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realtransforms.py
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realtransforms.py
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"""
Real spectrum tranforms (DCT, DST, MDCT)
"""
from __future__ import division, print_function, absolute_import
__all__ = ['dct', 'idct', 'dst', 'idst']
import numpy as np
from scipy.fftpack import _fftpack
from scipy.fftpack.basic import _datacopied
import atexit
atexit.register(_fftpack.destroy_ddct1_cache)
atexit.register(_fftpack.destroy_ddct2_cache)
atexit.register(_fftpack.destroy_dct1_cache)
atexit.register(_fftpack.destroy_dct2_cache)
atexit.register(_fftpack.destroy_ddst1_cache)
atexit.register(_fftpack.destroy_ddst2_cache)
atexit.register(_fftpack.destroy_dst1_cache)
atexit.register(_fftpack.destroy_dst2_cache)
def dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False):
"""
Return the Discrete Cosine Transform of arbitrary type sequence x.
Parameters
----------
x : array_like
The input array.
type : {1, 2, 3}, optional
Type of the DCT (see Notes). Default type is 2.
n : int, optional
Length of the transform.
axis : int, optional
Axis over which to compute the transform.
norm : {None, 'ortho'}, optional
Normalization mode (see Notes). Default is None.
overwrite_x : bool, optional
If True the contents of x can be destroyed. (default=False)
Returns
-------
y : ndarray of real
The transformed input array.
See Also
--------
idct : Inverse DCT
Notes
-----
For a single dimension array ``x``, ``dct(x, norm='ortho')`` is equal to
MATLAB ``dct(x)``.
There are theoretically 8 types of the DCT, only the first 3 types are
implemented in scipy. 'The' DCT generally refers to DCT type 2, and 'the'
Inverse DCT generally refers to DCT type 3.
**Type I**
There are several definitions of the DCT-I; we use the following
(for ``norm=None``)::
N-2
y[k] = x[0] + (-1)**k x[N-1] + 2 * sum x[n]*cos(pi*k*n/(N-1))
n=1
Only None is supported as normalization mode for DCT-I. Note also that the
DCT-I is only supported for input size > 1
**Type II**
There are several definitions of the DCT-II; we use the following
(for ``norm=None``)::
N-1
y[k] = 2* sum x[n]*cos(pi*k*(2n+1)/(2*N)), 0 <= k < N.
n=0
If ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor `f`::
f = sqrt(1/(4*N)) if k = 0,
f = sqrt(1/(2*N)) otherwise.
Which makes the corresponding matrix of coefficients orthonormal
(``OO' = Id``).
**Type III**
There are several definitions, we use the following
(for ``norm=None``)::
N-1
y[k] = x[0] + 2 * sum x[n]*cos(pi*(k+0.5)*n/N), 0 <= k < N.
n=1
or, for ``norm='ortho'`` and 0 <= k < N::
N-1
y[k] = x[0] / sqrt(N) + sqrt(2/N) * sum x[n]*cos(pi*(k+0.5)*n/N)
n=1
The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up
to a factor `2N`. The orthonormalized DCT-III is exactly the inverse of
the orthonormalized DCT-II.
References
----------
.. [1] 'A Fast Cosine Transform in One and Two Dimensions', by J.
Makhoul, `IEEE Transactions on acoustics, speech and signal
processing` vol. 28(1), pp. 27-34,
http://dx.doi.org/10.1109/TASSP.1980.1163351 (1980).
.. [2] Wikipedia, "Discrete cosine transform",
http://en.wikipedia.org/wiki/Discrete_cosine_transform
Examples
--------
The Type 1 DCT is equivalent to the FFT (though faster) for real,
even-symmetrical inputs. The output is also real and even-symmetrical.
Half of the FFT input is used to generate half of the FFT output:
>>> fft(array([4., 3., 5., 10., 5., 3.])).real
array([ 30., -8., 6., -2., 6., -8.])
>>> dct(array([4., 3., 5., 10.]), 1)
array([ 30., -8., 6., -2.])
"""
if type == 1 and norm is not None:
raise NotImplementedError(
"Orthonormalization not yet supported for DCT-I")
return _dct(x, type, n, axis, normalize=norm, overwrite_x=overwrite_x)
def idct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False):
"""
Return the Inverse Discrete Cosine Transform of an arbitrary type sequence.
