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""" Real spectrum transforms (DCT, DST, MDCT) """
"""Handle shape and axes arguments for dctn, idctn, dstn, idstn.""" if shape is None: if axes is None: shape = x.shape else: shape = np.take(x.shape, axes) shape = tuple(shape) for dim in shape: if dim < 1: raise ValueError("Invalid number of DCT data points " "(%s) specified." % (shape,))
if axes is None: axes = list(range(-x.ndim, 0)) elif np.isscalar(axes): axes = [axes, ] if len(axes) != len(shape): raise ValueError("when given, axes and shape arguments " "have to be of the same length") if len(np.unique(axes)) != len(axes): raise ValueError("All axes must be unique.")
return shape, axes
""" Return multidimensional Discrete Cosine Transform along the specified axes.
Parameters ---------- x : array_like The input array. type : {1, 2, 3}, optional Type of the DCT (see Notes). Default type is 2. shape : tuple of ints, 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 ``scipy.take(x.shape, axes, axis=0)``. If ``shape[i] > x.shape[i]``, the i-th dimension is padded with zeros. If ``shape[i] < x.shape[i]``, the i-th dimension is truncated to length ``shape[i]``. axes : tuple 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.fftpack import dctn, idctn >>> y = np.random.randn(16, 16) >>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho')) True
""" x = np.asanyarray(x) shape, axes = _init_nd_shape_and_axes(x, shape, axes) for n, ax in zip(shape, axes): x = dct(x, type=type, n=n, axis=ax, norm=norm, overwrite_x=overwrite_x) return x
""" Return multidimensional Discrete Cosine Transform along the specified axes.
Parameters ---------- x : array_like The input array. type : {1, 2, 3}, optional Type of the DCT (see Notes). Default type is 2. shape : tuple of ints, 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 ``scipy.take(x.shape, axes, axis=0)``. If ``shape[i] > x.shape[i]``, the i-th dimension is padded with zeros. If ``shape[i] < x.shape[i]``, the i-th dimension is truncated to length ``shape[i]``. axes : tuple 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 -------- >>> from scipy.fftpack import dctn, idctn >>> y = np.random.randn(16, 16) >>> np.allclose(y, idctn(dctn(y, norm='ortho'), norm='ortho')) True """ x = np.asanyarray(x) shape, axes = _init_nd_shape_and_axes(x, shape, axes) for n, ax in zip(shape, axes): x = idct(x, type=type, n=n, axis=ax, norm=norm, overwrite_x=overwrite_x) return x
""" Return multidimensional Discrete Sine Transform along the specified axes.
Parameters ---------- x : array_like The input array. type : {1, 2, 3}, optional Type of the DCT (see Notes). Default type is 2. shape : tuple of ints, 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 ``scipy.take(x.shape, axes, axis=0)``. If ``shape[i] > x.shape[i]``, the i-th dimension is padded with zeros. If ``shape[i] < x.shape[i]``, the i-th dimension is truncated to length ``shape[i]``. axes : tuple 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 -------- >>> from scipy.fftpack import dstn, idstn >>> y = np.random.randn(16, 16) >>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho')) True
""" x = np.asanyarray(x) shape, axes = _init_nd_shape_and_axes(x, shape, axes) for n, ax in zip(shape, axes): x = dst(x, type=type, n=n, axis=ax, norm=norm, overwrite_x=overwrite_x) return x
""" Return multidimensional Discrete Sine Transform along the specified axes.
Parameters ---------- x : array_like The input array. type : {1, 2, 3}, optional Type of the DCT (see Notes). Default type is 2. shape : tuple of ints, 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 ``scipy.take(x.shape, axes, axis=0)``. If ``shape[i] > x.shape[i]``, the i-th dimension is padded with zeros. If ``shape[i] < x.shape[i]``, the i-th dimension is truncated to length ``shape[i]``. axes : tuple 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 DST
Notes ----- For full details of the IDST types and normalization modes, as well as references, see `idst`.
Examples -------- >>> from scipy.fftpack import dstn, idstn >>> y = np.random.randn(16, 16) >>> np.allclose(y, idstn(dstn(y, norm='ortho'), norm='ortho')) True """ x = np.asanyarray(x) shape, axes = _init_nd_shape_and_axes(x, shape, axes) for n, ax in zip(shape, axes): x = idst(x, type=type, n=n, axis=ax, norm=norm, overwrite_x=overwrite_x) return x
""" Return the Discrete Cosine Transform of arbitrary type sequence x.
Parameters ---------- x : array_like The input array. type : {1, 2, 3}, 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 3 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``)::
N-2 y[k] = x[0] + (-1)**k x[N-1] + 2 * sum x[n]*cos(pi*k*n/(N-1)) n=1
Only None is supported as normalization mode for DCT-I. Note also that 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``)::
N-1 y[k] = 2* sum x[n]*cos(pi*k*(2n+1)/(2*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.
Which makes the corresponding matrix of coefficients orthonormal (``OO' = Id``).
**Type III**
There are several definitions, we use the following (for ``norm=None``)::
N-1 y[k] = x[0] + 2 * sum x[n]*cos(pi*(k+0.5)*n/N), 0 <= k < N. n=1
or, for ``norm='ortho'`` and 0 <= k < N::
N-1 y[k] = x[0] / sqrt(N) + sqrt(2/N) * sum x[n]*cos(pi*(k+0.5)*n/N) n=1
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.
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, http://dx.doi.org/10.1109/TASSP.1980.1163351 (1980). .. [2] Wikipedia, "Discrete cosine transform", http://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 >>> 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.])
""" if type == 1 and norm is not None: raise NotImplementedError( "Orthonormalization not yet supported for DCT-I") return _dct(x, type, n, axis, normalize=norm, overwrite_x=overwrite_x)
""" Return the Inverse Discrete Cosine Transform of an arbitrary type sequence.
Parameters ---------- x : array_like The input array. type : {1, 2, 3}, 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. 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 >>> 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.])
""" 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 _dct(x, _TP[type], n, axis, normalize=norm, overwrite_x=overwrite_x)
try: name = {'float64':'ddct%d', 'float32':'dct%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
try: nm = {None:0, 'ortho':1}[normalize] except KeyError: raise ValueError("Unknown normalize mode %s" % normalize) return nm
tmp = _asfarray(x) copy_made = _datacopied(tmp, x) if n is None: n = tmp.shape[axis] elif n != tmp.shape[axis]: tmp, copy_made2 = _fix_shape(tmp, n, axis) copy_made = copy_made or copy_made2 if n < 1: raise ValueError("Invalid number of %s data points " "(%d) specified." % (dct_or_dst, n)) return tmp, n, copy_made
f = _get_dct_fun(type, x0.dtype) return _eval_fun(f, x0, n, axis, nm, overwrite_x)
f = _get_dst_fun(type, x0.dtype) return _eval_fun(f, x0, n, axis, nm, overwrite_x)
if axis == -1 or axis == len(tmp.shape) - 1: return f(tmp, n, nm, overwrite_x)
tmp = np.swapaxes(tmp, axis, -1) tmp = f(tmp, n, nm, overwrite_x) return np.swapaxes(tmp, axis, -1)
""" 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)
""" 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)
""" 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)
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
""" 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) |