class DWT(LinearOperator):
"""One dimensional Wavelet operator.
Apply 1D-Wavelet Transform along a specific direction ``dir`` of a
multi-dimensional array of size ``dims``.
Note that the Wavelet operator is an overload of the ``pywt``
implementation of the 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
dir : :obj:`int`, optional
Direction along which DWT 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 >=0).
dtype : :obj:`str`, optional
Type of elements in input array.
Attributes
----------
shape : :obj:`tuple`
Operator shape
explicit : :obj:`bool`
Operator contains a matrix that can be solved explicitly
(True) or not (False)
Raises
------
ModuleNotFoundError
If ``pywt`` is not installed
ValueError
If ``wavelet`` does not belong to ``pywt.families``
Notes
-----
The Wavelet operator applies the multilevel Discrete Wavelet Transform
(DWT) in forward mode and the multilevel Inverse Discrete Wavelet Transform
(IDWT) in adjoint mode.
Wavelet transforms can be used to compress signals and present
a key advantage over Fourier transforms in that they captures both
frequency and location information in time. Consider using this operator
as sparsifying transform when using L1 solvers.
"""
def __init__(self, dims, dir=0, wavelet='haar', level=1, dtype='float64'):
if pywt is None:
raise ModuleNotFoundError(pywt_message)
_checkwavelet(wavelet)
if isinstance(dims, int):
dims = (dims, )
# define padding for length to be power of 2
ndimpow2 = max(2**ceil(log(dims[dir], 2)), 2**level)
pad = [(0, 0)] * len(dims)
pad[dir] = (0, ndimpow2 - dims[dir])
self.pad = Pad(dims, pad)
self.dims = dims
self.dir = dir
self.dimsd = list(dims)
self.dimsd[self.dir] = ndimpow2
# apply transform to find out slices
_, self.sl = \
pywt.coeffs_to_array(pywt.wavedecn(np.ones(self.dimsd),
wavelet=wavelet,
level=level,
mode='periodization',
axes=(self.dir,)),
axes=(self.dir,))
self.wavelet = wavelet
self.waveletadj = _adjointwavelet(wavelet)
self.level = level
self.reshape = True if len(self.dims) > 1 else False
self.shape = (int(np.prod(self.dimsd)), int(np.prod(self.dims)))
self.dtype = np.dtype(dtype)
self.explicit = False
def _matvec(self, x):
x = self.pad.matvec(x)
if self.reshape:
x = np.reshape(x, self.dimsd)
y = pywt.coeffs_to_array(pywt.wavedecn(x,
wavelet=self.wavelet,
level=self.level,
mode='periodization',
axes=(self.dir, )),
axes=(self.dir, ))[0]
return y.ravel()
def _rmatvec(self, x):
if self.reshape:
x = np.reshape(x, self.dimsd)
x = pywt.array_to_coeffs(x, self.sl, output_format='wavedecn')
y = pywt.waverecn(x,
wavelet=self.waveletadj,
mode='periodization',
axes=(self.dir, ))
y = self.pad.rmatvec(y.ravel())
return y
class Seislet(LinearOperator):
r"""Two dimensional Seislet operator.
Apply 2D-Seislet Transform to an input array given an
estimate of its local ``slopes``. In forward mode, the input array is
reshaped into a two-dimensional array of size :math:`n_x \times n_t` and
the transform is performed along the first (spatial) axis (see Notes for
more details).
Parameters
----------
slopes : :obj:`numpy.ndarray`
Slope field of size :math:`n_x \times n_t`
sampling : :obj:`tuple`, optional
Sampling steps in x- and t-axis.
level : :obj:`int`, optional
Number of scaling levels (must be >=0).
kind : :obj:`str`, optional
Basis function used for predict and update steps: ``haar`` or
``linear``.
inv : :obj:`int`, optional
Apply inverse transform when invoking the adjoint (``True``)
or not (``False``). Note that in some scenario it may be more
appropriate to use the exact inverse as adjoint of the Seislet
operator even if this is not an orthogonal operator and the dot-test
would not be satisfied (see Notes for details). Otherwise, the user
can access the inverse directly as method of this class.
dtype : :obj:`str`, optional
Type of elements in input array.