Parameters
----------
x : array_like
The input array.
type : {1, 2, 3}, optional
Type of the DCT (see Notes). Default type is 2.
n : int, optional
Length of the transform.
axis : int, optional
Axis over which to compute the transform.
norm : {None, 'ortho'}, optional
Normalization mode (see Notes). Default is None.
overwrite_x : bool, optional
If True the contents of x can be destroyed. (default=False)
Returns
-------
idct : ndarray of real
The transformed input array.
See Also
--------
dct : Forward DCT
Notes
-----
For a single dimension array `x`, ``idct(x, norm='ortho')`` is equal to
MATLAB ``idct(x)``.
'The' IDCT is the IDCT of type 2, which is the same as DCT of type 3.
IDCT of type 1 is the DCT of type 1, IDCT of type 2 is the DCT of type
3, and IDCT of type 3 is the DCT of type 2. For the definition of these
types, see `dct`.
Examples
--------
The Type 1 DCT is equivalent to the DFT for real, even-symmetrical
inputs. The output is also real and even-symmetrical. Half of the IFFT
input is used to generate half of the IFFT output:
>>> ifft(array([ 30., -8., 6., -2., 6., -8.])).real
array([ 4., 3., 5., 10., 5., 3.])
>>> idct(array([ 30., -8., 6., -2.]), 1) / 6
array([ 4., 3., 5., 10.])
"""
if type == 1 and norm is not None:
raise NotImplementedError(
"Orthonormalization not yet supported for IDCT-I")
# Inverse/forward type table
_TP = {1:1, 2:3, 3:2}
return _dct(x, _TP[type], n, axis, normalize=norm, overwrite_x=overwrite_x)
def _dct(x, type, n=None, axis=-1, overwrite_x=False, normalize=None):
"""
Return Discrete Cosine Transform of arbitrary type sequence x.
Parameters
----------
x : array-like
input array.
n : int, optional
Length of the transform.
axis : int, optional
Axis along which the dct is computed. (default=-1)
overwrite_x : bool, optional
If True the contents of x can be destroyed. (default=False)
Returns
-------
z : real ndarray
"""
tmp = np.asarray(x)
if not np.isrealobj(tmp):
raise TypeError("1st argument must be real sequence")
if n is None:
n = tmp.shape[axis]
else:
raise NotImplementedError("Padding/truncating not yet implemented")
if tmp.dtype == np.double:
if type == 1:
f = _fftpack.ddct1
elif type == 2:
f = _fftpack.ddct2
elif type == 3:
f = _fftpack.ddct3
else:
raise ValueError("Type %d not understood" % type)
elif tmp.dtype == np.float32:
if type == 1:
f = _fftpack.dct1
elif type == 2:
f = _fftpack.dct2
elif type == 3:
f = _fftpack.dct3
else:
raise ValueError("Type %d not understood" % type)
else:
raise ValueError("dtype %s not supported" % tmp.dtype)
if normalize:
if normalize == "ortho":
nm = 1
else:
raise ValueError("Unknown normalize mode %s" % normalize)
else:
nm = 0
if type == 1 and n < 2:
raise ValueError("DCT-I is not defined for size < 2")
overwrite_x = overwrite_x or _datacopied(tmp, x)
if axis == -1 or axis == len(tmp.shape) - 1:
return f(tmp, n, nm, overwrite_x)
#else:
# raise NotImplementedError("Axis arg not yet implemented")
tmp = np.swapaxes(tmp, axis, -1)
tmp = f(tmp, n, nm, overwrite_x)
return np.swapaxes(tmp, axis, -1)
###########
def dst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False):
"""
Return the Discrete Sine Transform of arbitrary type sequence x.
Parameters
----------
x : array_like
The input array.
type : {1, 2, 3}, optional
Type of the DST (see Notes). Default type is 2.
n : int, optional
Length of the transform.
axis : int, optional
Axis over which to compute the transform.
norm : {None, 'ortho'}, optional
Normalization mode (see Notes). Default is None.
overwrite_x : bool, optional
If True the contents of x can be destroyed. (default=False)
Returns
-------
dst : ndarray of reals
The transformed input array.
See Also
--------
idst : Inverse DST
Notes
-----
For a single dimension array ``x``.
There are theoretically 8 types of the DST for different combinations of
even/odd boundary conditions and boundary off sets [1]_, only the first
3 types are implemented in scipy.
**Type I**
There are several definitions of the DST-I; we use the following
for ``norm=None``. DST-I assumes the input is odd around n=-1 and n=N. ::
N-1
y[k] = 2 * sum x[n]*sin(pi*(k+1)*(n+1)/(N+1))
n=0
Only None is supported as normalization mode for DCT-I. Note also that the
DCT-I is only supported for input size > 1
The (unnormalized) DCT-I is its own inverse, up to a factor `2(N+1)`.
