694 lines
25 KiB
Python
694 lines
25 KiB
Python
![]() |
from ._basic import _dispatch
|
||
|
from scipy._lib.uarray import Dispatchable
|
||
|
import numpy as np
|
||
|
|
||
|
__all__ = ['dct', 'idct', 'dst', 'idst', 'dctn', 'idctn', 'dstn', 'idstn']
|
||
|
|
||
|
|
||
|
@_dispatch
|
||
|
def dctn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False,
|
||
|
workers=None, *, orthogonalize=None):
|
||
|
"""
|
||
|
Return multidimensional Discrete Cosine Transform along the specified axes.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
x : array_like
|
||
|
The input array.
|
||
|
type : {1, 2, 3, 4}, optional
|
||
|
Type of the DCT (see Notes). Default type is 2.
|
||
|
s : int or array_like of ints or None, optional
|
||
|
The shape of the result. If both `s` and `axes` (see below) are None,
|
||
|
`s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is
|
||
|
``numpy.take(x.shape, axes, axis=0)``.
|
||
|
If ``s[i] > x.shape[i]``, the ith dimension is padded with zeros.
|
||
|
If ``s[i] < x.shape[i]``, the ith dimension is truncated to length
|
||
|
``s[i]``.
|
||
|
If any element of `s` is -1, the size of the corresponding dimension of
|
||
|
`x` is used.
|
||
|
axes : int or array_like of ints or None, optional
|
||
|
Axes over which the DCT is computed. If not given, the last ``len(s)``
|
||
|
axes are used, or all axes if `s` is also not specified.
|
||
|
norm : {"backward", "ortho", "forward"}, optional
|
||
|
Normalization mode (see Notes). Default is "backward".
|
||
|
overwrite_x : bool, optional
|
||
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
workers : int, optional
|
||
|
Maximum number of workers to use for parallel computation. If negative,
|
||
|
the value wraps around from ``os.cpu_count()``.
|
||
|
See :func:`~scipy.fft.fft` for more details.
|
||
|
orthogonalize : bool, optional
|
||
|
Whether to use the orthogonalized DCT variant (see Notes).
|
||
|
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
|
||
|
|
||
|
.. versionadded:: 1.8.0
|
||
|
|
||
|
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
|
||
|
--------
|
||
|
>>> import numpy as np
|
||
|
>>> from scipy.fft import dctn, idctn
|
||
|
>>> rng = np.random.default_rng()
|
||
|
>>> y = rng.standard_normal((16, 16))
|
||
|
>>> np.allclose(y, idctn(dctn(y)))
|
||
|
True
|
||
|
|
||
|
"""
|
||
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
||
|
|
||
|
@_dispatch
|
||
|
def idctn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False,
|
||
|
workers=None, orthogonalize=None):
|
||
|
"""
|
||
|
Return multidimensional Inverse Discrete Cosine Transform along the specified axes.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
x : array_like
|
||
|
The input array.
|
||
|
type : {1, 2, 3, 4}, optional
|
||
|
Type of the DCT (see Notes). Default type is 2.
|
||
|
s : int or array_like of ints or None, optional
|
||
|
The shape of the result. If both `s` and `axes` (see below) are
|
||
|
None, `s` is ``x.shape``; if `s` is None but `axes` is
|
||
|
not None, then `s` is ``numpy.take(x.shape, axes, axis=0)``.
|
||
|
If ``s[i] > x.shape[i]``, the ith dimension is padded with zeros.
|
||
|
If ``s[i] < x.shape[i]``, the ith dimension is truncated to length
|
||
|
``s[i]``.
|
||
|
If any element of `s` is -1, the size of the corresponding dimension of
|
||
|
`x` is used.
|
||
|
axes : int or array_like of ints or None, optional
|
||
|
Axes over which the IDCT is computed. If not given, the last ``len(s)``
|
||
|
axes are used, or all axes if `s` is also not specified.
|
||
|
norm : {"backward", "ortho", "forward"}, optional
|
||
|
Normalization mode (see Notes). Default is "backward".
|
||
|
overwrite_x : bool, optional
|
||
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
workers : int, optional
|
||
|
Maximum number of workers to use for parallel computation. If negative,
|
||
|
the value wraps around from ``os.cpu_count()``.
