717 lines
23 KiB
Python
717 lines
23 KiB
Python
"""
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Real spectrum transforms (DCT, DST, MDCT)
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"""
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from __future__ import division, print_function, absolute_import
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__all__ = ['dct', 'idct', 'dst', 'idst', 'dctn', 'idctn', 'dstn', 'idstn']
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import numpy as np
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from scipy.fftpack import _fftpack
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from scipy.fftpack.basic import _datacopied, _fix_shape, _asfarray
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import atexit
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atexit.register(_fftpack.destroy_ddct1_cache)
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atexit.register(_fftpack.destroy_ddct2_cache)
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atexit.register(_fftpack.destroy_dct1_cache)
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atexit.register(_fftpack.destroy_dct2_cache)
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atexit.register(_fftpack.destroy_ddst1_cache)
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atexit.register(_fftpack.destroy_ddst2_cache)
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atexit.register(_fftpack.destroy_dst1_cache)
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atexit.register(_fftpack.destroy_dst2_cache)
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def _init_nd_shape_and_axes(x, shape, axes):
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"""Handle shape and axes arguments for dctn, idctn, dstn, idstn."""
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if shape is None:
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if axes is None:
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shape = x.shape
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else:
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shape = np.take(x.shape, axes)
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shape = tuple(shape)
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for dim in shape:
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if dim < 1:
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raise ValueError("Invalid number of DCT data points "
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"(%s) specified." % (shape,))
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if axes is None:
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axes = list(range(-x.ndim, 0))
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elif np.isscalar(axes):
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axes = [axes, ]
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if len(axes) != len(shape):
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raise ValueError("when given, axes and shape arguments "
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"have to be of the same length")
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if len(np.unique(axes)) != len(axes):
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raise ValueError("All axes must be unique.")
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return shape, axes
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def dctn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False):
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"""
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Return multidimensional Discrete Cosine Transform along the specified axes.
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Parameters
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----------
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x : array_like
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The input array.
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type : {1, 2, 3}, optional
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Type of the DCT (see Notes). Default type is 2.
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shape : tuple of ints, optional
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The shape of the result. If both `shape` and `axes` (see below) are
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None, `shape` is ``x.shape``; if `shape` is None but `axes` is
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not None, then `shape` is ``scipy.take(x.shape, axes, axis=0)``.
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If ``shape[i] > x.shape[i]``, the i-th dimension is padded with zeros.
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If ``shape[i] < x.shape[i]``, the i-th dimension is truncated to
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length ``shape[i]``.
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axes : tuple or None, optional
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Axes along which the DCT is computed; the default is over all axes.
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norm : {None, 'ortho'}, optional
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Normalization mode (see Notes). Default is None.
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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Returns
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-------
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y : ndarray of real
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The transformed input array.
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See Also
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--------
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idctn : Inverse multidimensional DCT
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Notes
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-----
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For full details of the DCT types and normalization modes, as well as
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references, see `dct`.
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Examples
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--------
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>>> from scipy.fftpack import dctn, idctn
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>>> y = np.random.randn(16, 16)
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>>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho'))
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True
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"""
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x = np.asanyarray(x)
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shape, axes = _init_nd_shape_and_axes(x, shape, axes)
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for n, ax in zip(shape, axes):
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x = dct(x, type=type, n=n, axis=ax, norm=norm, overwrite_x=overwrite_x)
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return x
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def idctn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False):
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"""
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Return multidimensional Discrete Cosine Transform along the specified axes.
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Parameters
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----------
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x : array_like
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The input array.
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type : {1, 2, 3}, optional
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Type of the DCT (see Notes). Default type is 2.
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shape : tuple of ints, optional
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The shape of the result. If both `shape` and `axes` (see below) are
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None, `shape` is ``x.shape``; if `shape` is None but `axes` is
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not None, then `shape` is ``scipy.take(x.shape, axes, axis=0)``.
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If ``shape[i] > x.shape[i]``, the i-th dimension is padded with zeros.
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If ``shape[i] < x.shape[i]``, the i-th dimension is truncated to
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length ``shape[i]``.
