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 @cindex DWT, see wavelet transforms
 @cindex wavelet transforms
 @cindex transforms, wavelet

 This chapter describes functions for performing Discrete Wavelet
 Transforms (DWTs).  The library includes wavelets for real data in both
 one and two dimensions.  The wavelet functions are declared in the header
 files @file{gsl_wavelet.h} and @file{gsl_wavelet2d.h}.

 @menu
 * DWT Definitions::
 * DWT Initialization::
 * DWT Transform Functions::
 * DWT Examples::
 * DWT References::
 @end menu

 @node DWT Definitions
 @section Definitions
 @cindex DWT, mathematical definition

 The continuous wavelet transform and its inverse are defined by
 the relations,
 @iftex
 @tex
 \beforedisplay
 $$
 w(s, \tau) = \int_{-\infty}^\infty f(t) * \psi^*_{s,\tau}(t) dt
 $$
 \afterdisplay
 @end tex
 @end iftex
 @ifinfo

 @example
 w(s,\tau) = \int f(t) * \psi^*_@{s,\tau@}(t) dt
 @end example

 @end ifinfo
 @noindent
 and,
 @tex
 \beforedisplay
 $$
 f(t) = \int_0^\infty ds \int_{-\infty}^\infty w(s, \tau) * \psi_{s,\tau}(t) d\tau
 $$
 \afterdisplay
 @end tex
 @ifinfo

 @example
 f(t) = \int \int_@{-\infty@}^\infty w(s, \tau) * \psi_@{s,\tau@}(t) d\tau ds
 @end example

 @end ifinfo
 @noindent
 where the basis functions @c{$\psi_{s,\tau}$}
 @math{\psi_@{s,\tau@}} are obtained by scaling
 and translation from a single function, referred to as the @dfn{mother
 wavelet}.

 The discrete version of the wavelet transform acts on equally-spaced
 samples, with fixed scaling and translation steps (@math{s},
 @math{\tau}).  The frequency and time axes are sampled @dfn{dyadically}
 on scales of @math{2^j} through a level parameter @math{j}.
 @c  The wavelet @math{\psi}
 @c  can be expressed in terms of a scaling function @math{\varphi},
 @c
 @c  @tex
 @c  \beforedisplay
 @c  $$
 @c  \psi(2^{j-1},t) = \sum_{k=0}^{2^j-1} g_j(k) * \bar{\varphi}(2^j t-k)
 @c  $$
 @c  \afterdisplay
 @c  @end tex
 @c  @ifinfo
 @c  @example
 @c  \psi(2^@{j-1@},t) = \sum_@{k=0@}^@{2^j-1@} g_j(k) * \bar@{\varphi@}(2^j t-k)
 @c  @end example
 @c  @end ifinfo
 @c  @noindent
 @c  and
 @c
 @c  @tex
 @c  \beforedisplay
 @c  $$
 @c  \varphi(2^{j-1},t) = \sum_{k=0}^{2^j-1} h_j(k) * \bar{\varphi}(2^j t-k)
 @c  $$
 @c  \afterdisplay
 @c  @end tex
 @c  @ifinfo
 @c  @example
 @c  \varphi(2^@{j-1@},t) = \sum_@{k=0@}^@{2^j-1@} h_j(k) * \bar@{\varphi@}(2^j t-k)
 @c  @end example
 @c  @end ifinfo
 @c  @noindent
 @c  The functions @math{\psi} and @math{\varphi} are related through the
 @c  coefficients
 @c  @c{$g_{n} = (-1)^n h_{L-1-n}$}
 @c  @math{g_@{n@} = (-1)^n h_@{L-1-n@}}
 @c  for @c{$n=0 \dots L-1$}
 @c  @math{n=0 ... L-1},
 @c  where @math{L} is the total number of coefficients.  The two sets of
 @c  coefficients @math{h_j} and @math{g_i} define the scaling function and
 @c the wavelet.
 The resulting family of functions @c{$\{\psi_{j,n}\}$}
 @math{@{\psi_@{j,n@}@}}
 constitutes an orthonormal
 basis for square-integrable signals.

 The discrete wavelet transform is an @math{O(N)} algorithm, and is also
 referred to as the @dfn{fast wavelet transform}.

 @node DWT Initialization
 @section Initialization
 @cindex DWT initialization

 The @code{gsl_wavelet} structure contains the filter coefficients
 defining the wavelet and any associated offset parameters.