|
||
|
See :func:`~scipy.fft.fft` for more details.
|
||
|
orthogonalize : bool, optional
|
||
|
Whether to use the orthogonalized IDCT variant (see Notes).
|
||
|
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
|
||
|
|
||
|
.. versionadded:: 1.8.0
|
||
|
|
||
|
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
|
||
|
--------
|
||
|
>>> import numpy as np
|
||
|
>>> from scipy.fft import dctn, idctn
|
||
|
>>> rng = np.random.default_rng()
|
||
|
>>> y = rng.standard_normal((16, 16))
|
||
|
>>> np.allclose(y, idctn(dctn(y)))
|
||
|
True
|
||
|
|
||
|
"""
|
||
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
||
|
|
||
|
@_dispatch
|
||
|
def dstn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False,
|
||
|
workers=None, orthogonalize=None):
|
||
|
"""
|
||
|
Return multidimensional Discrete Sine Transform along the specified axes.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
x : array_like
|
||
|
The input array.
|
||
|
type : {1, 2, 3, 4}, optional
|
||
|
Type of the DST (see Notes). Default type is 2.
|
||
|
s : int or array_like of ints or None, optional
|
||
|
The shape of the result. If both `s` and `axes` (see below) are None,
|
||
|
`s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is
|
||
|
``numpy.take(x.shape, axes, axis=0)``.
|
||
|
If ``s[i] > x.shape[i]``, the ith dimension is padded with zeros.
|
||
|
If ``s[i] < x.shape[i]``, the ith dimension is truncated to length
|
||
|
``s[i]``.
|
||
|
If any element of `shape` is -1, the size of the corresponding dimension
|
||
|
of `x` is used.
|
||
|
axes : int or array_like of ints or None, optional
|
||
|
Axes over which the DST is computed. If not given, the last ``len(s)``
|
||
|
axes are used, or all axes if `s` is also not specified.
|
||
|
norm : {"backward", "ortho", "forward"}, optional
|
||
|
Normalization mode (see Notes). Default is "backward".
|
||
|
overwrite_x : bool, optional
|
||
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
workers : int, optional
|
||
|
Maximum number of workers to use for parallel computation. If negative,
|
||
|
the value wraps around from ``os.cpu_count()``.
|
||
|
See :func:`~scipy.fft.fft` for more details.
|
||
|
orthogonalize : bool, optional
|
||
|
Whether to use the orthogonalized DST variant (see Notes).
|
||
|
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
|
||
|
|
||
|
.. versionadded:: 1.8.0
|
||
|
|
||
|
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
|
||
|
--------
|
||
|
>>> import numpy as np
|
||
|
>>> from scipy.fft import dstn, idstn
|
||
|
>>> rng = np.random.default_rng()
|
||
|
>>> y = rng.standard_normal((16, 16))
|
||
|
>>> np.allclose(y, idstn(dstn(y)))
|
||
|
True
|
||
|
|
||
|
"""
|
||
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
||
|
|
||
|
@_dispatch
|
||
|
def idstn(x, type=2, s=None, axes=None, norm=None, overwrite_x=False,
|
||
|
workers=None, orthogonalize=None):
|
||
|
"""
|
||
|
Return multidimensional Inverse Discrete Sine Transform along the specified axes.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
x : array_like
|
||
|
The input array.
|
||
|
type : {1, 2, 3, 4}, optional
|
||
|
Type of the DST (see Notes). Default type is 2.
|
||
|
s : int or array_like of ints or None, optional
|
||
|
The shape of the result. If both `s` and `axes` (see below) are None,
|
||
|
`s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is
|
||
|
``numpy.take(x.shape, axes, axis=0)``.
|
||
|
If ``s[i] > x.shape[i]``, the ith dimension is padded with zeros.
|
||
|
If ``s[i] < x.shape[i]``, the ith dimension is truncated to length
|
||
|
``s[i]``.
|
||
|
If any element of `s` is -1, the size of the corresponding dimension of
|
||
|
`x` is used.
|
||
|
axes : int or array_like of ints or None, optional
|
||
|
Axes over which the IDST is computed. If not given, the last ``len(s)``
|
||
|
axes are used, or all axes if `s` is also not specified.
