laywerrobot/lib/python3.6/site-packages/scipy/fftpack/realtransforms.py

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2020-08-27 21:55:39 +02:00
"""
Real spectrum transforms (DCT, DST, MDCT)
"""
from __future__ import division, print_function, absolute_import
__all__ = ['dct', 'idct', 'dst', 'idst', 'dctn', 'idctn', 'dstn', 'idstn']
import numpy as np
from scipy.fftpack import _fftpack
from scipy.fftpack.basic import _datacopied, _fix_shape, _asfarray
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 _init_nd_shape_and_axes(x, shape, axes):
"""Handle shape and axes arguments for dctn, idctn, dstn, idstn."""
if shape is None:
if axes is None:
shape = x.shape
else:
shape = np.take(x.shape, axes)
shape = tuple(shape)
for dim in shape:
if dim < 1:
raise ValueError("Invalid number of DCT data points "
"(%s) specified." % (shape,))
if axes is None:
axes = list(range(-x.ndim, 0))
elif np.isscalar(axes):
axes = [axes, ]
if len(axes) != len(shape):
raise ValueError("when given, axes and shape arguments "
"have to be of the same length")
if len(np.unique(axes)) != len(axes):
raise ValueError("All axes must be unique.")
return shape, axes
def dctn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False):
"""
Return multidimensional Discrete Cosine Transform along the specified axes.
Parameters
----------
x : array_like
The input array.
type : {1, 2, 3}, optional
Type of the DCT (see Notes). Default type is 2.
shape : tuple of ints, optional
The shape of the result. If both `shape` and `axes` (see below) are
None, `shape` is ``x.shape``; if `shape` is None but `axes` is
not None, then `shape` is ``scipy.take(x.shape, axes, axis=0)``.
If ``shape[i] > x.shape[i]``, the i-th dimension is padded with zeros.
If ``shape[i] < x.shape[i]``, the i-th dimension is truncated to
length ``shape[i]``.
axes : tuple or None, optional
Axes along which the DCT is computed; the default is over all axes.
norm : {None, 'ortho'}, optional
Normalization mode (see Notes). Default is None.
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
Returns
-------
y : ndarray of real
The transformed input array.
See Also
--------
idctn : Inverse multidimensional DCT
Notes
-----
For full details of the DCT types and normalization modes, as well as
references, see `dct`.
Examples
--------
>>> from scipy.fftpack import dctn, idctn
>>> y = np.random.randn(16, 16)
>>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho'))
True
"""
x = np.asanyarray(x)
shape, axes = _init_nd_shape_and_axes(x, shape, axes)
for n, ax in zip(shape, axes):
x = dct(x, type=type, n=n, axis=ax, norm=norm, overwrite_x=overwrite_x)
return x
def idctn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False):
"""
Return multidimensional Discrete Cosine Transform along the specified axes.
Parameters
----------
x : array_like
The input array.
type : {1, 2, 3}, optional
Type of the DCT (see Notes). Default type is 2.
shape : tuple of ints, optional
The shape of the result. If both `shape` and `axes` (see below) are
None, `shape` is ``x.shape``; if `shape` is None but `axes` is
not None, then `shape` is ``scipy.take(x.shape, axes, axis=0)``.
If ``shape[i] > x.shape[i]``, the i-th dimension is padded with zeros.
If ``shape[i] < x.shape[i]``, the i-th dimension is truncated to
length ``shape[i]``.
axes : tuple or None, optional
Axes along which the IDCT is computed; the default is over all axes.
norm : {None, 'ortho'}, optional
Normalization mode (see Notes). Default is None.
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
Returns
-------
y : ndarray of real
The transformed input array.
See Also
--------
dctn : multidimensional DCT
Notes
-----
For full details of the IDCT types and normalization modes, as well as
references, see `idct`.
Examples
--------
>>> from scipy.fftpack import dctn, idctn
>>> y = np.random.randn(16, 16)
>>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho'))
True
"""
x = np.asanyarray(x)
shape, axes = _init_nd_shape_and_axes(x, shape, axes)
for n, ax in zip(shape, axes):
x = idct(x, type=type, n=n, axis=ax, norm=norm,
overwrite_x=overwrite_x)
return x
def dstn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False):
"""
Return multidimensional Discrete Sine Transform along the specified axes.