 @deftypefun {gsl_wavelet *} gsl_wavelet_alloc (const gsl_wavelet_type * @var{T}, size_t @var{k})
 This function allocates and initializes a wavelet object of type
 @var{T}.  The parameter @var{k} selects the specific member of the
 wavelet family.  A null pointer is returned if insufficient memory is
 available or if a unsupported member is selected.
 @end deftypefun

 The following wavelet types are implemented:

 @deffn {Wavelet} gsl_wavelet_daubechies
 @deffnx {Wavelet} gsl_wavelet_daubechies_centered
 @cindex Daubechies wavelets
 @cindex maximal phase, Daubechies wavelets
 The is the Daubechies wavelet family of maximum phase with @math{k/2}
 vanishing moments.  The implemented wavelets are
 @c{$k=4, 6, \dots, 20$}
 @math{k=4, 6, @dots{}, 20}, with @var{k} even.
 @end deffn

 @deffn {Wavelet} gsl_wavelet_haar
 @deffnx {Wavelet} gsl_wavelet_haar_centered
 @cindex Haar wavelets
 This is the Haar wavelet.  The only valid choice of @math{k} for the
 Haar wavelet is @math{k=2}.
 @end deffn

 @deffn {Wavelet} gsl_wavelet_bspline
 @deffnx {Wavelet} gsl_wavelet_bspline_centered
 @cindex biorthogonal wavelets
 @cindex B-spline wavelets
 This is the biorthogonal B-spline wavelet family of order @math{(i,j)}.
 The implemented values of @math{k = 100*i + j} are 103, 105, 202, 204,
 206, 208, 301, 303, 305 307, 309.
 @end deffn

 @noindent
 The centered forms of the wavelets align the coefficients of the various
 sub-bands on edges.  Thus the resulting visualization of the
 coefficients of the wavelet transform in the phase plane is easier to
 understand.

 @deftypefun {const char *} gsl_wavelet_name (const gsl_wavelet * @var{w})
 This function returns a pointer to the name of the wavelet family for
 @var{w}.
 @end deftypefun

 @c  @deftypefun {void} gsl_wavelet_print (const gsl_wavelet * @var{w})
 @c  This function prints the filter coefficients (@code{**h1}, @code{**g1}, @code{**h2}, @code{**g2}) of the wavelet object @var{w}.
 @c  @end deftypefun

 @deftypefun {void} gsl_wavelet_free (gsl_wavelet * @var{w})
 This function frees the wavelet object @var{w}.
 @end deftypefun

 The @code{gsl_wavelet_workspace} structure contains scratch space of the
 same size as the input data and is used to hold intermediate results
 during the transform.

 @deftypefun {gsl_wavelet_workspace *} gsl_wavelet_workspace_alloc (size_t @var{n})
 This function allocates a workspace for the discrete wavelet transform.
 To perform a one-dimensional transform on @var{n} elements, a workspace
 of size @var{n} must be provided.  For two-dimensional transforms of
 @var{n}-by-@var{n} matrices it is sufficient to allocate a workspace of
 size @var{n}, since the transform operates on individual rows and
 columns.
 @end deftypefun

 @deftypefun {void} gsl_wavelet_workspace_free (gsl_wavelet_workspace * @var{work})
 This function frees the allocated workspace @var{work}.
 @end deftypefun

 @node DWT Transform Functions
 @section Transform Functions

 This sections describes the actual functions performing the discrete
 wavelet transform.  Note that the transforms use periodic boundary
 conditions.  If the signal is not periodic in the sample length then
 spurious coefficients will appear at the beginning and end of each level
 of the transform.

 @menu
 * DWT in one dimension::
 * DWT in two dimension::
 @end menu

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

 @deftypefun int gsl_wavelet_transform (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{stride}, size_t @var{n}, gsl_wavelet_direction @var{dir}, gsl_wavelet_workspace * @var{work})
 @deftypefunx int gsl_wavelet_transform_forward (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{stride}, size_t @var{n}, gsl_wavelet_workspace * @var{work})
 @deftypefunx int gsl_wavelet_transform_inverse (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{stride}, size_t @var{n}, gsl_wavelet_workspace * @var{work})

 These functions compute in-place forward and inverse discrete wavelet
 transforms of length @var{n} with stride @var{stride} on the array
 @var{data}. The length of the transform @var{n} is 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 length @var{n} must be provided.