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axes : tuple or None, optional
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Axes along which the IDCT is computed; the default is over all axes.
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norm : {None, 'ortho'}, optional
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Normalization mode (see Notes). Default is None.
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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Returns
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-------
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y : ndarray of real
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The transformed input array.
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See Also
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--------
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dctn : multidimensional DCT
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Notes
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-----
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For full details of the IDCT types and normalization modes, as well as
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references, see `idct`.
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Examples
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--------
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>>> from scipy.fftpack import dctn, idctn
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>>> y = np.random.randn(16, 16)
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>>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho'))
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True
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"""
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x = np.asanyarray(x)
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shape, axes = _init_nd_shape_and_axes(x, shape, axes)
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for n, ax in zip(shape, axes):
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x = idct(x, type=type, n=n, axis=ax, norm=norm,
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overwrite_x=overwrite_x)
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return x
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def dstn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False):
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"""
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Return multidimensional Discrete Sine Transform along the specified axes.
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Parameters
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----------
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x : array_like
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The input array.
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type : {1, 2, 3}, optional
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Type of the DCT (see Notes). Default type is 2.
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shape : tuple of ints, optional
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The shape of the result. If both `shape` and `axes` (see below) are
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None, `shape` is ``x.shape``; if `shape` is None but `axes` is
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not None, then `shape` is ``scipy.take(x.shape, axes, axis=0)``.
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If ``shape[i] > x.shape[i]``, the i-th dimension is padded with zeros.
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If ``shape[i] < x.shape[i]``, the i-th dimension is truncated to
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length ``shape[i]``.
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axes : tuple or None, optional
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Axes along which the DCT is computed; the default is over all axes.
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norm : {None, 'ortho'}, optional
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Normalization mode (see Notes). Default is None.
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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Returns
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-------
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y : ndarray of real
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The transformed input array.
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See Also
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--------
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idstn : Inverse multidimensional DST
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Notes
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-----
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For full details of the DST types and normalization modes, as well as
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references, see `dst`.
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Examples
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--------
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>>> from scipy.fftpack import dstn, idstn
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>>> y = np.random.randn(16, 16)
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>>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho'))
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True
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"""
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x = np.asanyarray(x)
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shape, axes = _init_nd_shape_and_axes(x, shape, axes)
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for n, ax in zip(shape, axes):
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x = dst(x, type=type, n=n, axis=ax, norm=norm, overwrite_x=overwrite_x)
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return x
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def idstn(x, type=2, shape=None, axes=None, norm=None, overwrite_x=False):
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"""
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Return multidimensional Discrete Sine Transform along the specified axes.
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Parameters
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----------
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x : array_like
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The input array.
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type : {1, 2, 3}, optional
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Type of the DCT (see Notes). Default type is 2.
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shape : tuple of ints, optional
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The shape of the result. If both `shape` and `axes` (see below) are
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None, `shape` is ``x.shape``; if `shape` is None but `axes` is
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not None, then `shape` is ``scipy.take(x.shape, axes, axis=0)``.
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If ``shape[i] > x.shape[i]``, the i-th dimension is padded with zeros.
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If ``shape[i] < x.shape[i]``, the i-th dimension is truncated to
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length ``shape[i]``.
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axes : tuple or None, optional
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Axes along which the IDCT is computed; the default is over all axes.
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norm : {None, 'ortho'}, optional
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Normalization mode (see Notes). Default is None.
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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Returns
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-------
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y : ndarray of real
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The transformed input array.
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See Also
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--------
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dctn : multidimensional DST
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Notes
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-----
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For full details of the IDST types and normalization modes, as well as
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references, see `idst`.
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Examples
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--------
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>>> from scipy.fftpack import dstn, idstn
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>>> y = np.random.randn(16, 16)
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>>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho'))
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True
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"""
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x = np.asanyarray(x)
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shape, axes = _init_nd_shape_and_axes(x, shape, axes)
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for n, ax in zip(shape, axes):
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x = idst(x, type=type, n=n, axis=ax, norm=norm,
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overwrite_x=overwrite_x)
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return x
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def dct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False):
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"""
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Return the Discrete Cosine Transform of arbitrary type sequence x.