|
||
|
norm : {"backward", "ortho", "forward"}, optional
|
||
|
Normalization mode (see Notes). Default is "backward".
|
||
|
overwrite_x : bool, optional
|
||
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
workers : int, optional
|
||
|
Maximum number of workers to use for parallel computation. If negative,
|
||
|
the value wraps around from ``os.cpu_count()``.
|
||
|
See :func:`~scipy.fft.fft` for more details.
|
||
|
orthogonalize : bool, optional
|
||
|
Whether to use the orthogonalized IDST variant (see Notes).
|
||
|
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
|
||
|
|
||
|
.. versionadded:: 1.8.0
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
y : ndarray of real
|
||
|
The transformed input array.
|
||
|
|
||
|
See Also
|
||
|
--------
|
||
|
dstn : multidimensional DST
|
||
|
|
||
|
Notes
|
||
|
-----
|
||
|
For full details of the IDST types and normalization modes, as well as
|
||
|
references, see `idst`.
|
||
|
|
||
|
Examples
|
||
|
--------
|
||
|
>>> import numpy as np
|
||
|
>>> from scipy.fft import dstn, idstn
|
||
|
>>> rng = np.random.default_rng()
|
||
|
>>> y = rng.standard_normal((16, 16))
|
||
|
>>> np.allclose(y, idstn(dstn(y)))
|
||
|
True
|
||
|
|
||
|
"""
|
||
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
||
|
|
||
|
@_dispatch
|
||
|
def dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, workers=None,
|
||
|
orthogonalize=None):
|
||
|
r"""Return the Discrete Cosine Transform of arbitrary type sequence x.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
x : array_like
|
||
|
The input array.
|
||
|
type : {1, 2, 3, 4}, 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 : {"backward", "ortho", "forward"}, optional
|
||
|
Normalization mode (see Notes). Default is "backward".
|
||
|
overwrite_x : bool, optional
|
||
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
workers : int, optional
|
||
|
Maximum number of workers to use for parallel computation. If negative,
|
||
|
the value wraps around from ``os.cpu_count()``.
|
||
|
See :func:`~scipy.fft.fft` for more details.
|
||
|
orthogonalize : bool, optional
|
||
|
Whether to use the orthogonalized DCT variant (see Notes).
|
||
|
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
|
||
|
|
||
|
.. versionadded:: 1.8.0
|
||
|
|
||
|
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)``.
|
||
|
|
||
|
.. warning:: For ``type in {1, 2, 3}``, ``norm="ortho"`` breaks the direct
|
||
|
correspondence with the direct Fourier transform. To recover
|
||
|
it you must specify ``orthogonalize=False``.
|
||
|
|
||
|
For ``norm="ortho"`` both the `dct` and `idct` are scaled by the same
|
||
|
overall factor in both directions. By default, the transform is also
|
||
|
orthogonalized which for types 1, 2 and 3 means the transform definition is
|
||
|
modified to give orthogonality of the DCT matrix (see below).
|
||
|
|
||
|
For ``norm="backward"``, there is no scaling on `dct` and the `idct` is
|
||
|
scaled by ``1/N`` where ``N`` is the "logical" size of the DCT. For
|
||
|
``norm="forward"`` the ``1/N`` normalization is applied to the forward
|
||
|
`dct` instead and the `idct` is unnormalized.
|
||
|
|
||
|
There are, theoretically, 8 types of the DCT, only the first 4 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="backward"``)
|
||
|
|
||
|
.. math::
|
||
|
|
||
|
y_k = x_0 + (-1)^k x_{N-1} + 2 \sum_{n=1}^{N-2} x_n \cos\left(
|
||
|
\frac{\pi k n}{N-1} \right)
|
||
|
|
||
|
If ``orthogonalize=True``, ``x[0]`` and ``x[N-1]`` are multiplied by a
|
||
|
scaling factor of :math:`\sqrt{2}`, and ``y[0]`` and ``y[N-1]`` are divided
|
||
|
by :math:`\sqrt{2}`. When combined with ``norm="ortho"``, this makes the
|
||
|
corresponding matrix of coefficients orthonormal (``O @ O.T = np.eye(N)``).
