775 lines
29 KiB
Python
775 lines
29 KiB
Python
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import warnings
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from itertools import product
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import numpy as np
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from ._c99_config import _have_c99_complex
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from ._extensions._dwt import idwt_single
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from ._extensions._swt import swt_max_level, swt as _swt, swt_axis as _swt_axis
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from ._extensions._pywt import Wavelet, Modes, _check_dtype
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from ._multidim import idwt2, idwtn
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from ._utils import _as_wavelet, _wavelets_per_axis
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__all__ = ["swt", "swt_max_level", 'iswt', 'swt2', 'iswt2', 'swtn', 'iswtn']
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def _rescale_wavelet_filterbank(wavelet, sf):
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wav = Wavelet(wavelet.name + 'r',
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[np.asarray(f) * sf for f in wavelet.filter_bank])
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# copy attributes from the original wavelet
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wav.orthogonal = wavelet.orthogonal
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wav.biorthogonal = wavelet.biorthogonal
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return wav
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def swt(data, wavelet, level=None, start_level=0, axis=-1,
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trim_approx=False, norm=False):
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"""
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Multilevel 1D stationary wavelet transform.
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Parameters
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----------
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data :
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Input signal
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wavelet :
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Wavelet to use (Wavelet object or name)
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level : int, optional
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The number of decomposition steps to perform.
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start_level : int, optional
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The level at which the decomposition will begin (it allows one to
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skip a given number of transform steps and compute
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coefficients starting from start_level) (default: 0)
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axis: int, optional
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Axis over which to compute the SWT. If not given, the
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last axis is used.
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trim_approx : bool, optional
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If True, approximation coefficients at the final level are retained.
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norm : bool, optional
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If True, transform is normalized so that the energy of the coefficients
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will be equal to the energy of ``data``. In other words,
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``np.linalg.norm(data.ravel())`` will equal the norm of the
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concatenated transform coefficients when ``trim_approx`` is True.
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Returns
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-------
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coeffs : list
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List of approximation and details coefficients pairs in order
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similar to wavedec function::
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[(cAn, cDn), ..., (cA2, cD2), (cA1, cD1)]
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where n equals input parameter ``level``.
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If ``start_level = m`` is given, then the beginning m steps are
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skipped::
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[(cAm+n, cDm+n), ..., (cAm+1, cDm+1), (cAm, cDm)]
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If ``trim_approx`` is ``True``, then the output list is exactly as in
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``pywt.wavedec``, where the first coefficient in the list is the
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approximation coefficient at the final level and the rest are the
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detail coefficients::
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[cAn, cDn, ..., cD2, cD1]
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Notes
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-----
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The implementation here follows the "algorithm a-trous" and requires that
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the signal length along the transformed axis be a multiple of ``2**level``.
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If this is not the case, the user should pad up to an appropriate size
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using a function such as ``numpy.pad``.
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A primary benefit of this transform in comparison to its decimated
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counterpart (``pywt.wavedecn``), is that it is shift-invariant. This comes
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at cost of redundancy in the transform (the size of the output coefficients
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is larger than the input).
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When the following three conditions are true:
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1. The wavelet is orthogonal
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2. ``swt`` is called with ``norm=True``
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3. ``swt`` is called with ``trim_approx=True``
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the transform has the following additional properties that may be
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desirable in applications:
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1. energy is conserved
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2. variance is partitioned across scales
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When used with ``norm=True``, this transform is closely related to the
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multiple-overlap DWT (MODWT) as popularized for time-series analysis,
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although the underlying implementation is slightly different from the one
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published in [1]_. Specifically, the implementation used here requires a
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signal that is a multiple of ``2**level`` in length.
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References
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----------
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.. [1] DB Percival and AT Walden. Wavelet Methods for Time Series Analysis.
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Cambridge University Press, 2000.