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Parameters
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----------
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x : array_like
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The input array.
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type : {1, 2, 3}, optional
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Type of the DCT (see Notes). Default type is 2.
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n : int, optional
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Length of the transform. If ``n < x.shape[axis]``, `x` is
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truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The
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default results in ``n = x.shape[axis]``.
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axis : int, optional
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Axis along which the dct is computed; the default is over the
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last axis (i.e., ``axis=-1``).
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norm : {None, 'ortho'}, optional
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Normalization mode (see Notes). Default is None.
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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Returns
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-------
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y : ndarray of real
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The transformed input array.
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See Also
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--------
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idct : Inverse DCT
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Notes
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-----
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For a single dimension array ``x``, ``dct(x, norm='ortho')`` is equal to
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MATLAB ``dct(x)``.
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There are theoretically 8 types of the DCT, only the first 3 types are
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implemented in scipy. 'The' DCT generally refers to DCT type 2, and 'the'
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Inverse DCT generally refers to DCT type 3.
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**Type I**
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There are several definitions of the DCT-I; we use the following
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(for ``norm=None``)::
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N-2
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y[k] = x[0] + (-1)**k x[N-1] + 2 * sum x[n]*cos(pi*k*n/(N-1))
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n=1
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Only None is supported as normalization mode for DCT-I. Note also that the
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DCT-I is only supported for input size > 1
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**Type II**
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There are several definitions of the DCT-II; we use the following
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(for ``norm=None``)::
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N-1
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y[k] = 2* sum x[n]*cos(pi*k*(2n+1)/(2*N)), 0 <= k < N.
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n=0
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If ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor `f`::
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f = sqrt(1/(4*N)) if k = 0,
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f = sqrt(1/(2*N)) otherwise.
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Which makes the corresponding matrix of coefficients orthonormal
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(``OO' = Id``).
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**Type III**
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There are several definitions, we use the following
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(for ``norm=None``)::
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N-1
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y[k] = x[0] + 2 * sum x[n]*cos(pi*(k+0.5)*n/N), 0 <= k < N.
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n=1
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or, for ``norm='ortho'`` and 0 <= k < N::
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N-1
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y[k] = x[0] / sqrt(N) + sqrt(2/N) * sum x[n]*cos(pi*(k+0.5)*n/N)
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n=1
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The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up
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to a factor `2N`. The orthonormalized DCT-III is exactly the inverse of
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the orthonormalized DCT-II.
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References
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----------
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.. [1] 'A Fast Cosine Transform in One and Two Dimensions', by J.
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Makhoul, `IEEE Transactions on acoustics, speech and signal
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processing` vol. 28(1), pp. 27-34,
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http://dx.doi.org/10.1109/TASSP.1980.1163351 (1980).
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.. [2] Wikipedia, "Discrete cosine transform",
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http://en.wikipedia.org/wiki/Discrete_cosine_transform
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Examples
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--------
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The Type 1 DCT is equivalent to the FFT (though faster) for real,
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even-symmetrical inputs. The output is also real and even-symmetrical.
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Half of the FFT input is used to generate half of the FFT output:
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>>> from scipy.fftpack import fft, dct
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>>> fft(np.array([4., 3., 5., 10., 5., 3.])).real
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array([ 30., -8., 6., -2., 6., -8.])
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>>> dct(np.array([4., 3., 5., 10.]), 1)
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array([ 30., -8., 6., -2.])
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"""
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if type == 1 and norm is not None:
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raise NotImplementedError(
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"Orthonormalization not yet supported for DCT-I")
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return _dct(x, type, n, axis, normalize=norm, overwrite_x=overwrite_x)
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def idct(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False):
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"""
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Return the Inverse Discrete Cosine Transform of an arbitrary type sequence.
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Parameters
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----------
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x : array_like
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The input array.
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type : {1, 2, 3}, optional
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Type of the DCT (see Notes). Default type is 2.
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n : int, optional
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Length of the transform. If ``n < x.shape[axis]``, `x` is
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truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The
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default results in ``n = x.shape[axis]``.