Parameters
----------
x : array_like
The input array.
type : {1, 2, 3}, optional
Type of the DCT (see Notes). Default type is 2.
shape : tuple of ints, optional
The shape of the result. If both `shape` and `axes` (see below) are
None, `shape` is ``x.shape``; if `shape` is None but `axes` is
not None, then `shape` is ``scipy.take(x.shape, axes, axis=0)``.
If ``shape[i] > x.shape[i]``, the i-th dimension is padded with zeros.
If ``shape[i] < x.shape[i]``, the i-th dimension is truncated to
length ``shape[i]``.
axes : tuple or None, optional
Axes along which the DCT is computed; the default is over all axes.
norm : {None, 'ortho'}, optional
Normalization mode (see Notes). Default is None.
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
Returns
-------
y : ndarray of real
The transformed input array.
See Also
--------
idstn : Inverse multidimensional DST
Notes
-----
For full details of the DST types and normalization modes, as well as
references, see `dst`.
Examples
--------
>>> from scipy.fftpack import dstn, idstn
>>> y = np.random.randn(16, 16)
>>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho'))
True
"""
x = np.asanyarray(x)
shape, axes = _init_nd_shape_and_axes(x, shape, axes)
for n, ax in zip(shape, axes):
x = dst(x, type=type, n=n, axis=ax, norm=norm, overwrite_x=overwrite_x)
return x
def idstn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False):
"""
Return multidimensional Discrete Sine Transform along the specified axes.
Parameters
----------
x : array_like
The input array.
type : {1, 2, 3}, optional
Type of the DCT (see Notes). Default type is 2.
shape : tuple of ints, optional
The shape of the result. If both `shape` and `axes` (see below) are
None, `shape` is ``x.shape``; if `shape` is None but `axes` is
not None, then `shape` is ``scipy.take(x.shape, axes, axis=0)``.
If ``shape[i] > x.shape[i]``, the i-th dimension is padded with zeros.
If ``shape[i] < x.shape[i]``, the i-th dimension is truncated to
length ``shape[i]``.
axes : tuple or None, optional
Axes along which the IDCT is computed; the default is over all axes.
norm : {None, 'ortho'}, optional
Normalization mode (see Notes). Default is None.
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
Returns
-------
y : ndarray of real
The transformed input array.
See Also
--------
dctn : multidimensional DST
Notes
-----
For full details of the IDST types and normalization modes, as well as
references, see `idst`.
Examples
--------
>>> from scipy.fftpack import dstn, idstn
>>> y = np.random.randn(16, 16)
>>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho'))
True
"""
x = np.asanyarray(x)
shape, axes = _init_nd_shape_and_axes(x, shape, axes)
for n, ax in zip(shape, axes):
x = idst(x, type=type, n=n, axis=ax, norm=norm,
overwrite_x=overwrite_x)
return x
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. If ``n < x.shape[axis]``, `x` is
truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The
default results in ``n = x.shape[axis]``.
axis : int, optional
Axis along which the dct is computed; the default is over the
last axis (i.e., ``axis=-1``).
norm : {None, 'ortho'}, optional
Normalization mode (see Notes). Default is None.
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is 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:
>>> from scipy.fftpack import fft, dct
>>> fft(np.array([4., 3., 5., 10., 5., 3.])).real
array([ 30., -8., 6., -2., 6., -8.])
>>> dct(np.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. If ``n < x.shape[axis]``, `x` is
truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The
default results in ``n = x.shape[axis]``.
axis : int, optional
Axis along which the idct is computed; the default is over the
last axis (i.e., ``axis=-1``).
norm : {None, 'ortho'}, optional
Normalization mode (see Notes). Default is None.
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is 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:
>>> from scipy.fftpack import ifft, idct
>>> ifft(np.array([ 30., -8., 6., -2., 6., -8.])).real
array([ 4., 3., 5., 10., 5., 3.])