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"""
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if not _have_c99_complex and np.iscomplexobj(data):
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data = np.asarray(data)
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coeffs_real = swt(data.real, wavelet, level, start_level, trim_approx)
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coeffs_imag = swt(data.imag, wavelet, level, start_level, trim_approx)
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if not trim_approx:
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coeffs_cplx = []
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for (cA_r, cD_r), (cA_i, cD_i) in zip(coeffs_real, coeffs_imag):
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coeffs_cplx.append((cA_r + 1j*cA_i, cD_r + 1j*cD_i))
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else:
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coeffs_cplx = [cr + 1j*ci
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for (cr, ci) in zip(coeffs_real, coeffs_imag)]
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return coeffs_cplx
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# accept array_like input; make a copy to ensure a contiguous array
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dt = _check_dtype(data)
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data = np.array(data, dtype=dt)
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wavelet = _as_wavelet(wavelet)
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if norm:
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if not wavelet.orthogonal:
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warnings.warn(
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"norm=True, but the wavelet is not orthogonal: \n"
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"\tThe conditions for energy preservation are not satisfied.")
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wavelet = _rescale_wavelet_filterbank(wavelet, 1/np.sqrt(2))
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if axis < 0:
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axis = axis + data.ndim
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if not 0 <= axis < data.ndim:
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raise ValueError("Axis greater than data dimensions")
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if level is None:
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level = swt_max_level(data.shape[axis])
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if data.ndim == 1:
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ret = _swt(data, wavelet, level, start_level, trim_approx)
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else:
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ret = _swt_axis(data, wavelet, level, start_level, axis, trim_approx)
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return ret
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def iswt(coeffs, wavelet, norm=False):
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"""
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Multilevel 1D inverse discrete stationary wavelet transform.
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Parameters
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----------
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coeffs : array_like
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Coefficients list of tuples::
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[(cAn, cDn), ..., (cA2, cD2), (cA1, cD1)]
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where cA is approximation, cD is details. Index 1 corresponds to
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``start_level`` from ``pywt.swt``.
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wavelet : Wavelet object or name string
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Wavelet to use
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norm : bool, optional
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Controls the normalization used by the inverse transform. This must
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be set equal to the value that was used by ``pywt.swt`` to preserve the
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energy of a round-trip transform.
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Returns
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-------
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1D array of reconstructed data.
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Examples
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--------
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>>> import pywt
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>>> coeffs = pywt.swt([1,2,3,4,5,6,7,8], 'db2', level=2)
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>>> pywt.iswt(coeffs, 'db2')
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array([ 1., 2., 3., 4., 5., 6., 7., 8.])
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"""
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# copy to avoid modification of input data
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# If swt was called with trim_approx=False, first element is a tuple
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trim_approx = not isinstance(coeffs[0], (tuple, list))
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if trim_approx:
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cA = coeffs[0]
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coeffs = coeffs[1:]
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else:
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cA = coeffs[0][0]
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dt = _check_dtype(cA)
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output = np.array(cA, dtype=dt, copy=True)
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if not _have_c99_complex and np.iscomplexobj(output):
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# compute real and imaginary separately then combine
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if trim_approx:
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coeffs_real = [c.real for c in coeffs]
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coeffs_imag = [c.imag for c in coeffs]
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else:
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coeffs_real = [(cA.real, cD.real) for (cA, cD) in coeffs]
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coeffs_imag = [(cA.imag, cD.imag) for (cA, cD) in coeffs]
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return iswt(coeffs_real, wavelet) + 1j*iswt(coeffs_imag, wavelet)
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# num_levels, equivalent to the decomposition level, n
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num_levels = len(coeffs)
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wavelet = _as_wavelet(wavelet)
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if norm:
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wavelet = _rescale_wavelet_filterbank(wavelet, np.sqrt(2))
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mode = Modes.from_object('periodization')
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for j in range(num_levels, 0, -1):
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step_size = int(pow(2, j-1))
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last_index = step_size
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if trim_approx:
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cD = coeffs[-j]
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else:
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_, cD = coeffs[-j]
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cD = np.asarray(cD, dtype=_check_dtype(cD))
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if cD.dtype != output.dtype:
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# upcast to a common dtype (float64 or complex128)
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if output.dtype.kind == 'c' or cD.dtype.kind == 'c':
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dtype = np.complex128
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else:
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dtype = np.float64
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output = np.asarray(output, dtype=dtype)
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cD = np.asarray(cD, dtype=dtype)
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for first in range(last_index): # 0 to last_index - 1
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# Getting the indices that we will transform
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indices = np.arange(first, len(cD), step_size)
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# select the even indices
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even_indices = indices[0::2]
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# select the odd indices
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odd_indices = indices[1::2]
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# perform the inverse dwt on the selected indices,
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# making sure to use periodic boundary conditions
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# Note: indexing with an array of ints returns a contiguous
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# copy as required by idwt_single.