 For the forward transform, the elements of the original array are
 replaced by the discrete wavelet
 transform @c{$f_i \rightarrow w_{j,k}$}
 @math{f_i -> w_@{j,k@}}
 in a packed triangular storage layout,
 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
	@cindex DWT, see wavelet transforms
	@cindex wavelet transforms
	@cindex transforms, wavelet

	This chapter describes functions for performing Discrete Wavelet
	Transforms (DWTs). The library includes wavelets for real data in both
	one and two dimensions. The wavelet functions are declared in the header
	files @file{gsl_wavelet.h} and @file{gsl_wavelet2d.h}.

	@menu
	* DWT Definitions::
	* DWT Initialization::
	* DWT Transform Functions::
	* DWT Examples::
	* DWT References::
	@end menu

	@node DWT Definitions
	@section Definitions
	@cindex DWT, mathematical definition

	The continuous wavelet transform and its inverse are defined by
	the relations,
	@iftex
	@tex
	\beforedisplay
	$$
	w(s, \tau) = \int_{-\infty}^\infty f(t) * \psi^*_{s,\tau}(t) dt
	$$
	\afterdisplay
	@end tex
	@end iftex
	@ifinfo

	@example
	w(s,\tau) = \int f(t) * \psi^*_@{s,\tau@}(t) dt
	@end example

	@end ifinfo
	@noindent
	and,
	@tex
	\beforedisplay
	$$
	f(t) = \int_0^\infty ds \int_{-\infty}^\infty w(s, \tau) * \psi_{s,\tau}(t) d\tau
	$$
	\afterdisplay
	@end tex
	@ifinfo

	@example
	f(t) = \int \int_@{-\infty@}^\infty w(s, \tau) * \psi_@{s,\tau@}(t) d\tau ds
	@end example

	@end ifinfo
	@noindent
	where the basis functions @c{$\psi_{s,\tau}$}
	@math{\psi_@{s,\tau@}} are obtained by scaling
	and translation from a single function, referred to as the @dfn{mother
	wavelet}.

	The discrete version of the wavelet transform acts on equally-spaced
	samples, with fixed scaling and translation steps (@math{s},
	@math{\tau}). The frequency and time axes are sampled @dfn{dyadically}
	on scales of @math{2^j} through a level parameter @math{j}.
	@c The wavelet @math{\psi}
	@c can be expressed in terms of a scaling function @math{\varphi},
	@c
	@c @tex
	@c \beforedisplay
	@c $$
	@c \psi(2^{j-1},t) = \sum_{k=0}^{2^j-1} g_j(k) * \bar{\varphi}(2^j t-k)
	@c $$
	@c \afterdisplay
	@c @end tex
	@c @ifinfo
	@c @example
	@c \psi(2^@{j-1@},t) = \sum_@{k=0@}^@{2^j-1@} g_j(k) * \bar@{\varphi@}(2^j t-k)
	@c @end example
	@c @end ifinfo
	@c @noindent
	@c and
	@c
	@c @tex
	@c \beforedisplay
	@c $$
	@c \varphi(2^{j-1},t) = \sum_{k=0}^{2^j-1} h_j(k) * \bar{\varphi}(2^j t-k)
	@c $$
	@c \afterdisplay
	@c @end tex
	@c @ifinfo
	@c @example
	@c \varphi(2^@{j-1@},t) = \sum_@{k=0@}^@{2^j-1@} h_j(k) * \bar@{\varphi@}(2^j t-k)
	@c @end example
	@c @end ifinfo
	@c @noindent
	@c The functions @math{\psi} and @math{\varphi} are related through the
	@c coefficients
	@c @c{$g_{n} = (-1)^n h_{L-1-n}$}
	@c @math{g_@{n@} = (-1)^n h_@{L-1-n@}}
	@c for @c{$n=0 \dots L-1$}
	@c @math{n=0 ... L-1},
	@c where @math{L} is the total number of coefficients. The two sets of
	@c coefficients @math{h_j} and @math{g_i} define the scaling function and
	@c the wavelet.
	The resulting family of functions @c{$\{\psi_{j,n}\}$}
	@math{@{\psi_@{j,n@}@}}
	constitutes an orthonormal
	basis for square-integrable signals.

	The discrete wavelet transform is an @math{O(N)} algorithm, and is also
	referred to as the @dfn{fast wavelet transform}.

	@node DWT Initialization
	@section Initialization
	@cindex DWT initialization

	The @code{gsl_wavelet} structure contains the filter coefficients
	defining the wavelet and any associated offset parameters.