>>> idct(np.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 _get_dct_fun(type, dtype):
try:
name = {'float64':'ddct%d', 'float32':'dct%d'}[dtype.name]
except KeyError:
raise ValueError("dtype %s not supported" % dtype)
try:
f = getattr(_fftpack, name % type)
except AttributeError as e:
raise ValueError(str(e) + ". Type %d not understood" % type)
return f
def _get_norm_mode(normalize):
try:
nm = {None:0, 'ortho':1}[normalize]
except KeyError:
raise ValueError("Unknown normalize mode %s" % normalize)
return nm
def __fix_shape(x, n, axis, dct_or_dst):
tmp = _asfarray(x)
copy_made = _datacopied(tmp, x)
if n is None:
n = tmp.shape[axis]
elif n != tmp.shape[axis]:
tmp, copy_made2 = _fix_shape(tmp, n, axis)
copy_made = copy_made or copy_made2
if n < 1:
raise ValueError("Invalid number of %s data points "
"(%d) specified." % (dct_or_dst, n))
return tmp, n, copy_made
def _raw_dct(x0, type, n, axis, nm, overwrite_x):
f = _get_dct_fun(type, x0.dtype)
return _eval_fun(f, x0, n, axis, nm, overwrite_x)
def _raw_dst(x0, type, n, axis, nm, overwrite_x):
f = _get_dst_fun(type, x0.dtype)
return _eval_fun(f, x0, n, axis, nm, overwrite_x)
def _eval_fun(f, tmp, n, axis, nm, overwrite_x):
if axis == -1 or axis == len(tmp.shape) - 1:
return f(tmp, n, nm, overwrite_x)
tmp = np.swapaxes(tmp, axis, -1)
tmp = f(tmp, n, nm, overwrite_x)
return np.swapaxes(tmp, axis, -1)
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. If ``n < x.shape[axis]``, `x` is
truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The
default results in ``n = x.shape[axis]``.
axis : int, optional
Axis along which the dct is computed; the default is over the
last axis (i.e., ``axis=-1``).
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is False.
Returns
-------
z : ndarray
"""
x0, n, copy_made = __fix_shape(x, n, axis, 'DCT')
if type == 1 and n < 2:
raise ValueError("DCT-I is not defined for size < 2")
overwrite_x = overwrite_x or copy_made
nm = _get_norm_mode(normalize)
if np.iscomplexobj(x0):
return (_raw_dct(x0.real, type, n, axis, nm, overwrite_x) + 1j *
_raw_dct(x0.imag, type, n, axis, nm, overwrite_x))
else:
return _raw_dct(x0, type, n, axis, nm, overwrite_x)
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. If ``n < x.shape[axis]``, `x` is
truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The
default results in ``n = x.shape[axis]``.
axis : int, optional
Axis along which the dst is computed; the default is over the
last axis (i.e., ``axis=-1``).
norm : {None, 'ortho'}, optional
Normalization mode (see Notes). Default is None.
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is 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. If ``n < x.shape[axis]``, `x` is
truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The
default results in ``n = x.shape[axis]``.
axis : int, optional
Axis along which the idst is computed; the default is over the
last axis (i.e., ``axis=-1``).
norm : {None, 'ortho'}, optional
Normalization mode (see Notes). Default is None.
overwrite_x : bool, optional
If True, the contents of `x` can be destroyed; the default is 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 _get_dst_fun(type, dtype):
try:
name = {'float64':'ddst%d', 'float32':'dst%d'}[dtype.name]
except KeyError:
raise ValueError("dtype %s not supported" % dtype)
try:
f = getattr(_fftpack, name % type)
except AttributeError as e:
raise ValueError(str(e) + ". Type %d not understood" % type)
return f
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
"""
x0, n, copy_made = __fix_shape(x, n, axis, 'DST')
if type == 1 and n < 2:
raise ValueError("DST-I is not defined for size < 2")
overwrite_x = overwrite_x or copy_made
nm = _get_norm_mode(normalize)
if np.iscomplexobj(x0):
return (_raw_dst(x0.real, type, n, axis, nm, overwrite_x) + 1j *
_raw_dst(x0.imag, type, n, axis, nm, overwrite_x))
else:
return _raw_dst(x0, type, n, axis, nm, overwrite_x)