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x1 = idwt_single(output[even_indices],
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cD[even_indices],
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wavelet, mode)
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x2 = idwt_single(output[odd_indices],
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cD[odd_indices],
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wavelet, mode)
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# perform a circular shift right
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x2 = np.roll(x2, 1)
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# average and insert into the correct indices
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output[indices] = (x1 + x2)/2.
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return output
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def swt2(data, wavelet, level, start_level=0, axes=(-2, -1),
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trim_approx=False, norm=False):
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"""
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Multilevel 2D stationary wavelet transform.
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Parameters
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----------
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data : array_like
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2D array with input data
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wavelet : Wavelet object or name string, or 2-tuple of wavelets
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Wavelet to use. This can also be a tuple of wavelets to apply per
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axis in ``axes``.
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level : int
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The number of decomposition steps to perform.
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start_level : int, optional
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The level at which the decomposition will start (default: 0)
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axes : 2-tuple of ints, optional
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Axes over which to compute the SWT. Repeated elements are not allowed.
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trim_approx : bool, optional
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If True, approximation coefficients at the final level are retained.
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norm : bool, optional
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If True, transform is normalized so that the energy of the coefficients
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will be equal to the energy of ``data``. In other words,
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``np.linalg.norm(data.ravel())`` will equal the norm of the
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concatenated transform coefficients when ``trim_approx`` is True.
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Returns
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-------
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coeffs : list
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Approximation and details coefficients (for ``start_level = m``).
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If ``trim_approx`` is ``True``, approximation coefficients are
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retained for all levels::
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[
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(cA_m+level,
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(cH_m+level, cV_m+level, cD_m+level)
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),
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...,
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(cA_m+1,
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(cH_m+1, cV_m+1, cD_m+1)
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),
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(cA_m,
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(cH_m, cV_m, cD_m)
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)
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]
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where cA is approximation, cH is horizontal details, cV is
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vertical details, cD is diagonal details and m is ``start_level``.
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If ``trim_approx`` is ``False``, approximation coefficients are only
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retained at the final level of decomposition. This matches the format
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used by ``pywt.wavedec2``::
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[
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cA_m+level,
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(cH_m+level, cV_m+level, cD_m+level),
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...,
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(cH_m+1, cV_m+1, cD_m+1),
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(cH_m, cV_m, cD_m),
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]
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Notes
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-----
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The implementation here follows the "algorithm a-trous" and requires that
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the signal length along the transformed axes be a multiple of ``2**level``.
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If this is not the case, the user should pad up to an appropriate size
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using a function such as ``numpy.pad``.
|
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A primary benefit of this transform in comparison to its decimated
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counterpart (``pywt.wavedecn``), is that it is shift-invariant. This comes
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at cost of redundancy in the transform (the size of the output coefficients
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is larger than the input).
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When the following three conditions are true:
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1. The wavelet is orthogonal
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2. ``swt2`` is called with ``norm=True``
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3. ``swt2`` is called with ``trim_approx=True``
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the transform has the following additional properties that may be
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desirable in applications:
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1. energy is conserved
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2. variance is partitioned across scales
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"""
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axes = tuple(axes)
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data = np.asarray(data)
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if len(axes) != 2:
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raise ValueError("Expected 2 axes")
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if len(axes) != len(set(axes)):
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raise ValueError("The axes passed to swt2 must be unique.")
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if data.ndim < len(np.unique(axes)):
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raise ValueError("Input array has fewer dimensions than the specified "
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"axes")
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coefs = swtn(data, wavelet, level, start_level, axes, trim_approx, norm)
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ret = []
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if trim_approx:
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ret.append(coefs[0])
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coefs = coefs[1:]
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for c in coefs:
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if trim_approx:
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ret.append((c['da'], c['ad'], c['dd']))
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else:
|
||
|
|
ret.append((c['aa'], (c['da'], c['ad'], c['dd'])))
|
||
|
|
return ret
|
||
|
|
|
||
|
|
|
||
|
|
def iswt2(coeffs, wavelet, norm=False):
|
||
|
|
"""
|
||
|
|
Multilevel 2D inverse discrete stationary wavelet transform.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
coeffs : list
|
||
|
|
Approximation and details coefficients::
|
||
|
|
|
||
|
|
[
|
||
|
|
(cA_n,
|
||
|
|
(cH_n, cV_n, cD_n)
|
||
|
|
),
|
||
|
|
...,
|
||
|
|
(cA_2,
|
||
|
|
(cH_2, cV_2, cD_2)
|
||
|
|
),
|
||
|
|
(cA_1,
|
||
|
|
(cH_1, cV_1, cD_1)
|
||
|
|
)
|
||
|
|
]
|
||
|
|
|
||
|
|
where cA is approximation, cH is horizontal details, cV is
|
||
|
|
vertical details, cD is diagonal details and n is the number of
|
||
|
|
levels. Index 1 corresponds to ``start_level`` from ``pywt.swt2``.