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axis : int, optional
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Axis along which the idct is computed; the default is over the
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last axis (i.e., ``axis=-1``).
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norm : {None, 'ortho'}, optional
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Normalization mode (see Notes). Default is None.
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overwrite_x : bool, optional
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If True, the contents of `x` can be destroyed; the default is False.
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Returns
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-------
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idct : ndarray of real
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The transformed input array.
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See Also
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--------
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dct : Forward DCT
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Notes
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-----
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For a single dimension array `x`, ``idct(x, norm='ortho')`` is equal to
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MATLAB ``idct(x)``.
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'The' IDCT is the IDCT of type 2, which is the same as DCT of type 3.
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IDCT of type 1 is the DCT of type 1, IDCT of type 2 is the DCT of type
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3, and IDCT of type 3 is the DCT of type 2. For the definition of these
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types, see `dct`.
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Examples
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--------
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The Type 1 DCT is equivalent to the DFT for real, even-symmetrical
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inputs. The output is also real and even-symmetrical. Half of the IFFT
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input is used to generate half of the IFFT output:
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>>> from scipy.fftpack import ifft, idct
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>>> ifft(np.array([ 30., -8., 6., -2., 6., -8.])).real
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array([ 4., 3., 5., 10., 5., 3.])
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>>> idct(np.array([ 30., -8., 6., -2.]), 1) / 6
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array([ 4., 3., 5., 10.])
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"""
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if type == 1 and norm is not None:
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raise NotImplementedError(
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"Orthonormalization not yet supported for IDCT-I")
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# Inverse/forward type table
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_TP = {1:1, 2:3, 3:2}
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return _dct(x, _TP[type], n, axis, normalize=norm, overwrite_x=overwrite_x)
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def _get_dct_fun(type, dtype):
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try:
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name = {'float64':'ddct%d', 'float32':'dct%d'}[dtype.name]
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except KeyError:
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raise ValueError("dtype %s not supported" % dtype)
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try:
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f = getattr(_fftpack, name % type)
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except AttributeError as e:
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raise ValueError(str(e) + ". Type %d not understood" % type)
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return f
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def _get_norm_mode(normalize):
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try:
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nm = {None:0, 'ortho':1}[normalize]
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except KeyError:
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raise ValueError("Unknown normalize mode %s" % normalize)
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return nm
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def __fix_shape(x, n, axis, dct_or_dst):
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tmp = _asfarray(x)
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copy_made = _datacopied(tmp, x)
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if n is None:
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n = tmp.shape[axis]
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elif n != tmp.shape[axis]:
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tmp, copy_made2 = _fix_shape(tmp, n, axis)
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copy_made = copy_made or copy_made2
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if n < 1:
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raise ValueError("Invalid number of %s data points "
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"(%d) specified." % (dct_or_dst, n))
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return tmp, n, copy_made
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def _raw_dct(x0, type, n, axis, nm, overwrite_x):
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f = _get_dct_fun(type, x0.dtype)
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return _eval_fun(f, x0, n, axis, nm, overwrite_x)
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def _raw_dst(x0, type, n, axis, nm, overwrite_x):
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f = _get_dst_fun(type, x0.dtype)
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return _eval_fun(f, x0, n, axis, nm, overwrite_x)
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def _eval_fun(f, tmp, n, axis, nm, overwrite_x):
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if axis == -1 or axis == len(tmp.shape) - 1:
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return f(tmp, n, nm, overwrite_x)
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|
|
|
tmp = np.swapaxes(tmp, axis, -1)
|
|
tmp = f(tmp, n, nm, overwrite_x)
|
|
return np.swapaxes(tmp, axis, -1)
|
|
|
|
|
|
def _dct(x, type, n=None, axis=-1, overwrite_x=False, normalize=None):
|
|
"""
|
|
Return Discrete Cosine Transform of arbitrary type sequence x.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
input array.
|
|
n : int, optional
|
|
Length of the transform. If ``n < x.shape[axis]``, `x` is
|
|
truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The
|
|
default results in ``n = x.shape[axis]``.
|
|
axis : int, optional
|
|
Axis along which the dct is computed; the default is over the
|
|
last axis (i.e., ``axis=-1``).
