ó ƒ(ph*cãóô•SSKJr SSKJr SSKr/SQr\SSS.Sjj5r\SSj5r\SS j5r \SS j5r \SS j5r \SS j5r \SS j5r \SSj5rg)é)Ú _dispatché)Ú DispatchableN)ÚdctÚidctÚdstÚidstÚdctnÚidctnÚdstnÚidstn)Ú orthogonalizecó8•[U[R54$)aŸ 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 of the input is padded with zeros. If ``s[i] < x.shape[i]``, the ith dimension of the input 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 ©rÚnpÚndarray©ÚxÚtypeÚsÚaxesÚnormÚ overwrite_xÚworkersrs ÚL/var/www/html/venv/lib/python3.13/site-packages/scipy/fft/_realtransforms.pyr r ó€ô| ˜œBŸJ™JÓ 'Ð )Ð)ócó8•[U[R54$)a£ 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 of the input is padded with zeros. If ``s[i] < x.shape[i]``, the ith dimension of the input 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 rrs rr r Irrcó8•[U[R54$)a¢ Return multidimensional Discrete Sine Transform along the specified axes. Parameters ---------- x : array_like The input array. type : {1, 2, 3, 4}, optional Type of the DST (see Notes). Default type is 2. s : int or array_like of ints or None, optional The shape of the result. If both `s` and `axes` (see below) are None, `s` is ``x.shape``; if `s` is None but `axes` is not None, then `s` is ``numpy.take(x.shape, axes, axis=0)``. If ``s[i] > x.shape[i]``, the ith dimension of the input is padded with zeros. If ``s[i] < x.shape[i]``, the ith dimension of the input is truncated to length ``s[i]``. If any element of `shape` is -1, the size of the corresponding dimension of `x` is used. axes : int or array_like of ints or None, optional Axes over which the DST is computed. If not given, the last ``len(s)`` axes are used, or all axes if `s` is also not specified. norm : {"backward", "ortho", "forward"}, optional Normalization mode (see Notes). Default is "backward". overwrite_x : bool, optional If True, the contents of `x` can be destroyed; the default is False. workers : int, optional Maximum number of workers to use for parallel computation. If negative, the value wraps around from ``os.cpu_count()``. See :func:`~scipy.fft.fft` for more details. orthogonalize : bool, optional Whether to use the orthogonalized DST variant (see Notes). Defaults to ``True`` when ``norm="ortho"`` and ``False`` otherwise. .. versionadded:: 1.8.0 Returns ------- y : ndarray of real The transformed input array. See Also -------- idstn : Inverse multidimensional DST Notes ----- For full details of the DST types and normalization modes, as well as references, see `dst`. Examples -------- >>> import numpy as np >>> from scipy.fft import dstn, idstn >>> rng = np.random.default_rng() >>> y = rng.standard_normal((16, 16)) >>> np.allclose(y, idstn(dstn(y))) True rrs rr r Šrrcó8•[U[R54$)a¡ 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 of the input is padded with zeros. If ``s[i] < x.shape[i]``, the ith dimension of the input 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 rrs rr r Ërrcó8•[U[R54$)a7Return 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.]) r©rrÚnÚaxisrrrrs rrr s€ôf ˜œBŸJ™JÓ 'Ð )Ð)rcó8•[U[R54$)a 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.]) rr"s rrr¢s€ôZ ˜œBŸJ™JÓ 'Ð )Ð)rcó8•[U[R54$)aˆ 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[-1]`` 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[-1]`` 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 rr"s rrròs€ôH ˜œBŸJ™JÓ 'Ð )Ð)rcó8•[U[R54$)a• 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. 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