dwt.texi

This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY without ev

where @var{j} is the index of the level 
@c{$j = 0 \dots J-1$}
@math{j = 0 ... J-1}
and
@var{k} is the index of the coefficient within each level,
@c{$k = 0 \dots 2^j - 1$}
@math{k = 0 ... (2^j)-1}.  
The total number of levels is @math{J = \log_2(n)}.  The output data
has the following form,
@tex
\beforedisplay
$$
(s_{-1,0}, d_{0,0}, d_{1,0}, d_{1,1}, d_{2,0},\cdots, d_{j,k},\cdots, d_{J-1,2^{J-1} - 1}) 
$$
\afterdisplay
@end tex
@ifinfo

@example
(s_@{-1,0@}, d_@{0,0@}, d_@{1,0@}, d_@{1,1@}, d_@{2,0@}, ..., 
  d_@{j,k@}, ..., d_@{J-1,2^@{J-1@}-1@}) 
@end example

@end ifinfo
@noindent
where the first element is the smoothing coefficient @c{$s_{-1,0}$}
@math{s_@{-1,0@}}, followed by the detail coefficients @c{$d_{j,k}$}
@math{d_@{j,k@}} for each level
@math{j}.  The backward transform inverts these coefficients to obtain 
the original data.

These functions return a status of @code{GSL_SUCCESS} upon successful
completion.  @code{GSL_EINVAL} is returned if @var{n} is not an integer
power of 2 or if insufficient workspace is provided.
@end deftypefun

@node DWT in two dimension
@subsection Wavelet transforms in two dimension
@cindex DWT, two dimensional

The library provides functions to perform two-dimensional discrete
wavelet transforms on square matrices.  The matrix dimensions must be an
integer power of two.  There are two possible orderings of the rows and
columns in the two-dimensional wavelet transform, referred to as the
``standard'' and ``non-standard'' forms.

The ``standard'' transform performs a complete discrete wavelet
transform on the rows of the matrix, followed by a separate complete
discrete wavelet transform on the columns of the resulting
row-transformed matrix.  This procedure uses the same ordering as a
two-dimensional fourier transform.

The ``non-standard'' transform is performed in interleaved passes on the
rows and columns of the matrix for each level of the transform.  The
first level of the transform is applied to the matrix rows, and then to
the matrix columns.  This procedure is then repeated across the rows and
columns of the data for the subsequent levels of the transform, until
the full discrete wavelet transform is complete.  The non-standard form
of the discrete wavelet transform is typically used in image analysis.

The functions described in this section are declared in the header file
@file{gsl_wavelet2d.h}.

@deftypefun {int} gsl_wavelet2d_transform (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{tda}, size_t @var{size1}, size_t @var{size2}, gsl_wavelet_direction @var{dir}, gsl_wavelet_workspace * @var{work})
@deftypefunx {int} gsl_wavelet2d_transform_forward (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{tda}, size_t @var{size1}, size_t @var{size2}, gsl_wavelet_workspace * @var{work})
@deftypefunx {int} gsl_wavelet2d_transform_inverse (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{tda}, size_t @var{size1}, size_t @var{size2}, gsl_wavelet_workspace * @var{work})

These functions compute two-dimensional in-place forward and inverse
discrete wavelet transforms in standard and non-standard forms on the
array @var{data} stored in row-major form with dimensions @var{size1}
and @var{size2} and physical row length @var{tda}.  The dimensions must
be equal (square matrix) and are restricted to powers of two.  For the
@code{transform} version of the function the argument @var{dir} can be
either @code{forward} (@math{+1}) or @code{backward} (@math{-1}).  A
workspace @var{work} of the appropriate size must be provided.  On exit,
the appropriate elements of the array @var{data} are replaced by their
two-dimensional wavelet transform.

The functions return a status of @code{GSL_SUCCESS} upon successful
completion.  @code{GSL_EINVAL} is returned if @var{size1} and
@var{size2} are not equal and integer powers of 2, or if insufficient
workspace is provided.
@end deftypefun

@deftypefun {int} gsl_wavelet2d_transform_matrix (const gsl_wavelet * @var{w}, gsl_matrix * @var{m}, gsl_wavelet_direction @var{dir}, gsl_wavelet_workspace * @var{work})
@deftypefunx {int} gsl_wavelet2d_transform_matrix_forward (const gsl_wavelet * @var{w}, gsl_matrix * @var{m}, gsl_wavelet_workspace * @var{work})
@deftypefunx {int} gsl_wavelet2d_transform_matrix_inverse (const gsl_wavelet * @var{w}, gsl_matrix * @var{m}, gsl_wavelet_workspace * @var{work})
These functions compute the two-dimensional in-place wavelet transform
on a matrix @var{a}.
@end deftypefun