|
|
overwrite_x : bool, optional
|
|
If True, the contents of `x` can be destroyed; the default is False.
|
|
|
|
Returns
|
|
-------
|
|
z : ndarray
|
|
|
|
"""
|
|
x0, n, copy_made = __fix_shape(x, n, axis, 'DCT')
|
|
if type == 1 and n < 2:
|
|
raise ValueError("DCT-I is not defined for size < 2")
|
|
overwrite_x = overwrite_x or copy_made
|
|
nm = _get_norm_mode(normalize)
|
|
if np.iscomplexobj(x0):
|
|
return (_raw_dct(x0.real, type, n, axis, nm, overwrite_x) + 1j *
|
|
_raw_dct(x0.imag, type, n, axis, nm, overwrite_x))
|
|
else:
|
|
return _raw_dct(x0, type, n, axis, nm, overwrite_x)
|
|
|
|
|
|
def dst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False):
|
|
"""
|
|
Return the Discrete Sine Transform of arbitrary type sequence x.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
The input array.
|
|
type : {1, 2, 3}, optional
|
|
Type of the DST (see Notes). Default type is 2.
|
|
n : int, optional
|
|
Length of the transform. If ``n < x.shape[axis]``, `x` is
|
|
truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The
|
|
default results in ``n = x.shape[axis]``.
|
|
axis : int, optional
|
|
Axis along which the dst is computed; the default is over the
|
|
last axis (i.e., ``axis=-1``).
|
|
norm : {None, 'ortho'}, optional
|
|
Normalization mode (see Notes). Default is None.
|
|
overwrite_x : bool, optional
|
|
If True, the contents of `x` can be destroyed; the default is False.
|
|
|
|
Returns
|
|
-------
|
|
dst : ndarray of reals
|
|
The transformed input array.
|
|
|
|
See Also
|
|
--------
|
|
idst : Inverse DST
|
|
|
|
Notes
|
|
-----
|
|
For a single dimension array ``x``.
|
|
|
|
There are theoretically 8 types of the DST for different combinations of
|
|
even/odd boundary conditions and boundary off sets [1]_, only the first
|
|
3 types are implemented in scipy.
|
|
|
|
**Type I**
|
|
|
|
There are several definitions of the DST-I; we use the following
|
|
for ``norm=None``. DST-I assumes the input is odd around n=-1 and n=N. ::
|
|
|
|
N-1
|
|
y[k] = 2 * sum x[n]*sin(pi*(k+1)*(n+1)/(N+1))
|
|
n=0
|
|
|
|
Only None is supported as normalization mode for DCT-I. Note also that the
|
|
DCT-I is only supported for input size > 1
|
|
The (unnormalized) DCT-I is its own inverse, up to a factor `2(N+1)`.
|
|
|
|
**Type II**
|
|
|
|
There are several definitions of the DST-II; we use the following
|
|
for ``norm=None``. DST-II assumes the input is odd around n=-1/2 and
|
|
n=N-1/2; the output is odd around k=-1 and even around k=N-1 ::
|
|
|
|
N-1
|
|
y[k] = 2* sum x[n]*sin(pi*(k+1)*(n+0.5)/N), 0 <= k < N.
|
|
n=0
|
|
|
|
if ``norm='ortho'``, ``y[k]`` is multiplied by a scaling factor `f` ::
|
|
|
|
f = sqrt(1/(4*N)) if k == 0
|
|
f = sqrt(1/(2*N)) otherwise.
|
|
|
|
**Type III**
|
|
|
|
There are several definitions of the DST-III, we use the following
|
|
(for ``norm=None``). DST-III assumes the input is odd around n=-1
|
|
and even around n=N-1 ::
|
|
|
|
N-2
|
|
y[k] = x[N-1]*(-1)**k + 2* sum x[n]*sin(pi*(k+0.5)*(n+1)/N), 0 <= k < N.
|
|
n=0
|
|
|
|
The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up
|
|
to a factor `2N`. The orthonormalized DST-III is exactly the inverse of
|
|
the orthonormalized DST-II.
