428 lines
14 KiB
Python
428 lines
14 KiB
Python
|
from functools import partial, reduce
|
||
|
|
|
||
|
|
import numpy as np
|
||
|
|
|
||
|
|
from ._multilevel import (_prep_axes_wavedecn, wavedec, wavedec2, wavedecn,
|
||
|
|
waverec, waverec2, waverecn)
|
||
|
|
from ._swt import iswt, iswt2, iswtn, swt, swt2, swt_max_level, swtn
|
||
|
|
from ._utils import _modes_per_axis, _wavelets_per_axis
|
||
|
|
|
||
|
|
__all__ = ["mra", "mra2", "mran", "imra", "imra2", "imran"]
|
||
|
|
|
||
|
|
|
||
|
|
def mra(data, wavelet, level=None, axis=-1, transform='swt',
|
||
|
|
mode='periodization'):
|
||
|
|
"""Forward 1D multiresolution analysis.
|
||
|
|
|
||
|
|
It is a projection onto the wavelet subspaces.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
data: array_like
|
||
|
|
Input data
|
||
|
|
wavelet : Wavelet object or name string
|
||
|
|
Wavelet to use
|
||
|
|
level : int, optional
|
||
|
|
Decomposition level (must be >= 0). If level is None (default) then it
|
||
|
|
will be calculated using the `dwt_max_level` function.
|
||
|
|
axis: int, optional
|
||
|
|
Axis over which to compute the DWT. If not given, the last axis is
|
||
|
|
used. Currently only available when ``transform='dwt'``.
|
||
|
|
transform : {'dwt', 'swt'}
|
||
|
|
Whether to use the DWT or SWT for the transforms.
|
||
|
|
mode : str, optional
|
||
|
|
Signal extension mode, see `Modes` (default: 'symmetric'). This option
|
||
|
|
is only used when transform='dwt'.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
[cAn, {details_level_n}, ... {details_level_1}] : list
|
||
|
|
For more information, see the detailed description in `wavedec`
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
imra, swt
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
This is sometimes referred to as an additive decomposition because the
|
||
|
|
inverse transform (``imra``) is just the sum of the coefficient arrays
|
||
|
|
[1]_. The decomposition using ``transform='dwt'`` corresponds to section
|
||
|
|
2.2 while that using an undecimated transform (``transform='swt'``) is
|
||
|
|
described in section 3.2 and appendix A.
|
||
|
|
|
||
|
|
This transform does not share the variance partition property of ``swt``
|
||
|
|
with `norm=True`. It does however, result in coefficients that are
|
||
|
|
temporally aligned regardless of the symmetry of the wavelet used.
|
||
|
|
|
||
|
|
The redundancy of this transform is ``(level + 1)``.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Donald B. Percival and Harold O. Mofjeld. Analysis of Subtidal
|
||
|
|
Coastal Sea Level Fluctuations Using Wavelets. Journal of the American
|
||
|
|
Statistical Association Vol. 92, No. 439 (Sep., 1997), pp. 868-880.