|
||
|
|
||
|
.. note::
|
||
|
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="backward"``)
|
||
|
|
||
|
.. math::
|
||
|
|
||
|
y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi k(2n+1)}{2N} \right)
|
||
|
|
||
|
If ``orthogonalize=True``, ``y[0]`` is divided by :math:`\sqrt{2}` which,
|
||
|
when combined with ``norm="ortho"``, makes the corresponding matrix of
|
||
|
coefficients orthonormal (``O @ O.T = np.eye(N)``).
|
||
|
|
||
|
**Type III**
|
||
|
|
||
|
There are several definitions, we use the following (for
|
||
|
``norm="backward"``)
|
||
|
|
||
|
.. math::
|
||
|
|
||
|
y_k = x_0 + 2 \sum_{n=1}^{N-1} x_n \cos\left(\frac{\pi(2k+1)n}{2N}\right)
|
||
|
|
||
|
If ``orthogonalize=True``, ``x[0]`` terms are multiplied by
|
||
|
:math:`\sqrt{2}` which, when combined with ``norm="ortho"``, makes the
|
||
|
corresponding matrix of coefficients orthonormal (``O @ O.T = np.eye(N)``).
|
||
|
|
||
|
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.
|
||
|
|
||
|
**Type IV**
|
||
|
|
||
|
There are several definitions of the DCT-IV; we use the following
|
||
|
(for ``norm="backward"``)
|
||
|
|
||
|
.. math::
|
||
|
|
||
|
y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi(2k+1)(2n+1)}{4N} \right)
|
||
|
|
||
|
``orthogonalize`` has no effect here, as the DCT-IV matrix is already
|
||
|
orthogonal up to a scale factor of ``2N``.
|
||
|
|
||
|
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,
|
||
|
:doi:`10.1109/TASSP.1980.1163351` (1980).
|
||
|
.. [2] Wikipedia, "Discrete cosine transform",
|
||
|
https://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.fft import fft, dct
|
||
|
>>> import numpy as np
|
||
|
>>> 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.])
|
||
|
|
||
|
"""
|
||
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
||
|
|
||
|
@_dispatch
|
||
|
def idct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False,
|
||
|
workers=None, orthogonalize=None):
|
||
|
"""
|
||
|
Return the Inverse Discrete Cosine Transform of an arbitrary type sequence.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
x : array_like
|
||
|
The input array.
|
||
|
type : {1, 2, 3, 4}, 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 : {"backward", "ortho", "forward"}, optional
|
||
|
Normalization mode (see Notes). Default is "backward".
|
||
|
overwrite_x : bool, optional
|
||
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
workers : int, optional
|
||
|
Maximum number of workers to use for parallel computation. If negative,
|
||
|
the value wraps around from ``os.cpu_count()``.
|
||
|
See :func:`~scipy.fft.fft` for more details.
|
||
|
orthogonalize : bool, optional
|
||
|
Whether to use the orthogonalized IDCT variant (see Notes).
|
||
|
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
|
||
|
|
||
|
.. versionadded:: 1.8.0
|
||
|
|
||
|
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)``.
|
||
|
|
||
|
.. warning:: For ``type in {1, 2, 3}``, ``norm="ortho"`` breaks the direct
|
||
|
correspondence with the inverse direct Fourier transform. To
|
||
|
recover it you must specify ``orthogonalize=False``.
|
||
|
|
||
|
For ``norm="ortho"`` both the `dct` and `idct` are scaled by the same
|
||
|
overall factor in both directions. By default, the transform is also
|
||
|
orthogonalized which for types 1, 2 and 3 means the transform definition is
|
||
|
modified to give orthogonality of the IDCT matrix (see `dct` for the full
|
||
|
definitions).
|
||
|
|
||
|
'The' IDCT is the IDCT-II, which is the same as the normalized DCT-III.
|
||
|
|
||
|
The IDCT is equivalent to a normal DCT except for the normalization and
|
||
|
type. DCT type 1 and 4 are their own inverse and DCTs 2 and 3 are each
|
||
|
other's inverses.
|
||
|
|
||
|
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.fft import ifft, idct
|
||
|
>>> import numpy as np
|
||
|
>>> 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)
|
||
|
array([ 4., 3., 5., 10.])
