7)=6$;

HHH coefficients if $mod(j,7)=0$,

for $j=1,\dots ,7n_{fwd}$. See the C09 Introduction for a description of how these coefficients are produced.

Note that for computational efficiency reasons these coefficients are stored as $dwtlvfr[\lceil j/7\rceil -1]\times dwtlvm[\lceil j/7\rceil -1]\times dwtlvn[\lceil j/7\rceil -1]$ in $c$.

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 $\text{i}=1,2,\dots ,n_{fwd}$.

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 $\text{i}=1,2,\dots ,n_{fwd}$.

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

The number of coefficients in the third dimension for each coefficient type at each level. $dwtlvfr[\text{i}-1]$ contains the number of coefficients in the third dimension (for each coefficient type computed) at the ($n_{fwd}-\text{i}+1$)th level of resolution, for $\text{i}=1,2,\dots ,n_{fwd}$.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n=⟨value⟩$, the value of $\text{n}$ on initialization (see dim3_init()).

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m=⟨value⟩$, the value of $\text{m}$ on initialization (see dim3_init()).

(errno $1$)

On entry, $\text{fr}=⟨value⟩$.

Constraint: $\text{fr}=⟨value⟩$, the value of $\text{fr}$ on initialization (see dim3_init()).

(errno $5$)

On entry, $nwl=⟨value⟩$ and $\text{nwlmax}=⟨value⟩$ in dim3_init().

Constraint: $nwl\le \text{nwlmax}$ in dim3_init().

(errno $5$)

On entry, $nwl=⟨value⟩$.

Constraint: $nwl\ge 1$.

(errno $6$)

Either the communication array $comm$[‘icomm’] has been corrupted or there has not been a prior call to the initialization function dim3_init().

(errno $6$)

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

Notes

dim3_multi_fwd computes the multi-level DWT of three-dimensional data. For a given wavelet and end extension method, dim3_multi_fwd will compute a multi-level transform of a three-dimensional array $A$, 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 dim3_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 data, $A$. Level 1 consists of the first set of coefficients computed: the seven sets of detail 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 $n_{fwd}$, all eight types of coefficients are stored. All coefficients are packed into a single array.

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)