|
||
|
|
https://doi.org/10.2307/2965551
|
||
|
|
|
||
|
|
"""
|
||
|
|
if transform == 'swt':
|
||
|
|
if mode != 'periodization':
|
||
|
|
raise ValueError(
|
||
|
|
"transform swt only supports mode='periodization'")
|
||
|
|
kwargs = dict(wavelet=wavelet, axis=axis, norm=True)
|
||
|
|
forward = partial(swt, level=level, trim_approx=True, **kwargs)
|
||
|
|
inverse = partial(iswt, **kwargs)
|
||
|
|
is_swt = True
|
||
|
|
elif transform == 'dwt':
|
||
|
|
kwargs = dict(wavelet=wavelet, mode=mode, axis=axis)
|
||
|
|
forward = partial(wavedec, level=level, **kwargs)
|
||
|
|
inverse = partial(waverec, **kwargs)
|
||
|
|
is_swt = False
|
||
|
|
else:
|
||
|
|
raise ValueError("unrecognized transform: {}".format(transform))
|
||
|
|
|
||
|
|
wav_coeffs = forward(data)
|
||
|
|
|
||
|
|
mra_coeffs = []
|
||
|
|
nc = len(wav_coeffs)
|
||
|
|
|
||
|
|
if is_swt:
|
||
|
|
# replicate same zeros array to save memory
|
||
|
|
z = np.zeros_like(wav_coeffs[0])
|
||
|
|
tmp = [z, ] * nc
|
||
|
|
else:
|
||
|
|
# zero arrays have variable size in DWT case
|
||
|
|
tmp = [np.zeros_like(c) for c in wav_coeffs]
|
||
|
|
|
||
|
|
for j in range(nc):
|
||
|
|
# tmp has arrays of zeros except for the jth entry
|
||
|
|
tmp[j] = wav_coeffs[j]
|
||
|
|
|
||
|
|
# reconstruct
|
||
|
|
rec = inverse(tmp)
|
||
|
|
if rec.shape != data.shape:
|
||
|
|
# trim any excess coefficients
|
||
|
|
rec = rec[tuple([slice(sz) for sz in data.shape])]
|
||
|
|
mra_coeffs.append(rec)
|
||
|
|
|
||
|
|
# restore zeros
|
||
|
|
if is_swt:
|
||
|
|
tmp[j] = z
|
||
|
|
else:
|
||
|
|
tmp[j] = np.zeros_like(tmp[j])
|
||
|
|
return mra_coeffs
|
||
|
|
|
||
|
|
|
||
|
|
def imra(mra_coeffs):
|
||
|
|
"""Inverse 1D multiresolution analysis via summation.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
mra_coeffs : list of ndarray
|
||
|
|
Multiresolution analysis coefficients as returned by `mra`.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
rec : ndarray
|
||
|
|
The reconstructed signal.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
mra
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Donald B. Percival and Harold O. Mofjeld. Analysis of Subtidal
|
||
|
|
Coastal Sea Level Fluctuations Using Wavelets. Journal of the American
|
||
|
|
Statistical Association Vol. 92, No. 439 (Sep., 1997), pp. 868-880.
|
||
|
|
https://doi.org/10.2307/2965551
|
||
|
|
"""
|
||
|
|
return reduce(lambda x, y: x + y, mra_coeffs)
|
||
|
|
|
||
|
|
|
||
|
|
def mra2(data, wavelet, level=None, axes=(-2, -1), transform='swt2',
|
||
|
|
mode='periodization'):
|
||
|
|
"""Forward 2D multiresolution analysis.
|
||
|
|
|
||
|
|
It is a projection onto wavelet subspaces.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
data: array_like
|
||
|
|
Input data
|
||
|
|
wavelet : Wavelet object or name string, or 2-tuple of wavelets
|
||
|
|
Wavelet to use. This can also be a tuple containing a wavelet to
|
||
|
|
apply along each axis in `axes`.
|
||
|
|
level : int, optional
|
||
|
|
Decomposition level (must be >= 0). If level is None (default) then it
|
||
|
|
will be calculated using the `dwt_max_level` function.
|
||
|
|
axes : 2-tuple of ints, optional
|
||
|
|
Axes over which to compute the DWT. Repeated elements are not allowed.
|
||
|
|
Currently only available when ``transform='dwt2'``.
|
||
|
|
transform : {'dwt2', 'swt2'}
|
||
|
|
Whether to use the DWT or SWT for the transforms.
|
||
|
|
mode : str or 2-tuple of str, optional
|
||
|
|
Signal extension mode, see `Modes` (default: 'symmetric'). This option
|
||
|
|
is only used when transform='dwt2'.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
coeffs : list
|
||
|
|
For more information, see the detailed description in `wavedec2`
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
This is sometimes referred to as an additive decomposition because the
|
||
|
|
inverse transform (``imra2``) is just the sum of the coefficient arrays
|
||
|
|
[1]_. The decomposition using ``transform='dwt'`` corresponds to section
|
||
|
|
2.2 while that using an undecimated transform (``transform='swt'``) is
|
||
|
|
described in section 3.2 and appendix A.