Attributes
----------
shape : :obj:`tuple`
Operator shape
explicit : :obj:`bool`
Operator contains a matrix that can be solved explicitly
(True) or not (False)
Raises
------
NotImplementedError
If ``kind`` is different from haar or linear
ValueError
If ``sampling`` has more or less than two elements.
Notes
-----
The Seislet transform [1]_ is implemented using the lifting scheme.
In its simplest form (i.e., corresponding to the Haar basis function for
the Wavelet transform) the input dataset is separated into even
(:math:`\mathbf{e}`) and odd (:math:`\mathbf{o}`) traces. Even traces are
used to forward predict the odd traces using local slopes and the
new odd traces (also referred to as residual) is defined as:
.. math::
\mathbf{o}^{i+1} = \mathbf{r}^i = \mathbf{o}^i - P(\mathbf{e}^i)
where :math:`P = P^+` is the slope-based forward prediction operator
(which is here implemented as a sinc-based resampling).
The residual is then updated and summed to the even traces to obtain the
new even traces (also referred to as coarse representation):
.. math::
\mathbf{e}^{i+1} = \mathbf{c}^i = \mathbf{e}^i + U(\mathbf{o}^{i+1})
where :math:`U = P^- / 2` is the update operator which performs a
slope-based backward prediction. At this point
:math:`\mathbf{e}^{i+1}` becomes the new data and the procedure is repeated
`level` times (at maximum until :math:`\mathbf{e}^{i+1}` is a single trace.
The Seislet transform is effectively composed of all residuals and
the coarsest data representation.
In the inverse transform the two operations are reverted. Starting from the
coarsest scale data representation :math:`\mathbf{c}` and residual
:math:`\mathbf{r}`, the even and odd parts of the previous scale are
reconstructed as:
.. math::
\mathbf{e}^i = \mathbf{c}^i - U(\mathbf{r}^i)
= \mathbf{e}^{i+1} - U(\mathbf{o}^{i+1})
and:
.. math::
\mathbf{o}^i = \mathbf{r}^i + P(\mathbf{e}^i)
= \mathbf{o}^{i+1} + P(\mathbf{e}^i)
A new data is formed by interleaving :math:`\mathbf{e}^i` and
:math:`\mathbf{o}^i` and the procedure repeated until the new data as the
same number of traces as the original one.
Finally the adjoint operator can be easily derived by writing the lifting
scheme in a matricial form:
.. math::
\begin{bmatrix}
\mathbf{r}_1 \\ \mathbf{r}_2 \\ ... \\ \mathbf{r}_N \\
\mathbf{c}_1 \\ \mathbf{c}_2 \\ ... \\ \mathbf{c}_N
\end{bmatrix} =
\begin{bmatrix}
\mathbf{I} & \mathbf{0} & ... & \mathbf{0} & -\mathbf{P} & \mathbf{0} & ... & \mathbf{0} \\
\mathbf{0} & \mathbf{I} & ... & \mathbf{0} & \mathbf{0} & -\mathbf{P} & ... & \mathbf{0} \\
... & ... & ... & ... & ... & ... & ... & ... \\
\mathbf{0} & \mathbf{0} & ... & \mathbf{I} & \mathbf{0} & \mathbf{0} & ... & -\mathbf{P} \\
\mathbf{U} & \mathbf{0} & ... & \mathbf{0} & \mathbf{I}-\mathbf{UP} & \mathbf{0} & ... & \mathbf{0} \\
\mathbf{0} & \mathbf{U} & ... & \mathbf{0} & \mathbf{0} & \mathbf{I}-\mathbf{UP} & ... & \mathbf{0} \\