|
||
|
|
||
|
"""
|
||
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
||
|
|
||
|
@_dispatch
|
||
|
def dst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False, workers=None,
|
||
|
orthogonalize=None):
|
||
|
r"""
|
||
|
Return the Discrete Sine Transform of arbitrary type sequence x.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
x : array_like
|
||
|
The input array.
|
||
|
type : {1, 2, 3, 4}, 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 : {"backward", "ortho", "forward"}, optional
|
||
|
Normalization mode (see Notes). Default is "backward".
|
||
|
overwrite_x : bool, optional
|
||
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
workers : int, optional
|
||
|
Maximum number of workers to use for parallel computation. If negative,
|
||
|
the value wraps around from ``os.cpu_count()``.
|
||
|
See :func:`~scipy.fft.fft` for more details.
|
||
|
orthogonalize : bool, optional
|
||
|
Whether to use the orthogonalized DST variant (see Notes).
|
||
|
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
|
||
|
|
||
|
.. versionadded:: 1.8.0
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
dst : ndarray of reals
|
||
|
The transformed input array.
|
||
|
|
||
|
See Also
|
||
|
--------
|
||
|
idst : Inverse DST
|
||
|
|
||
|
Notes
|
||
|
-----
|
||
|
.. warning:: For ``type in {2, 3}``, ``norm="ortho"`` breaks the direct
|
||
|
correspondence with the direct Fourier transform. To recover
|
||
|
it you must specify ``orthogonalize=False``.
|
||
|
|
||
|
For ``norm="ortho"`` both the `dst` and `idst` are scaled by the same
|
||
|
overall factor in both directions. By default, the transform is also
|
||
|
orthogonalized which for types 2 and 3 means the transform definition is
|
||
|
modified to give orthogonality of the DST matrix (see below).
|
||
|
|
||
|
For ``norm="backward"``, there is no scaling on the `dst` and the `idst` is
|
||
|
scaled by ``1/N`` where ``N`` is the "logical" size of the DST.
|
||
|
|
||
|
There are, theoretically, 8 types of the DST for different combinations of
|
||
|
even/odd boundary conditions and boundary off sets [1]_, only the first
|
||
|
4 types are implemented in SciPy.
|
||
|
|
||
|
**Type I**
|
||
|
|
||
|
There are several definitions of the DST-I; we use the following for
|
||
|
``norm="backward"``. DST-I assumes the input is odd around :math:`n=-1` and
|
||
|
:math:`n=N`.
|
||
|
|
||
|
.. math::
|
||
|
|
||
|
y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(k+1)(n+1)}{N+1}\right)
|
||
|
|
||
|
Note that the DST-I is only supported for input size > 1.
|
||
|
The (unnormalized) DST-I is its own inverse, up to a factor :math:`2(N+1)`.
|
||
|
The orthonormalized DST-I is exactly its own inverse.
|
||
|
|
||
|
``orthogonalize`` has no effect here, as the DST-I matrix is already
|
||
|
orthogonal up to a scale factor of ``2N``.
|
||
|
|
||
|
**Type II**
|
||
|
|
||
|
There are several definitions of the DST-II; we use the following for
|
||
|
``norm="backward"``. DST-II assumes the input is odd around :math:`n=-1/2` and
|
||
|
:math:`n=N-1/2`; the output is odd around :math:`k=-1` and even around :math:`k=N-1`
|
||
|
|
||
|
.. math::
|
||
|
|
||
|
y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(k+1)(2n+1)}{2N}\right)
|
||
|
|
||
|
If ``orthogonalize=True``, ``y[0]`` is divided :math:`\sqrt{2}` which, when
|
||
|
combined with ``norm="ortho"``, makes the corresponding matrix of
|
||
|
coefficients orthonormal (``O @ O.T = np.eye(N)``).