|
||
|
|
wavelet : Wavelet object or name string, or 2-tuple of wavelets
|
||
|
|
Wavelet to use. This can also be a 2-tuple of wavelets to apply per
|
||
|
|
axis.
|
||
|
|
norm : bool, optional
|
||
|
|
Controls the normalization used by the inverse transform. This must
|
||
|
|
be set equal to the value that was used by ``pywt.swt2`` to preserve
|
||
|
|
the energy of a round-trip transform.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
2D array of reconstructed data.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import pywt
|
||
|
|
>>> coeffs = pywt.swt2([[1,2,3,4],[5,6,7,8],
|
||
|
|
... [9,10,11,12],[13,14,15,16]],
|
||
|
|
... 'db1', level=2)
|
||
|
|
>>> pywt.iswt2(coeffs, 'db1')
|
||
|
|
array([[ 1., 2., 3., 4.],
|
||
|
|
[ 5., 6., 7., 8.],
|
||
|
|
[ 9., 10., 11., 12.],
|
||
|
|
[ 13., 14., 15., 16.]])
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
# If swt was called with trim_approx=False, first element is a tuple
|
||
|
|
trim_approx = not isinstance(coeffs[0], (tuple, list))
|
||
|
|
if trim_approx:
|
||
|
|
cA = coeffs[0]
|
||
|
|
coeffs = coeffs[1:]
|
||
|
|
else:
|
||
|
|
cA = coeffs[0][0]
|
||
|
|
|
||
|
|
# copy to avoid modification of input data
|
||
|
|
dt = _check_dtype(cA)
|
||
|
|
output = np.array(cA, dtype=dt, copy=True)
|
||
|
|
|
||
|
|
if output.ndim != 2:
|
||
|
|
raise ValueError(
|
||
|
|
"iswt2 only supports 2D arrays. see iswtn for a general "
|
||
|
|
"n-dimensionsal ISWT")
|
||
|
|
# num_levels, equivalent to the decomposition level, n
|
||
|
|
num_levels = len(coeffs)
|
||
|
|
wavelets = _wavelets_per_axis(wavelet, axes=(0, 1))
|
||
|
|
if norm:
|
||
|
|
wavelets = [_rescale_wavelet_filterbank(wav, np.sqrt(2))
|
||
|
|
for wav in wavelets]
|
||
|
|
|
||
|
|
for j in range(num_levels):
|
||
|
|
step_size = int(pow(2, num_levels-j-1))
|
||
|
|
last_index = step_size
|
||
|
|
if trim_approx:
|
||
|
|
(cH, cV, cD) = coeffs[j]
|
||
|
|
else:
|
||
|
|
_, (cH, cV, cD) = coeffs[j]
|
||
|
|
# We are going to assume cH, cV, and cD are of equal size
|
||
|
|
if (cH.shape != cV.shape) or (cH.shape != cD.shape):
|
||
|
|
raise RuntimeError(
|
||
|
|
"Mismatch in shape of intermediate coefficient arrays")
|
||
|
|
|
||
|
|
# make sure output shares the common dtype
|
||
|
|
# (conversion of dtype for individual coeffs is handled within idwt2 )
|
||
|
|
common_dtype = np.result_type(*(
|
||
|
|
[dt, ] + [_check_dtype(c) for c in [cH, cV, cD]]))
|
||
|
|
if output.dtype != common_dtype:
|
||
|
|
output = output.astype(common_dtype)
|
||
|
|
|
||
|
|
for first_h in range(last_index): # 0 to last_index - 1
|
||
|
|
for first_w in range(last_index): # 0 to last_index - 1
|
||
|
|
# Getting the indices that we will transform
|
||
|
|
indices_h = slice(first_h, cH.shape[0], step_size)
|
||
|
|
indices_w = slice(first_w, cH.shape[1], step_size)
|
||
|
|
|
||
|
|
even_idx_h = slice(first_h, cH.shape[0], 2*step_size)
|
||
|
|
even_idx_w = slice(first_w, cH.shape[1], 2*step_size)
