__all__ = ["SWT"]
import warnings
from math import ceil
import numpy as np
from pylops import LinearOperator
from pylops.basicoperators import Pad
from pylops.utils import deps
from pylops.utils._internal import _value_or_sized_to_tuple
from pylops.utils.typing import DTypeLike, InputDimsLike, NDArray
from .dwt import _adjointwavelet, _checklevel, _checkwavelet
pywt_message = deps.pywt_import("the swt module")
if pywt_message is None:
import pywt
[docs]
class SWT(LinearOperator):
"""One dimensional Stationary Wavelet operator.
Apply 1D-Stationary Wavelet Transform along an ``axis`` of a
multi-dimensional array of size ``dims``.
Note that the Stationary Wavelet operator is an overload of the ``pywt``
implementation of the stationary wavelet transform. Refer to
https://pywavelets.readthedocs.io for a detailed description of the
input parameters.
Parameters
----------
dims : :obj:`int` or :obj:`tuple`
Number of samples for each dimension
axis : :obj:`int`, optional
Axis along which SWT is applied
wavelet : :obj:`str`, optional
Name of wavelet type. Use :func:`pywt.wavelist(kind='discrete')` for
a list of available wavelets.
level : :obj:`int`, optional
Number of scaling levels (must be >=1).
dtype : :obj:`str`, optional
Type of elements in input array.
name : :obj:`str`, optional
Name of operator (to be used by :func:`pylops.utils.describe.describe`)
Attributes
----------
pad : :obj:`pylops.basicoperators.Pad`
Padding operator used to pad the input signal to the next multiple
of ``2**level`` length.
waveletadj : :obj:`str`
Name of the adjoint wavelet type.
dims : :obj:`tuple`
Shape of the array after the adjoint, but before flattening.
For example, ``x_reshaped = (Op.H * y.ravel()).reshape(Op.dims)``.
dimsd : :obj:`tuple`
Shape of the array after the forward, but before flattening.
For example, ``y_reshaped = (Op * x.ravel()).reshape(Op.dimsd)``.
shape : :obj:`tuple`
Operator shape.
Raises
------
ModuleNotFoundError
If ``pywt`` is not installed
ValueError
If ``wavelet`` does not belong to ``pywt.families``
ValueError
If ``level`` is smaller than 1
Notes
-----
The Stationary Wavelet operator applies the multilevel Stationary Wavelet
Transform (SWT) in forward mode and the multilevel Inverse Stationary
Wavelet Transform (ISWT) in adjoint mode.
Contrarily to the Discrete Wavelet Transform (see
:class:`pylops.signalprocessing.DWT`), the SWT does not decimate the
coefficients at each level; as a result, the transform is
shift-invariant but redundant. All coefficients have the same size of
the (padded) input signal and are stacked along a new leading axis
of the output, i.e., ``dimsd = (level + 1, *dims)`` in the order
``[cA_n, cD_n, ..., cD_2, cD_1]``.
The transform is computed with ``trim_approx=True`` and ``norm=True``
such that the adjoint is exactly given by the ISWT (with the dual
wavelet in the case of biorthogonal wavelets). For orthogonal wavelets,
the operator is also a tight frame, i.e.,
:math:`\\mathbf{S}^H \\mathbf{S} = \\mathbf{I}`.
"""
def __init__(
self,
dims: int | InputDimsLike,
axis: int = -1,
wavelet: str = "haar",
level: int = 1,
dtype: DTypeLike = "float64",
name: str = "S",
) -> None:
if pywt_message is not None:
raise ModuleNotFoundError(pywt_message)
_checkwavelet(wavelet)
_checklevel(level, minlevel=1)
dims = _value_or_sized_to_tuple(dims)
# define padding for length to be multiple of 2**level
ndimpad = ceil(dims[axis] / 2**level) * 2**level
pad = [(0, 0)] * len(dims)
pad[axis] = (0, ndimpad - dims[axis])
self.pad = Pad(dims, pad)
self.axis = axis
self.dimspad = list(dims)
self.dimspad[self.axis] = ndimpad
dimsd = [level + 1] + self.dimspad
super().__init__(dtype=np.dtype(dtype), dims=dims, dimsd=dimsd, name=name)
self.wavelet = wavelet
self.waveletadj = _adjointwavelet(wavelet)
self.level = level
def _matvec(self, x: NDArray) -> NDArray:
x = self.pad.matvec(x)
x = np.reshape(x, self.dimspad)
# norm=True is required for the adjoint to be exact, but pywt warns
# that energy is not preserved for non-orthogonal wavelets: this
# does not affect the adjointness, so the warning is suppressed
with warnings.catch_warnings():
warnings.simplefilter("ignore", UserWarning)
y = pywt.swt(
x,
wavelet=self.wavelet,
level=self.level,
axis=self.axis,
trim_approx=True,
norm=True,
)
y = np.stack(y)
return y.ravel()
def _rmatvec(self, x: NDArray) -> NDArray:
x = np.reshape(x, self.dimsd)
# suppress pywt warning for norm=True with non-orthogonal wavelets
# (see _matvec)
with warnings.catch_warnings():
warnings.simplefilter("ignore", UserWarning)
y = pywt.iswt(list(x), wavelet=self.waveletadj, norm=True, axis=self.axis)
y = self.pad.rmatvec(y.ravel())
return y
PyLops