def dctn(x,
type=2,
s=None,
axes=None,
norm=None,
overwrite_x=False,
workers=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.
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.fft import dctn, idctn
>>> y = np.random.randn(16, 16)
>>> np.allclose(y, idctn(dctn(y)))
True
"""
return (Dispatchable(x, np.ndarray), )
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, 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 : {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-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
>>> 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), )
def idst(x,
type=2,
n=None,
axis=-1,
norm=None,
overwrite_x=False,
workers=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.
Returns
-------
idst : ndarray of real
The transformed input array.
See Also
--------
dst : Forward DST
Notes
-----
'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), )
def dctn(x, type=2, s=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, 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
``scipy.take(x.shape, axes, axis=0)``.
If ``s[i] > x.shape[i]``, the i-th dimension is padded with zeros.
If ``s[i] < x.shape[i]``, the i-th 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 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.fft import dctn, idctn
>>> y = np.random.randn(16, 16)
>>> np.allclose(y, idctn(dctn(y)))
True
"""
return (Dispatchable(x, np.ndarray), )
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, 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 : {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-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), )
def ifht(A, dln, mu, offset=0.0, bias=0.0):
"""ifht multimethod."""
return (Dispatchable(A, np.ndarray), )
def dst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False):
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 : {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``.
For ``norm=None``, there is no scaling on the `dst` and the `idst` is
scaled by ``1/N`` where ``N`` is the "logical" size of the DST. For
``norm='ortho'`` both directions are scaled by the same factor
``1/sqrt(N)``.
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=None``. DST-I assumes the input is odd around `n=-1` and `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 `2(N+1)`.
The orthonormalized DST-I is exactly its own inverse.
**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 :math:`k=-1` and even around `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 ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor ``f``
.. math::
f = \begin{cases}
\sqrt{\frac{1}{4N}} & \text{if }k = 0, \\
\sqrt{\frac{1}{2N}} & \text{otherwise} \end{cases}
**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`
.. 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)
The (unnormalized) DST-III is the inverse of the (unnormalized) DST-II, up
to a factor `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=None``). DST-IV assumes the input is odd around `n=-0.5` and even
around `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)
The (unnormalized) DST-IV is its own inverse, up to a factor `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), )
def dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False):
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 : {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)``.
For ``norm=None``, 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='ortho'``
both directions are scaled by the same factor ``1/sqrt(N)``.
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=None``)
.. 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 ``norm='ortho'``, ``x[0]`` and ``x[N-1]`` are multiplied by a scaling
factor of :math:`\sqrt{2}`, and ``y[k]`` is multiplied by a scaling factor
``f``
.. math::
f = \begin{cases}
\frac{1}{2}\sqrt{\frac{1}{N-1}} & \text{if }k=0\text{ or }N-1, \\
\frac{1}{2}\sqrt{\frac{2}{N-1}} & \text{otherwise} \end{cases}
.. 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=None``)
.. math::
y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi k(2n+1)}{2N} \right)
If ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor ``f``
.. math::
f = \begin{cases}
\sqrt{\frac{1}{4N}} & \text{if }k=0, \\
\sqrt{\frac{1}{2N}} & \text{otherwise} \end{cases}
Which 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=None``)
.. math::
y_k = x_0 + 2 \sum_{n=1}^{N-1} x_n \cos\left(\frac{\pi(2k+1)n}{2N}\right)
or, for ``norm='ortho'``
.. math::
y_k = \frac{x_0}{\sqrt{N}} + \sqrt{\frac{2}{N}} \sum_{n=1}^{N-1} x_n
\cos\left(\frac{\pi(2k+1)n}{2N}\right)
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=None``)
.. math::
y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi(2k+1)(2n+1)}{4N} \right)
If ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor ``f``
.. math::
f = \frac{1}{\sqrt{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
>>> 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), )
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), )
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), )
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
>>> 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), )
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
>>> 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), )