|
||
|
|
||
|
**Type III**
|
||
|
|
||
|
There are several definitions of the DST-III, we use the following (for
|
||
|
``norm="backward"``). DST-III assumes the input is odd around :math:`n=-1` and
|
||
|
even around :math:`n=N-1`
|
||
|
|
||
|
.. math::
|
||
|
|
||
|
y_k = (-1)^k x_{N-1} + 2 \sum_{n=0}^{N-2} x_n \sin\left(
|
||
|
\frac{\pi(2k+1)(n+1)}{2N}\right)
|
||
|
|
||
|
If ``orthogonalize=True``, ``x[0]`` is multiplied by :math:`\sqrt{2}`
|
||
|
which, when combined with ``norm="ortho"``, makes the corresponding matrix
|
||
|
of coefficients orthonormal (``O @ O.T = np.eye(N)``).
|
||
|
|
||
|
The (unnormalized) DST-III is the inverse of the (unnormalized) DST-II, up
|
||
|
to a factor :math:`2N`. The orthonormalized DST-III is exactly the inverse of the
|
||
|
orthonormalized DST-II.
|
||
|
|
||
|
**Type IV**
|
||
|
|
||
|
There are several definitions of the DST-IV, we use the following (for
|
||
|
``norm="backward"``). DST-IV assumes the input is odd around :math:`n=-0.5` and
|
||
|
even around :math:`n=N-0.5`
|
||
|
|
||
|
.. math::
|
||
|
|
||
|
y_k = 2 \sum_{n=0}^{N-1} x_n \sin\left(\frac{\pi(2k+1)(2n+1)}{4N}\right)
|
||
|
|
||
|
``orthogonalize`` has no effect here, as the DST-IV matrix is already
|
||
|
orthogonal up to a scale factor of ``2N``.
|
||
|
|
||
|
The (unnormalized) DST-IV is its own inverse, up to a factor :math:`2N`. The
|
||
|
orthonormalized DST-IV is exactly its own inverse.
|
||
|
|
||
|
References
|
||
|
----------
|
||
|
.. [1] Wikipedia, "Discrete sine transform",
|
||
|
https://en.wikipedia.org/wiki/Discrete_sine_transform
|
||
|
|
||
|
"""
|
||
|
return (Dispatchable(x, np.ndarray),)
|
||
|
|
||
|
|
||
|
@_dispatch
|
||
|
def idst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False,
|
||
|
workers=None, orthogonalize=None):
|
||
|
"""
|
||
|
Return the Inverse Discrete Sine Transform of an arbitrary type sequence.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
x : array_like
|
||
|
The input array.
|
||
|
type : {1, 2, 3, 4}, 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 : {"backward", "ortho", "forward"}, optional
|
||
|
Normalization mode (see Notes). Default is "backward".
|
||
|
overwrite_x : bool, optional
|
||
|
If True, the contents of `x` can be destroyed; the default is False.
|
||
|
workers : int, optional
|
||
|
Maximum number of workers to use for parallel computation. If negative,
|
||
|
the value wraps around from ``os.cpu_count()``.
|
||
|
See :func:`~scipy.fft.fft` for more details.
|
||
|
orthogonalize : bool, optional
|
||
|
Whether to use the orthogonalized IDST variant (see Notes).
|
||
|
Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise.
|
||
|
|
||
|
.. versionadded:: 1.8.0
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
idst : ndarray of real
|
||
|
The transformed input array.
|
||
|
|
||
|
See Also
|
||
|
--------
|
||
|
dst : Forward DST
|
||
|
|
||
|
Notes
|
||
|
-----
|
||
|
.. warning:: For ``type in {2, 3}``, ``norm="ortho"`` breaks the direct
|
||
|
correspondence with the inverse direct Fourier transform.
|
||
|
|
||
|
For ``norm="ortho"`` both the `dst` and `idst` are scaled by the same
|
||
|
overall factor in both directions. By default, the transform is also
|
||
|
orthogonalized which for types 2 and 3 means the transform definition is
|
||
|
modified to give orthogonality of the DST matrix (see `dst` for the full
|
||
|
definitions).
|
||
|
|
||
|
'The' IDST is the IDST-II, which is the same as the normalized DST-III.
|
||
|
|
||
|
The IDST is equivalent to a normal DST except for the normalization and
|
||
|
type. DST type 1 and 4 are their own inverse and DSTs 2 and 3 are each
|
||
|
other's inverses.
|
||
|
|
||
|
"""
|
||
|
return (Dispatchable(x, np.ndarray),)
|