|
||
|
|
|
||
|
|
This transform does not share the variance partition property of ``swt2``
|
||
|
|
with `norm=True`. It does however, result in coefficients that are
|
||
|
|
temporally aligned regardless of the symmetry of the wavelet used.
|
||
|
|
|
||
|
|
The redundancy of this transform is ``3 * level + 1``.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
imra2, swt2
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Donald B. Percival and Harold O. Mofjeld. Analysis of Subtidal
|
||
|
|
Coastal Sea Level Fluctuations Using Wavelets. Journal of the American
|
||
|
|
Statistical Association Vol. 92, No. 439 (Sep., 1997), pp. 868-880.
|
||
|
|
https://doi.org/10.2307/2965551
|
||
|
|
"""
|
||
|
|
if transform == 'swt2':
|
||
|
|
if mode != 'periodization':
|
||
|
|
raise ValueError(
|
||
|
|
"transform swt only supports mode='periodization'")
|
||
|
|
if level is None:
|
||
|
|
level = min(swt_max_level(s) for s in data.shape)
|
||
|
|
kwargs = dict(wavelet=wavelet, axes=axes, norm=True)
|
||
|
|
forward = partial(swt2, level=level, trim_approx=True, **kwargs)
|
||
|
|
inverse = partial(iswt2, **kwargs)
|
||
|
|
elif transform == 'dwt2':
|
||
|
|
kwargs = dict(wavelet=wavelet, mode=mode, axes=axes)
|
||
|
|
forward = partial(wavedec2, level=level, **kwargs)
|
||
|
|
inverse = partial(waverec2, **kwargs)
|
||
|
|
else:
|
||
|
|
raise ValueError("unrecognized transform: {}".format(transform))
|
||
|
|
|
||
|
|
wav_coeffs = forward(data)
|
||
|
|
|
||
|
|
mra_coeffs = []
|
||
|
|
nc = len(wav_coeffs)
|
||
|
|
z = np.zeros_like(wav_coeffs[0])
|
||
|
|
tmp = [z]
|
||
|
|
for j in range(1, nc):
|
||
|
|
tmp.append([np.zeros_like(c) for c in wav_coeffs[j]])
|
||
|
|
|
||
|
|
# tmp has arrays of zeros except for the jth entry
|
||
|
|
tmp[0] = wav_coeffs[0]
|
||
|
|
# reconstruct
|
||
|
|
rec = inverse(tmp)
|
||
|
|
if rec.shape != data.shape:
|
||
|
|
# trim any excess coefficients
|
||
|
|
rec = rec[tuple([slice(sz) for sz in data.shape])]
|
||
|
|
mra_coeffs.append(rec)
|
||
|
|
# restore zeros
|
||
|
|
tmp[0] = z
|
||
|
|
|
||
|
|
for j in range(1, nc):
|
||
|
|
dcoeffs = []
|
||
|
|
for n in range(3):
|
||
|
|
# tmp has arrays of zeros except for the jth entry
|
||
|
|
z = tmp[j][n]
|
||
|
|
tmp[j][n] = wav_coeffs[j][n]
|
||
|
|
# reconstruct
|
||
|
|
rec = inverse(tmp)
|
||
|
|
if rec.shape != data.shape:
|
||
|
|
# trim any excess coefficients
|
||
|
|
rec = rec[tuple([slice(sz) for sz in data.shape])]
|
||
|
|
dcoeffs.append(rec)
|
||
|
|
# restore zeros
|
||
|
|
tmp[j][n] = z
|
||
|
|
mra_coeffs.append(tuple(dcoeffs))
|
||
|
|
return mra_coeffs
|
||
|
|
|
||
|
|
|
||
|
|
def imra2(mra_coeffs):
|
||
|
|
"""Inverse 2D multiresolution analysis via summation.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
mra_coeffs : list
|
||
|
|
Multiresolution analysis coefficients as returned by `mra2`.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
rec : ndarray
|
||
|
|
The reconstructed signal.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
mra2
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Donald B. Percival and Harold O. Mofjeld. Analysis of Subtidal
|
||
|
|
Coastal Sea Level Fluctuations Using Wavelets. Journal of the American
|
||
|
|
Statistical Association Vol. 92, No. 439 (Sep., 1997), pp. 868-880.