@deftypefun {int} gsl_wavelet2d_nstransform (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{tda}, size_t @var{size1}, size_t @var{size2}, gsl_wavelet_direction @var{dir}, gsl_wavelet_workspace * @var{work})
@deftypefunx {int} gsl_wavelet2d_nstransform_forward (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{tda}, size_t @var{size1}, size_t @var{size2}, gsl_wavelet_workspace * @var{work})
@deftypefunx {int} gsl_wavelet2d_nstransform_inverse (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{tda}, size_t @var{size1}, size_t @var{size2}, gsl_wavelet_workspace * @var{work})
These functions compute the two-dimensional wavelet transform in
non-standard form.
@end deftypefun

@deftypefun {int} gsl_wavelet2d_nstransform_matrix (const gsl_wavelet * @var{w}, gsl_matrix * @var{m}, gsl_wavelet_direction @var{dir}, gsl_wavelet_workspace * @var{work})
@deftypefunx {int} gsl_wavelet2d_nstransform_matrix_forward (const gsl_wavelet * @var{w}, gsl_matrix * @var{m}, gsl_wavelet_workspace * @var{work})
@deftypefunx {int} gsl_wavelet2d_nstransform_matrix_inverse (const gsl_wavelet * @var{w}, gsl_matrix * @var{m}, gsl_wavelet_workspace * @var{work})
These functions compute the non-standard form of the two-dimensional
in-place wavelet transform on a matrix @var{a}.
@end deftypefun

@node DWT Examples
@section Examples

The following program demonstrates the use of the one-dimensional
wavelet transform functions.  It computes an approximation to an input
signal (of length 256) using the 20 largest components of the wavelet
transform, while setting the others to zero.

@example
@verbatiminclude examples/dwt.c
@end example

@noindent
The output can be used with the @sc{gnu} plotutils @code{graph} program,

@example
$ ./a.out ecg.dat > dwt.dat
$ graph -T ps -x 0 256 32 -h 0.3 -a dwt.dat > dwt.ps
@end example

@iftex
The graphs below show an original and compressed version of a sample ECG
recording from the MIT-BIH Arrhythmia Database, part of the PhysioNet
archive of public-domain of medical datasets.

@sp 1
@center @image{dwt-orig,3.4in} 
@center @image{dwt-samp,3.4in} 
@quotation
Original (upper) and wavelet-compressed (lower) ECG signals, using the
20 largest components of the Daubechies(4) discrete wavelet transform.
@end quotation
@end iftex

@node DWT References
@section References and Further Reading

The mathematical background to wavelet transforms is covered in the
original lectures by Daubechies,

@itemize @asis
@item
Ingrid Daubechies.
Ten Lectures on Wavelets.
@cite{CBMS-NSF Regional Conference Series in Applied Mathematics} (1992), 
SIAM, ISBN 0898712742.
@end itemize

@noindent
An easy to read introduction to the subject with an emphasis on the
application of the wavelet transform in various branches of science is,

@itemize @asis
@item
Paul S. Addison. @cite{The Illustrated Wavelet Transform Handbook}.
Institute of Physics Publishing (2002), ISBN 0750306920.
@end itemize

@noindent
For extensive coverage of signal analysis by wavelets, wavelet packets
and local cosine bases see,

@itemize @asis
@item
S. G. Mallat.  @cite{A wavelet tour of signal processing} (Second
edition). Academic Press (1999), ISBN 012466606X.
@end itemize

@noindent
The concept of multiresolution analysis underlying the wavelet transform
is described in,

@itemize @asis
@item
S. G. Mallat.
Multiresolution Approximations and Wavelet Orthonormal Bases of L@math{^2}(R).
@cite{Transactions of the American Mathematical Society}, 315(1), 1989, 69--87.
@end itemize

@itemize @asis
@item
S. G. Mallat.
A Theory for Multiresolution Signal Decomposition---The Wavelet Representation.
@cite{IEEE Transactions on Pattern Analysis and Machine Intelligence}, 11, 1989,
674--693. 
@end itemize

@noindent
The coefficients for the individual wavelet families implemented by the
library can be found in the following papers,

@itemize @asis
@item
I. Daubechies.
Orthonormal Bases of Compactly Supported Wavelets.
@cite{Communications on Pure and Applied Mathematics}, 41 (1988) 909--996.
@end itemize

@itemize @asis
@item
A. Cohen, I. Daubechies, and J.-C. Feauveau.
Biorthogonal Bases of Compactly Supported Wavelets.
@cite{Communications on Pure and Applied Mathematics}, 45 (1992)
485--560.
@end itemize

@noindent
The PhysioNet archive of physiological datasets can be found online at
@uref{http://www.physionet.org/} and is described in the following
paper,

@itemize @asis
@item
Goldberger et al.  
PhysioBank, PhysioToolkit, and PhysioNet: Components
of a New Research Resource for Complex Physiologic
Signals. 
@cite{Circulation} 101(23):e215-e220 2000.
@end itemize