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<H3><A NAME="SECTION002092100000000000000">
TRANSF/WAVE</A>
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<DIV ALIGN="CENTER">
TRANSF/WAVE Image Wavelet [Algo] [Nbr_Scale] [Fc]
</DIV>
This command creates a file which contains the wavelet transform. The
suffixe of a wavelet transform file is ``.wave''. It is automatically
added to the name passed to the command. Several algorithms are
proposed:
<DL COMPACT>
<DT>1.
<DD><EM>à trous</EM> algorithm with a linear scaling function.
The wavelet function is the difference between two resolutions
(see <A HREF="node317.html#sec_trou">14.4.3</A>).
<DT>2.
<DD><EM>à trous</EM> with a B3-spline scaling function (default value).
The wavelet function is the difference between two resolutions
(see <A HREF="node317.html#sec_trou">14.4.3</A>).
<DT>3.
<DD>algorithm using the Fourier transform, without any reduction
of the samples between two scales. The Fourier transform of the scaling
function is a b3-spline and the wavelet function is the
difference between two resolutions (<A HREF="node321.html#sec_fft">14.4.5</A>).
<DT>4.
<DD>pyramidal algorithm in the direct space, with a linear scaling
function (see section <A HREF="node320.html#sec_pyr_dir">14.4.4</A>).
<DT>5.
<DD>pyramidal algorithm in the direct space, with a b3-spline
scaling function (see section <A HREF="node320.html#sec_pyr_dir">14.4.4</A>).
<DT>6.
<DD>algorithm using the Fourier transform with a reduction of
the samples between two scales. The Fourier transform of the scaling
function is a b3-spline the wavelet function is the difference
between two resolutions (<A HREF="node321.html#sec_fft">14.4.5</A>).
<DT>7.
<DD>algorithm using the Fourier transform with a reduction of the
samples between two scales. The Fourier transform of the scaling
function is a b3-spline. The wavelet function is the difference
between the square of two resolutions (<A HREF="node321.html#sec_fft">14.4.5</A>).
<DT>8.
<DD>Mallat's Algorithm with biorthogonal filters (<A HREF="node316.html#sec_mallat">14.4.2</A>).
</DL>The parameter <EM>Algo</EM> can be chosen between 1 and 8. If <EM>Algo</EM> is in
{1,2,3}, the number of data of the wavelet transform is equal to
the number of pixels multiplied by the number of scales (if the number
of pixels of the image is <I>N</I><SUP>2</SUP>, the number of wavelet coefficients is
<!-- MATH: $\mbox{Nbr\_Scale}.N^2$ -->
<IMG
WIDTH="134" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
SRC="img871.gif"
ALT="$\mbox{Nbr\_Scale}.N^2$">). Algorithms 4, 5, 6, and 7 are pyramidal (the number
of wavelet coefficients is
<!-- MATH: $\frac{4}{3}N^2$ -->
<IMG
WIDTH="49" HEIGHT="49" ALIGN="MIDDLE" BORDER="0"
SRC="img872.gif"
ALT="$\frac{4}{3}N^2$">), and the 8th algorithm
does not increase the number of data (the size of the wavelet
transform is <I>N</I><SUP>2</SUP>). Due to the discretisation and the undersampling,
the properties of these algorithms are not the same. The 8th algorithm
is more compact, but is not isotropic (see section
<A HREF="node316.html#sec_mallat">14.4.2</A>). Algorithms 3, 6, and 7 compute the wavelet transform in
the Fourier space (see section <A HREF="node321.html#sec_fft">14.4.5</A>) and the undersampling
respect Shannon's theorem. Pyramidal algorithms 4 and 5 compute the
wavelet transform in the direct space, but need an interative
reconstruction. Algorithms 1 and 2 are isotropic but increase the number
of data. The 2D-discrete wavelet transform is not restricted the
previous algorithms. Other algorithms exist (see for example
Feauveau's one
[<A
HREF="node370.html#feauveau">11</A>] which is not diadic). The interest of the wavelet transform
is that it is a very flexible tool. We can adapt the transform to our
problem. We prefer the 8th for image compression, 6 and 7 for image
restoration, 2 for data analysis, <I>etc.</I>. The wavelet function can be
derived too from the specific problem to resolve (see [<A
HREF="node370.html#starck1">35</A>]).
<P>
The parameter <EM>Nbr_Scale</EM> specifies the number of scales to compute.
The wavelet transform will contain
<!-- MATH: $\mbox{{\em Nbr\_Scale}}-1$ -->
<IMG
WIDTH="134" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img873.gif"
ALT="$\mbox{{\em Nbr\_Scale}}-1$">
wavelet
coefficients planes and one plane which will be the image at a very
low resolution.
The parameter <EM>Fc</EM> defines the cut-off frequency of the scaling
function (
<!-- MATH: $0 < F_c \leq 0.5$ -->
<IMG
WIDTH="126" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img874.gif"
ALT="$0 < F_c \leq 0.5$">). It is used only if the selected
wavelet transform algorithm
uses the FFT.
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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