	@deftypefun {gsl_wavelet } gsl_wavelet_alloc (const gsl_wavelet_type @var{T}, size_t @var{k})
	This function allocates and initializes a wavelet object of type
	@var{T}. The parameter @var{k} selects the specific member of the
	wavelet family. A null pointer is returned if insufficient memory is
	available or if a unsupported member is selected.
	@end deftypefun

	The following wavelet types are implemented:

	@deffn {Wavelet} gsl_wavelet_daubechies
	@deffnx {Wavelet} gsl_wavelet_daubechies_centered
	@cindex Daubechies wavelets
	@cindex maximal phase, Daubechies wavelets
	The is the Daubechies wavelet family of maximum phase with @math{k/2}
	vanishing moments. The implemented wavelets are
	@c{$k=4, 6, \dots, 20$}
	@math{k=4, 6, @dots{}, 20}, with @var{k} even.
	@end deffn

	@deffn {Wavelet} gsl_wavelet_haar
	@deffnx {Wavelet} gsl_wavelet_haar_centered
	@cindex Haar wavelets
	This is the Haar wavelet. The only valid choice of @math{k} for the
	Haar wavelet is @math{k=2}.
	@end deffn

	@deffn {Wavelet} gsl_wavelet_bspline
	@deffnx {Wavelet} gsl_wavelet_bspline_centered
	@cindex biorthogonal wavelets
	@cindex B-spline wavelets
	This is the biorthogonal B-spline wavelet family of order @math{(i,j)}.
	The implemented values of @math{k = 100*i + j} are 103, 105, 202, 204,
	206, 208, 301, 303, 305 307, 309.
	@end deffn

	@noindent
	The centered forms of the wavelets align the coefficients of the various
	sub-bands on edges. Thus the resulting visualization of the
	coefficients of the wavelet transform in the phase plane is easier to
	understand.

	@deftypefun {const char } gsl_wavelet_name (const gsl_wavelet @var{w})
	This function returns a pointer to the name of the wavelet family for
	@var{w}.
	@end deftypefun

	@c @deftypefun {void} gsl_wavelet_print (const gsl_wavelet * @var{w})
	@c This function prints the filter coefficients (@code{h1}, @code{g1}, @code{h2}, @code{g2}) of the wavelet object @var{w}.
	@c @end deftypefun

	@deftypefun {void} gsl_wavelet_free (gsl_wavelet * @var{w})
	This function frees the wavelet object @var{w}.
	@end deftypefun

	The @code{gsl_wavelet_workspace} structure contains scratch space of the
	same size as the input data and is used to hold intermediate results
	during the transform.

	@deftypefun {gsl_wavelet_workspace *} gsl_wavelet_workspace_alloc (size_t @var{n})
	This function allocates a workspace for the discrete wavelet transform.
	To perform a one-dimensional transform on @var{n} elements, a workspace
	of size @var{n} must be provided. For two-dimensional transforms of
	@var{n}-by-@var{n} matrices it is sufficient to allocate a workspace of
	size @var{n}, since the transform operates on individual rows and
	columns.
	@end deftypefun

	@deftypefun {void} gsl_wavelet_workspace_free (gsl_wavelet_workspace * @var{work})
	This function frees the allocated workspace @var{work}.
	@end deftypefun

	@node DWT Transform Functions
	@section Transform Functions

	This sections describes the actual functions performing the discrete
	wavelet transform. Note that the transforms use periodic boundary
	conditions. If the signal is not periodic in the sample length then
	spurious coefficients will appear at the beginning and end of each level
	of the transform.

	@menu
	* DWT in one dimension::
	* DWT in two dimension::
	@end menu

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

	@deftypefun int gsl_wavelet_transform (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{stride}, size_t @var{n}, gsl_wavelet_direction @var{dir}, gsl_wavelet_workspace * @var{work})
	@deftypefunx int gsl_wavelet_transform_forward (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{stride}, size_t @var{n}, gsl_wavelet_workspace * @var{work})
	@deftypefunx int gsl_wavelet_transform_inverse (const gsl_wavelet * @var{w}, double * @var{data}, size_t @var{stride}, size_t @var{n}, gsl_wavelet_workspace * @var{work})

	These functions compute in-place forward and inverse discrete wavelet
	transforms of length @var{n} with stride @var{stride} on the array
	@var{data}. The length of the transform @var{n} is 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 length @var{n} must be provided.

	For the forward transform, the elements of the original array are
	replaced by the discrete wavelet
	transform @c{$f_i \rightarrow w_{j,k}$}
	@math{f_i -> w_@{j,k@}}
	in a packed triangular storage layout,
	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