|
||
|
|
odd_idx_h = slice(first_h + step_size, cH.shape[0], 2*step_size)
|
||
|
|
odd_idx_w = slice(first_w + step_size, cH.shape[1], 2*step_size)
|
||
|
|
|
||
|
|
# perform the inverse dwt on the selected indices,
|
||
|
|
# making sure to use periodic boundary conditions
|
||
|
|
x1 = idwt2((output[even_idx_h, even_idx_w],
|
||
|
|
(cH[even_idx_h, even_idx_w],
|
||
|
|
cV[even_idx_h, even_idx_w],
|
||
|
|
cD[even_idx_h, even_idx_w])),
|
||
|
|
wavelets, 'periodization')
|
||
|
|
x2 = idwt2((output[even_idx_h, odd_idx_w],
|
||
|
|
(cH[even_idx_h, odd_idx_w],
|
||
|
|
cV[even_idx_h, odd_idx_w],
|
||
|
|
cD[even_idx_h, odd_idx_w])),
|
||
|
|
wavelets, 'periodization')
|
||
|
|
x3 = idwt2((output[odd_idx_h, even_idx_w],
|
||
|
|
(cH[odd_idx_h, even_idx_w],
|
||
|
|
cV[odd_idx_h, even_idx_w],
|
||
|
|
cD[odd_idx_h, even_idx_w])),
|
||
|
|
wavelets, 'periodization')
|
||
|
|
x4 = idwt2((output[odd_idx_h, odd_idx_w],
|
||
|
|
(cH[odd_idx_h, odd_idx_w],
|
||
|
|
cV[odd_idx_h, odd_idx_w],
|
||
|
|
cD[odd_idx_h, odd_idx_w])),
|
||
|
|
wavelets, 'periodization')
|
||
|
|
|
||
|
|
# perform a circular shifts
|
||
|
|
x2 = np.roll(x2, 1, axis=1)
|
||
|
|
x3 = np.roll(x3, 1, axis=0)
|
||
|
|
x4 = np.roll(x4, 1, axis=0)
|
||
|
|
x4 = np.roll(x4, 1, axis=1)
|
||
|
|
output[indices_h, indices_w] = (x1 + x2 + x3 + x4) / 4
|
||
|
|
|
||
|
|
return output
|
||
|
|
|
||
|
|
|
||
|
|
def swtn(data, wavelet, level, start_level=0, axes=None, trim_approx=False,
|
||
|
|
norm=False):
|
||
|
|
"""
|
||
|
|
n-dimensional stationary wavelet transform.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
data : array_like
|
||
|
|
n-dimensional array with input data.
|
||
|
|
wavelet : Wavelet object or name string, or tuple of wavelets
|
||
|
|
Wavelet to use. This can also be a tuple of wavelets to apply per
|
||
|
|
axis in ``axes``.
|
||
|
|
level : int
|
||
|
|
The number of decomposition steps to perform.
|
||
|
|
start_level : int, optional
|
||
|
|
The level at which the decomposition will start (default: 0)
|
||
|
|
axes : sequence of ints, optional
|
||
|
|
Axes over which to compute the SWT. A value of ``None`` (the
|
||
|
|
default) selects all axes. Axes may not be repeated.
|
||
|
|
trim_approx : bool, optional
|
||
|
|
If True, approximation coefficients at the final level are retained.
|
||
|
|
norm : bool, optional
|
||
|
|
If True, transform is normalized so that the energy of the coefficients
|
||
|
|
will be equal to the energy of ``data``. In other words,
|
||
|
|
``np.linalg.norm(data.ravel())`` will equal the norm of the
|
||
|
|
concatenated transform coefficients when ``trim_approx`` is True.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
[{coeffs_level_n}, ..., {coeffs_level_1}]: list of dict
|
||
|
|
Results for each level are arranged in a dictionary, where the key
|
||
|
|
specifies the transform type on each dimension and value is a
|
||
|
|
n-dimensional coefficients array.