**Type II**
There are several definitions of the DST-II; we use the following
for ``norm=None``. DST-II assumes the input is odd around n=-1/2 and
n=N-1/2; the output is odd around k=-1 and even around k=N-1 ::
N-1
y[k] = 2* sum x[n]*sin(pi*(k+1)*(n+0.5)/N), 0 <= k < N.
n=0
if ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor `f` ::
f = sqrt(1/(4*N)) if k == 0
f = sqrt(1/(2*N)) otherwise.
**Type III**
There are several definitions of the DST-III, we use the following
(for ``norm=None``). DST-III assumes the input is odd around n=-1
and even around n=N-1 ::
N-2
y[k] = x[N-1]*(-1)**k + 2* sum x[n]*sin(pi*(k+0.5)*(n+1)/N), 0 <= k < N.
n=0
The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up
to a factor `2N`. The orthonormalized DST-III is exactly the inverse of
the orthonormalized DST-II.
.. versionadded:: 0.11.0
References
----------
.. [1] Wikipedia, "Discrete sine transform",
http://en.wikipedia.org/wiki/Discrete_sine_transform
"""
if type == 1 and norm is not None:
raise NotImplementedError(
"Orthonormalization not yet supported for IDCT-I")
return _dst(x, type, n, axis, normalize=norm, overwrite_x=overwrite_x)
def idst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False):
"""
Return the Inverse Discrete Sine Transform of an arbitrary type sequence.
Parameters
----------
x : array_like
The input array.
type : {1, 2, 3}, optional
Type of the DST (see Notes). Default type is 2.
n : int, optional
Length of the transform.
axis : int, optional
Axis over which to compute the transform.
norm : {None, 'ortho'}, optional
Normalization mode (see Notes). Default is None.
overwrite_x : bool, optional
If True the contents of x can be destroyed. (default=False)
Returns
-------
idst : ndarray of real
The transformed input array.
See Also
--------
dst : Forward DST
Notes
-----
'The' IDST is the IDST of type 2, which is the same as DST of type 3.
IDST of type 1 is the DST of type 1, IDST of type 2 is the DST of type
3, and IDST of type 3 is the DST of type 2. For the definition of these
types, see `dst`.
.. versionadded:: 0.11.0
"""
if type == 1 and norm is not None:
raise NotImplementedError(
"Orthonormalization not yet supported for IDCT-I")
# Inverse/forward type table
_TP = {1:1, 2:3, 3:2}
return _dst(x, _TP[type], n, axis, normalize=norm, overwrite_x=overwrite_x)
def _dst(x, type, n=None, axis=-1, overwrite_x=False, normalize=None):
"""
Return Discrete Sine Transform of arbitrary type sequence x.
Parameters
----------
x : array-like
input array.
n : int, optional
Length of the transform.
axis : int, optional
Axis along which the dst is computed. (default=-1)
overwrite_x : bool, optional
If True the contents of x can be destroyed. (default=False)
Returns
-------
z : real ndarray
"""
tmp = np.asarray(x)
if not np.isrealobj(tmp):
raise TypeError("1st argument must be real sequence")
if n is None:
n = tmp.shape[axis]
else:
raise NotImplementedError("Padding/truncating not yet implemented")
if tmp.dtype == np.double:
if type == 1:
f = _fftpack.ddst1
elif type == 2:
f = _fftpack.ddst2
elif type == 3:
f = _fftpack.ddst3
else:
raise ValueError("Type %d not understood" % type)
elif tmp.dtype == np.float32:
if type == 1:
f = _fftpack.dst1
elif type == 2:
f = _fftpack.dst2
elif type == 3:
f = _fftpack.dst3
else:
raise ValueError("Type %d not understood" % type)
else:
raise ValueError("dtype %s not supported" % tmp.dtype)
if normalize:
if normalize == "ortho":
nm = 1
else:
raise ValueError("Unknown normalize mode %s" % normalize)
else:
nm = 0
if type == 1 and n < 2:
raise ValueError("DST-I is not defined for size < 2")
overwrite_x = overwrite_x or _datacopied(tmp, x)
if axis == -1 or axis == len(tmp.shape) - 1:
return f(tmp, n, nm, overwrite_x)
#else:
# raise NotImplementedError("Axis arg not yet implemented")
tmp = np.swapaxes(tmp, axis, -1)
tmp = f(tmp, n, nm, overwrite_x)
return np.swapaxes(tmp, axis, -1)