c09fzc 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 functions
c09fbc or
c09fdc.
c09fzc 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
c09fzc is preceded by a call to the initialization function
c09acc and either the forwards transform function
c09fac or multi-level forwards transform function
c09fcc.
Given an initial three-dimensional data set
$A$, a prior call to
c09fac or
c09fcc computes the approximation coefficients (at the highest requested level in the case of
c09fcc) and, seven sets of detail coefficients (at all levels in the case of
c09fcc) and stores these in compact form in a one-dimensional array
c.
c09fyc can then extract either the approximation coefficients or one of the sets of detail coefficients (at one of the levels following
c09fcc)
as three-dimensional data into the 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
c09fzc. Several extractions and insertions may be performed.
c09fbc or
c09fdc can then be used to reconstruct a manipulated data set
$\stackrel{~}{A}$. The dimensions of
the three-dimensional data stored in
d depend on the level extracted and are available from either: the arrays
dwtlvm,
dwtlvn and
dwtlvfr as returned by
c09fcc if this was called first; or, otherwise from
nwct,
nwcn and
nwcfr as returned by
c09acc. See
Section 2.1 in the
C09 Chapter Introduction for a discussion of the three-dimensional DWT.
None.
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 c09fac (i.e., when ${\mathbf{ilev}}=0$) this is equal to ${\mathbf{nwct}}/\left(8\times {\mathbf{nwcn}}\times {\mathbf{nwcfr}}\right)$ as returned by c09acc. Following a call to c09fcc transforming nwl levels, and when inserting at level ${\mathbf{ilev}}>0$, this is equal to ${\mathbf{dwtlvm}}\left[{\mathbf{nwl}}-{\mathbf{ilev}}\right]$.
- ${n}_{\mathrm{cn}}$ is the number of wavelet coefficients in the second dimension. Following a call to c09fac (i.e., when ${\mathbf{ilev}}=0$) this is equal to nwcn as returned by c09acc. Following a call to c09fcc transforming nwl levels, and when inserting at level ${\mathbf{ilev}}>0$, this is equal to ${\mathbf{dwtlvn}}\left[{\mathbf{nwl}}-{\mathbf{ilev}}\right]$.
- ${n}_{\mathrm{cfr}}$ is the number of wavelet coefficients in the third dimension. Following a call to c09fac (i.e., when ${\mathbf{ilev}}=0$) this is equal to nwcfr as returned by c09acc. Following a call to c09fcc transforming nwl levels, and when inserting at level ${\mathbf{ilev}}>0$, this is equal to ${\mathbf{dwtlvfr}}\left[{\mathbf{nwl}}-{\mathbf{ilev}}\right]$
Not applicable.
None.
The following example demonstrates using the coefficient extraction and insertion functions 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.