|
||
|
|
|
||
|
|
For example, for a 2D case the result at a given level will look
|
||
|
|
something like this::
|
||
|
|
|
||
|
|
{'aa': <coeffs> # A(LL) - approx. on 1st dim, approx. on 2nd dim
|
||
|
|
'ad': <coeffs> # V(LH) - approx. on 1st dim, det. on 2nd dim
|
||
|
|
'da': <coeffs> # H(HL) - det. on 1st dim, approx. on 2nd dim
|
||
|
|
'dd': <coeffs> # D(HH) - det. on 1st dim, det. on 2nd dim
|
||
|
|
}
|
||
|
|
|
||
|
|
For user-specified ``axes``, the order of the characters in the
|
||
|
|
dictionary keys map to the specified ``axes``.
|
||
|
|
|
||
|
|
If ``trim_approx`` is ``True``, the first element of the list contains
|
||
|
|
the array of approximation coefficients from the final level of
|
||
|
|
decomposition, while the remaining coefficient dictionaries contain
|
||
|
|
only detail coefficients. This matches the behavior of `pywt.wavedecn`.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The implementation here follows the "algorithm a-trous" and requires that
|
||
|
|
the signal length along the transformed axes be a multiple of ``2**level``.
|
||
|
|
If this is not the case, the user should pad up to an appropriate size
|
||
|
|
using a function such as ``numpy.pad``.
|
||
|
|
|
||
|
|
A primary benefit of this transform in comparison to its decimated
|
||
|
|
counterpart (``pywt.wavedecn``), is that it is shift-invariant. This comes
|
||
|
|
at cost of redundancy in the transform (the size of the output coefficients
|
||
|
|
is larger than the input).
|
||
|
|
|
||
|
|
When the following three conditions are true:
|
||
|
|
|
||
|
|
1. The wavelet is orthogonal
|
||
|
|
2. ``swtn`` is called with ``norm=True``
|
||
|
|
3. ``swtn`` is called with ``trim_approx=True``
|
||
|
|
|
||
|
|
the transform has the following additional properties that may be
|
||
|
|
desirable in applications:
|
||
|
|
|
||
|
|
1. energy is conserved
|
||
|
|
2. variance is partitioned across scales
|
||
|
|
|
||
|
|
"""
|
||
|
|
data = np.asarray(data)
|
||
|
|
if not _have_c99_complex and np.iscomplexobj(data):
|
||
|
|
real = swtn(data.real, wavelet, level, start_level, axes, trim_approx)
|
||
|
|
imag = swtn(data.imag, wavelet, level, start_level, axes, trim_approx)
|
||
|
|
if trim_approx:
|
||
|
|
cplx = [real[0] + 1j * imag[0]]
|
||
|
|
offset = 1
|
||
|
|
else:
|
||
|
|
cplx = []
|
||
|
|
offset = 0
|
||
|
|
for rdict, idict in zip(real[offset:], imag[offset:]):
|
||
|
|
cplx.append(
|
||
|
|
dict((k, rdict[k] + 1j * idict[k]) for k in rdict.keys()))
|
||
|
|
return cplx
|
||
|
|
|
||
|
|
if data.dtype == np.dtype('object'):
|
||
|
|
raise TypeError("Input must be a numeric array-like")
|
||
|
|
if data.ndim < 1:
|
||
|
|
raise ValueError("Input data must be at least 1D")
|
||
|
|
|
||
|
|
if axes is None:
|
||
|
|
axes = range(data.ndim)
|
||
|
|
axes = [a + data.ndim if a < 0 else a for a in axes]
|
||
|
|
if len(axes) != len(set(axes)):
|
||
|
|
raise ValueError("The axes passed to swtn must be unique.")
|
||
|
|
num_axes = len(axes)
|
||
|
|
|
||
|
|
wavelets = _wavelets_per_axis(wavelet, axes)
|
||
|
|
if norm:
|
||
|
|
if not np.all([wav.orthogonal for wav in wavelets]):
|
||
|
|
warnings.warn(
|
||
|
|
"norm=True, but the wavelets used are not orthogonal: \n"
|
||
|
|
"\tThe conditions for energy preservation are not satisfied.")