|
||
|
|
https://doi.org/10.2307/2965551
|
||
|
|
"""
|
||
|
|
rec = mra_coeffs[0]
|
||
|
|
for j in range(1, len(mra_coeffs)):
|
||
|
|
for n in range(3):
|
||
|
|
rec += mra_coeffs[j][n]
|
||
|
|
return rec
|
||
|
|
|
||
|
|
|
||
|
|
def mran(data, wavelet, level=None, axes=None, transform='swtn',
|
||
|
|
mode='periodization'):
|
||
|
|
"""Forward nD multiresolution analysis.
|
||
|
|
|
||
|
|
It is a projection onto the wavelet subspaces.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
data: array_like
|
||
|
|
Input data
|
||
|
|
wavelet : Wavelet object or name string, or tuple of wavelets
|
||
|
|
Wavelet to use. This can also be a tuple containing a wavelet to
|
||
|
|
apply along each axis in `axes`.
|
||
|
|
level : int, optional
|
||
|
|
Decomposition level (must be >= 0). If level is None (default) then it
|
||
|
|
will be calculated using the `dwt_max_level` function.
|
||
|
|
axes : tuple of ints, optional
|
||
|
|
Axes over which to compute the DWT. Repeated elements are not allowed.
|
||
|
|
transform : {'dwtn', 'swtn'}
|
||
|
|
Whether to use the DWT or SWT for the transforms.
|
||
|
|
mode : str or tuple of str, optional
|
||
|
|
Signal extension mode, see `Modes` (default: 'symmetric'). This option
|
||
|
|
is only used when transform='dwtn'.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
coeffs : list
|
||
|
|
For more information, see the detailed description in `wavedecn`.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
imran, swtn
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
This is sometimes referred to as an additive decomposition because the
|
||
|
|
inverse transform (``imran``) is just the sum of the coefficient arrays
|
||
|
|
[1]_. The decomposition using ``transform='dwt'`` corresponds to section
|
||
|
|
2.2 while that using an undecimated transform (``transform='swt'``) is
|
||
|
|
described in section 3.2 and appendix A.
|
||
|
|
|
||
|
|
This transform does not share the variance partition property of ``swtn``
|
||
|
|
with `norm=True`. It does however, result in coefficients that are
|
||
|
|
temporally aligned regardless of the symmetry of the wavelet used.
|
||
|
|
|
||
|
|
The redundancy of this transform is ``(2**n - 1) * level + 1`` where ``n``
|
||
|
|
corresponds to the number of axes transformed.
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Donald B. Percival and Harold O. Mofjeld. Analysis of Subtidal
|
||
|
|
Coastal Sea Level Fluctuations Using Wavelets. Journal of the American
|
||
|
|
Statistical Association Vol. 92, No. 439 (Sep., 1997), pp. 868-880.
