| """ |
| Real spectrum transforms (DCT, DST, MDCT) |
| """ |
|
|
| __all__ = ['dct', 'idct', 'dst', 'idst', 'dctn', 'idctn', 'dstn', 'idstn'] |
|
|
| from scipy.fft import _pocketfft |
| from ._helper import _good_shape |
|
|
| _inverse_typemap = {1: 1, 2: 3, 3: 2, 4: 4} |
|
|
|
|
| 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, 4}, optional |
| Type of the DCT (see Notes). Default type is 2. |
| shape : int or array_like of ints or None, 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 ``numpy.take(x.shape, axes, axis=0)``. |
| If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros. |
| If ``shape[i] < x.shape[i]``, the ith dimension is truncated to |
| length ``shape[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 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 |
| -------- |
| >>> import numpy as np |
| >>> from scipy.fftpack import dctn, idctn |
| >>> rng = np.random.default_rng() |
| >>> y = rng.standard_normal((16, 16)) |
| >>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho')) |
| True |
| |
| """ |
| shape = _good_shape(x, shape, axes) |
| return _pocketfft.dctn(x, type, shape, axes, norm, overwrite_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, 4}, optional |
| Type of the DCT (see Notes). Default type is 2. |
| shape : int or array_like of ints or None, 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 ``numpy.take(x.shape, axes, axis=0)``. |
| If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros. |
| If ``shape[i] < x.shape[i]``, the ith dimension is truncated to |
| length ``shape[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 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 |
| -------- |
| >>> import numpy as np |
| >>> from scipy.fftpack import dctn, idctn |
| >>> rng = np.random.default_rng() |
| >>> y = rng.standard_normal((16, 16)) |
| >>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho')) |
| True |
| |
| """ |
| type = _inverse_typemap[type] |
| shape = _good_shape(x, shape, axes) |
| return _pocketfft.dctn(x, type, shape, axes, norm, overwrite_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, 4}, optional |
| Type of the DST (see Notes). Default type is 2. |
| shape : int or array_like of ints or None, 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 ``numpy.take(x.shape, axes, axis=0)``. |
| If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros. |
| If ``shape[i] < x.shape[i]``, the ith dimension is truncated to |
| length ``shape[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 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 |
| -------- |
| >>> import numpy as np |
| >>> from scipy.fftpack import dstn, idstn |
| >>> rng = np.random.default_rng() |
| >>> y = rng.standard_normal((16, 16)) |
| >>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho')) |
| True |
| |
| """ |
| shape = _good_shape(x, shape, axes) |
| return _pocketfft.dstn(x, type, shape, axes, norm, overwrite_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, 4}, optional |
| Type of the DST (see Notes). Default type is 2. |
| shape : int or array_like of ints or None, 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 ``numpy.take(x.shape, axes, axis=0)``. |
| If ``shape[i] > x.shape[i]``, the ith dimension is padded with zeros. |
| If ``shape[i] < x.shape[i]``, the ith dimension is truncated to |
| length ``shape[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 along which the IDST 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 |
| -------- |
| 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.fftpack import dstn, idstn |
| >>> rng = np.random.default_rng() |
| >>> y = rng.standard_normal((16, 16)) |
| >>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho')) |
| True |
| |
| """ |
| type = _inverse_typemap[type] |
| shape = _good_shape(x, shape, axes) |
| return _pocketfft.dstn(x, type, shape, axes, norm, overwrite_x) |
|
|
|
|
| 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)``. |
| |
| 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} |
| |
| .. versionadded:: 1.2.0 |
| Orthonormalization in DCT-I. |
| |
| .. 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}} |
| |
| .. versionadded:: 1.2.0 |
| Support for DCT-IV. |
| |
| 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.fftpack 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 _pocketfft.dct(x, type, n, axis, norm, 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, 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 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. IDCT of type 4 is the DCT |
| of type 4. 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 |
| >>> 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) / 6 |
| array([ 4., 3., 5., 10.]) |
| |
| """ |
| type = _inverse_typemap[type] |
| return _pocketfft.dct(x, type, n, axis, norm, overwrite_x) |
|
|
|
|
| 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``. |
| |
| 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. |
| |
| .. versionadded:: 0.11.0 |
| |
| **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. |
| |
| .. versionadded:: 1.2.0 |
| Support for DST-IV. |
| |
| References |
| ---------- |
| .. [1] Wikipedia, "Discrete sine transform", |
| https://en.wikipedia.org/wiki/Discrete_sine_transform |
| |
| """ |
| return _pocketfft.dst(x, type, n, axis, norm, 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, 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 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 |
| |
| """ |
| type = _inverse_typemap[type] |
| return _pocketfft.dst(x, type, n, axis, norm, overwrite_x) |
|
|