|
||
|
|
wavelets = [_rescale_wavelet_filterbank(wav, 1/np.sqrt(2))
|
||
|
|
for wav in wavelets]
|
||
|
|
ret = []
|
||
|
|
for i in range(start_level, start_level + level):
|
||
|
|
coeffs = [('', data)]
|
||
|
|
for axis, wavelet in zip(axes, wavelets):
|
||
|
|
new_coeffs = []
|
||
|
|
for subband, x in coeffs:
|
||
|
|
cA, cD = _swt_axis(x, wavelet, level=1, start_level=i,
|
||
|
|
axis=axis)[0]
|
||
|
|
new_coeffs.extend([(subband + 'a', cA),
|
||
|
|
(subband + 'd', cD)])
|
||
|
|
coeffs = new_coeffs
|
||
|
|
|
||
|
|
coeffs = dict(coeffs)
|
||
|
|
ret.append(coeffs)
|
||
|
|
|
||
|
|
# data for the next level is the approximation coeffs from this level
|
||
|
|
data = coeffs['a' * num_axes]
|
||
|
|
if trim_approx:
|
||
|
|
coeffs.pop('a' * num_axes)
|
||
|
|
if trim_approx:
|
||
|
|
ret.append(data)
|
||
|
|
ret.reverse()
|
||
|
|
return ret
|
||
|
|
|
||
|
|
|
||
|
|
def iswtn(coeffs, wavelet, axes=None, norm=False):
|
||
|
|
"""
|
||
|
|
Multilevel nD inverse discrete stationary wavelet transform.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
coeffs : list
|
||
|
|
[{coeffs_level_n}, ..., {coeffs_level_1}]: list of dict
|
||
|
|
wavelet : Wavelet object or name string, or tuple of wavelets
|
||
|
|
Wavelet to use. This can also be a tuple of wavelets to apply per
|
||
|
|
axis in ``axes``.
|
||
|
|
axes : sequence of ints, optional
|
||
|
|
Axes over which to compute the inverse SWT. Axes may not be repeated.
|
||
|
|
The default is ``None``, which means transform all axes
|
||
|
|
(``axes = range(data.ndim)``).
|
||
|
|
norm : bool, optional
|
||
|
|
Controls the normalization used by the inverse transform. This must
|
||
|
|
be set equal to the value that was used by ``pywt.swtn`` to preserve
|
||
|
|
the energy of a round-trip transform.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
nD array of reconstructed data.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> import pywt
|
||
|
|
>>> coeffs = pywt.swtn([[1,2,3,4],[5,6,7,8],
|
||
|
|
... [9,10,11,12],[13,14,15,16]],
|
||
|
|
... 'db1', level=2)
|
||
|
|
>>> pywt.iswtn(coeffs, 'db1')
|
||
|
|
array([[ 1., 2., 3., 4.],
|
||
|
|
[ 5., 6., 7., 8.],
|
||
|
|
[ 9., 10., 11., 12.],
|
||
|
|
[ 13., 14., 15., 16.]])
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
# key length matches the number of axes transformed
|
||
|
|
ndim_transform = max(len(key) for key in coeffs[-1].keys())
|
||
|
|
|
||
|
|
trim_approx = not isinstance(coeffs[0], dict)
|
||
|
|
if trim_approx:
|
||
|
|
cA = coeffs[0]
|
||
|
|
coeffs = coeffs[1:]
|
||
|
|
else:
|
||
|
|
cA = coeffs[0]['a'*ndim_transform]
|
||
|
|
|
||
|
|
# copy to avoid modification of input data
|
||
|
|
dt = _check_dtype(cA)
|
||
|
|
output = np.array(cA, dtype=dt, copy=True)
|
||
|
|
ndim = output.ndim
|
||
|
|
|
||
|
|
if axes is None:
|
||
|
|
axes = range(output.ndim)
|
||
|
|
axes = [a + ndim if a < 0 else a for a in axes]
|
||
|
|
if len(axes) != len(set(axes)):
|
||
|
|
raise ValueError("The axes passed to swtn must be unique.")
|
||
|
|
if ndim_transform != len(axes):
|
||
|
|
raise ValueError("The number of axes used in iswtn must match the "
|
||
|
|
"number of dimensions transformed in swtn.")