|
||
|
|
https://doi.org/10.2307/2965551
|
||
|
|
"""
|
||
|
|
axes, axes_shapes, ndim_transform = _prep_axes_wavedecn(data.shape, axes)
|
||
|
|
wavelets = _wavelets_per_axis(wavelet, axes)
|
||
|
|
|
||
|
|
if transform == 'swtn':
|
||
|
|
if mode != 'periodization':
|
||
|
|
raise ValueError(
|
||
|
|
"transform swt only supports mode='periodization'")
|
||
|
|
if level is None:
|
||
|
|
level = min(swt_max_level(s) for s in data.shape)
|
||
|
|
kwargs = dict(wavelet=wavelets, axes=axes, norm=True)
|
||
|
|
forward = partial(swtn, level=level, trim_approx=True, **kwargs)
|
||
|
|
inverse = partial(iswtn, **kwargs)
|
||
|
|
elif transform == 'dwtn':
|
||
|
|
modes = _modes_per_axis(mode, axes)
|
||
|
|
kwargs = dict(wavelet=wavelets, mode=modes, axes=axes)
|
||
|
|
forward = partial(wavedecn, level=level, **kwargs)
|
||
|
|
inverse = partial(waverecn, **kwargs)
|
||
|
|
else:
|
||
|
|
raise ValueError("unrecognized transform: {}".format(transform))
|
||
|
|
|
||
|
|
wav_coeffs = forward(data)
|
||
|
|
|
||
|
|
mra_coeffs = []
|
||
|
|
nc = len(wav_coeffs)
|
||
|
|
z = np.zeros_like(wav_coeffs[0])
|
||
|
|
tmp = [z]
|
||
|
|
for j in range(1, nc):
|
||
|
|
tmp.append({k: np.zeros_like(v) for k, v in wav_coeffs[j].items()})
|
||
|
|
|
||
|
|
# tmp has arrays of zeros except for the jth entry
|
||
|
|
tmp[0] = wav_coeffs[0]
|
||
|
|
# reconstruct
|
||
|
|
rec = inverse(tmp)
|
||
|
|
if rec.shape != data.shape:
|
||
|
|
# trim any excess coefficients
|
||
|
|
rec = rec[tuple([slice(sz) for sz in data.shape])]
|
||
|
|
mra_coeffs.append(rec)
|
||
|
|
# restore zeros
|
||
|
|
tmp[0] = z
|
||
|
|
|
||
|
|
for j in range(1, nc):
|
||
|
|
dcoeffs = {}
|
||
|
|
dkeys = list(wav_coeffs[j].keys())
|
||
|
|
for k in dkeys:
|
||
|
|
# tmp has arrays of zeros except for the jth entry
|
||
|
|
z = tmp[j][k]
|
||
|
|
tmp[j][k] = wav_coeffs[j][k]
|
||
|
|
# tmp[j]['a' * len(k)] = z
|
||
|
|
# reconstruct
|
||
|
|
rec = inverse(tmp)
|
||
|
|
if rec.shape != data.shape:
|
||
|
|
# trim any excess coefficients
|
||
|
|
rec = rec[tuple([slice(sz) for sz in data.shape])]
|
||
|
|
dcoeffs[k] = rec
|
||
|
|
# restore zeros
|
||
|
|
tmp[j][k] = z
|
||
|
|
# tmp[j].pop('a' * len(k))
|
||
|
|
mra_coeffs.append(dcoeffs)
|
||
|
|
return mra_coeffs
|
||
|
|
|
||
|
|
|
||
|
|
def imran(mra_coeffs):
|
||
|
|
"""Inverse nD multiresolution analysis via summation.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
mra_coeffs : list
|
||
|
|
Multiresolution analysis coefficients as returned by `mra2`.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
rec : ndarray
|
||
|
|
The reconstructed signal.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
mran
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Donald B. Percival and Harold O. Mofjeld. Analysis of Subtidal
|
||
|
|
Coastal Sea Level Fluctuations Using Wavelets. Journal of the American
|
||
|
|
Statistical Association Vol. 92, No. 439 (Sep., 1997), pp. 868-880.
|
||
|
|
https://doi.org/10.2307/2965551
|
||
|
|
"""
|
||
|
|
rec = mra_coeffs[0]
|
||
|
|
for j in range(1, len(mra_coeffs)):
|
||
|
|
for k, v in mra_coeffs[j].items():
|
||
|
|
rec += v
|
||
|
|
return rec
|