|
|
|
|
.. versionadded:: 0.11.0
|
|
|
|
References
|
|
----------
|
|
.. [1] Wikipedia, "Discrete sine transform",
|
|
http://en.wikipedia.org/wiki/Discrete_sine_transform
|
|
|
|
"""
|
|
if type == 1 and norm is not None:
|
|
raise NotImplementedError(
|
|
"Orthonormalization not yet supported for IDCT-I")
|
|
return _dst(x, type, n, axis, normalize=norm, overwrite_x=overwrite_x)
|
|
|
|
|
|
def idst(x, type=2, n=None, axis=-1, norm=None, overwrite_x=False):
|
|
"""
|
|
Return the Inverse Discrete Sine Transform of an arbitrary type sequence.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
The input array.
|
|
type : {1, 2, 3}, optional
|
|
Type of the DST (see Notes). Default type is 2.
|
|
n : int, optional
|
|
Length of the transform. If ``n < x.shape[axis]``, `x` is
|
|
truncated. If ``n > x.shape[axis]``, `x` is zero-padded. The
|
|
default results in ``n = x.shape[axis]``.
|
|
axis : int, optional
|
|
Axis along which the idst is computed; the default is over the
|
|
last axis (i.e., ``axis=-1``).
|
|
norm : {None, 'ortho'}, optional
|
|
Normalization mode (see Notes). Default is None.
|
|
overwrite_x : bool, optional
|
|
If True, the contents of `x` can be destroyed; the default is False.
|
|
|
|
Returns
|
|
-------
|
|
idst : ndarray of real
|
|
The transformed input array.
|
|
|
|
See Also
|
|
--------
|
|
dst : Forward DST
|
|
|
|
Notes
|
|
-----
|
|
'The' IDST is the IDST of type 2, which is the same as DST of type 3.
|
|
|
|
IDST of type 1 is the DST of type 1, IDST of type 2 is the DST of type
|
|
3, and IDST of type 3 is the DST of type 2. For the definition of these
|
|
types, see `dst`.
|
|
|
|
.. versionadded:: 0.11.0
|
|
|
|
"""
|
|
if type == 1 and norm is not None:
|
|
raise NotImplementedError(
|
|
"Orthonormalization not yet supported for IDCT-I")
|
|
# Inverse/forward type table
|
|
_TP = {1:1, 2:3, 3:2}
|
|
return _dst(x, _TP[type], n, axis, normalize=norm, overwrite_x=overwrite_x)
|
|
|
|
|
|
def _get_dst_fun(type, dtype):
|
|
try:
|
|
name = {'float64':'ddst%d', 'float32':'dst%d'}[dtype.name]
|
|
except KeyError:
|
|
raise ValueError("dtype %s not supported" % dtype)
|
|
try:
|
|
f = getattr(_fftpack, name % type)
|
|
except AttributeError as e:
|
|
raise ValueError(str(e) + ". Type %d not understood" % type)
|
|
return f
|
|
|
|
|
|
def _dst(x, type, n=None, axis=-1, overwrite_x=False, normalize=None):
|
|
"""
|
|
Return Discrete Sine Transform of arbitrary type sequence x.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
input array.
|
|
n : int, optional
|
|
Length of the transform.
|
|
axis : int, optional
|
|
Axis along which the dst is computed. (default=-1)
|
|
overwrite_x : bool, optional
|
|
If True the contents of x can be destroyed. (default=False)
|
|
|
|
Returns
|
|
-------
|
|
z : real ndarray
|
|
|
|
"""
|
|
x0, n, copy_made = __fix_shape(x, n, axis, 'DST')
|
|
if type == 1 and n < 2:
|
|
raise ValueError("DST-I is not defined for size < 2")
|
|
overwrite_x = overwrite_x or copy_made
|
|
nm = _get_norm_mode(normalize)
|
|
if np.iscomplexobj(x0):
|
|
return (_raw_dst(x0.real, type, n, axis, nm, overwrite_x) + 1j *
|
|
_raw_dst(x0.imag, type, n, axis, nm, overwrite_x))
|
|
else:
|
|
return _raw_dst(x0, type, n, axis, nm, overwrite_x)
|