|
||
|
|
|
||
|
|
# num_levels, equivalent to the decomposition level, n
|
||
|
|
num_levels = len(coeffs)
|
||
|
|
wavelets = _wavelets_per_axis(wavelet, axes)
|
||
|
|
if norm:
|
||
|
|
wavelets = [_rescale_wavelet_filterbank(wav, np.sqrt(2))
|
||
|
|
for wav in wavelets]
|
||
|
|
|
||
|
|
# initialize various slice objects used in the loops below
|
||
|
|
# these will remain slice(None) only on axes that aren't transformed
|
||
|
|
indices = [slice(None), ]*ndim
|
||
|
|
even_indices = [slice(None), ]*ndim
|
||
|
|
odd_indices = [slice(None), ]*ndim
|
||
|
|
odd_even_slices = [slice(None), ]*ndim
|
||
|
|
|
||
|
|
for j in range(num_levels):
|
||
|
|
step_size = int(pow(2, num_levels-j-1))
|
||
|
|
last_index = step_size
|
||
|
|
if not trim_approx:
|
||
|
|
a = coeffs[j].pop('a'*ndim_transform) # will restore later
|
||
|
|
details = coeffs[j]
|
||
|
|
# make sure dtype matches the coarsest level approximation coefficients
|
||
|
|
common_dtype = np.result_type(*(
|
||
|
|
[dt, ] + [v.dtype for v in details.values()]))
|
||
|
|
if output.dtype != common_dtype:
|
||
|
|
output = output.astype(common_dtype)
|
||
|
|
|
||
|
|
# We assume all coefficient arrays are of equal size
|
||
|
|
shapes = [v.shape for k, v in details.items()]
|
||
|
|
if len(set(shapes)) != 1:
|
||
|
|
raise RuntimeError(
|
||
|
|
"Mismatch in shape of intermediate coefficient arrays")
|
||
|
|
|
||
|
|
# shape of a single coefficient array, excluding non-transformed axes
|
||
|
|
coeff_trans_shape = tuple([shapes[0][ax] for ax in axes])
|
||
|
|
|
||
|
|
# nested loop over all combinations of axis offsets at this level
|
||
|
|
for firsts in product(*([range(last_index), ]*ndim_transform)):
|
||
|
|
for first, sh, ax in zip(firsts, coeff_trans_shape, axes):
|
||
|
|
indices[ax] = slice(first, sh, step_size)
|
||
|
|
even_indices[ax] = slice(first, sh, 2*step_size)
|
||
|
|
odd_indices[ax] = slice(first+step_size, sh, 2*step_size)
|
||
|
|
|
||
|
|
# nested loop over all combinations of odd/even inidices
|
||
|
|
approx = output.copy()
|
||
|
|
output[tuple(indices)] = 0
|
||
|
|
ntransforms = 0
|
||
|
|
for odds in product(*([(0, 1), ]*ndim_transform)):
|
||
|
|
for o, ax in zip(odds, axes):
|
||
|
|
if o:
|
||
|
|
odd_even_slices[ax] = odd_indices[ax]
|
||
|
|
else:
|
||
|
|
odd_even_slices[ax] = even_indices[ax]
|
||
|
|
# extract the odd/even indices for all detail coefficients
|
||
|
|
details_slice = {}
|
||
|
|
for key, value in details.items():
|
||
|
|
details_slice[key] = value[tuple(odd_even_slices)]
|
||
|
|
details_slice['a'*ndim_transform] = approx[
|
||
|
|
tuple(odd_even_slices)]
|
||
|
|
|
||
|
|
# perform the inverse dwt on the selected indices,
|
||
|
|
# making sure to use periodic boundary conditions
|
||
|
|
x = idwtn(details_slice, wavelets, 'periodization', axes=axes)
|
||
|
|
for o, ax in zip(odds, axes):
|
||
|
|
# circular shift along any odd indexed axis
|
||
|
|
if o:
|
||
|
|
x = np.roll(x, 1, axis=ax)
|
||
|
|
output[tuple(indices)] += x
|
||
|
|
ntransforms += 1
|
||
|
|
output[tuple(indices)] /= ntransforms # normalize
|
||
|
|
if not trim_approx:
|
||
|
|
coeffs[j]['a'*ndim_transform] = a # restore approx coeffs to dict
|
||
|
|
return output
|