... & ... & ... & ... & ... & ... & ... & ... \\
\mathbf{0} & \mathbf{0} & ... & \mathbf{U} & \mathbf{0} & \mathbf{0} & ... & \mathbf{I}-\mathbf{UP} \\
\end{bmatrix}
\begin{bmatrix}
\mathbf{o}_1 \\ \mathbf{o}_2 \\ ... \\ \mathbf{o}_N \\
\mathbf{e}_1 \\ \mathbf{e}_2 \\ ... \\ \mathbf{e}_N \\
\end{bmatrix}
Transposing the operator leads to:
.. math::
\begin{bmatrix}
\mathbf{o}_1 \\ \mathbf{o}_2 \\ ... \\ \mathbf{o}_N \\
\mathbf{e}_1 \\ \mathbf{e}_2 \\ ... \\ \mathbf{e}_N \\
\end{bmatrix} =
\begin{bmatrix}
\mathbf{I} & \mathbf{0} & ... & \mathbf{0} & -\mathbf{U^T} & \mathbf{0} & ... & \mathbf{0} \\
\mathbf{0} & \mathbf{I} & ... & \mathbf{0} & \mathbf{0} & -\mathbf{U^T} & ... & \mathbf{0} \\
... & ... & ... & ... & ... & ... & ... & ... \\
\mathbf{0} & \mathbf{0} & ... & \mathbf{I} & \mathbf{0} & \mathbf{0} & ... & -\mathbf{U^T} \\
\mathbf{P^T} & \mathbf{0} & ... & \mathbf{0} & \mathbf{I}-\mathbf{P^TU^T} & \mathbf{0} & ... & \mathbf{0} \\
\mathbf{0} & \mathbf{P^T} & ... & \mathbf{0} & \mathbf{0} & \mathbf{I}-\mathbf{P^TU^T} & ... & \mathbf{0} \\
... & ... & ... & ... & ... & ... & ... & ... \\
\mathbf{0} & \mathbf{0} & ... & \mathbf{P^T} & \mathbf{0} & \mathbf{0} & ... & \mathbf{I}-\mathbf{P^TU^T} \\
\end{bmatrix}
\begin{bmatrix}
\mathbf{r}_1 \\ \mathbf{r}_2 \\ ... \\ \mathbf{r}_N \\
\mathbf{c}_1 \\ \mathbf{c}_2 \\ ... \\ \mathbf{c}_N
\end{bmatrix}
which can be written more easily in the following two steps:
.. math::
\mathbf{o} = \mathbf{r} + \mathbf{U}^H\mathbf{c}
and:
.. math::
\mathbf{e} = \mathbf{c} - \mathbf{P}^H(\mathbf{r} + \mathbf{U}^H(\mathbf{c})) =
\mathbf{c} - \mathbf{P}^H\mathbf{o}
Similar derivations follow for more complex wavelet bases.
.. [1] Fomel, S., Liu, Y., "Seislet transform and seislet frame",
Geophysics, 75, no. 3, V25-V38. 2010.
"""
def __init__(self,
slopes,
sampling=(1., 1.),
level=None,
kind='haar',
inv=False,
dtype='float64'):
if len(sampling) != 2:
raise ValueError('provide two sampling steps')
# define predict and update steps
if kind == 'haar':
self.predict = _predict_haar
elif kind == 'linear':
self.predict = _predict_lin
else:
raise NotImplementedError('kind should be haar or linear')
# define padding for length to be power of 2
dims = slopes.shape
ndimpow2 = 2**ceil(log(dims[0], 2))
pad = [(0, 0)] * len(dims)
pad[0] = (0, ndimpow2 - dims[0])
self.pad = Pad(dims, pad)
self.dims = list(dims)
self.dims[0] = ndimpow2
self.nx, self.nt = self.dims
# define levels
nlevels_max = int(np.log2(self.dims[0]))
self.levels_size = np.flip(np.array([2**i
for i in range(nlevels_max)]))
if level is not None:
self.levels_size = self.levels_size[:level]
else:
self.levels_size = self.levels_size[:-1]
level = nlevels_max - 1
self.level = level
self.levels_cum = np.cumsum(self.levels_size)
self.levels_cum = np.insert(self.levels_cum, 0, 0)
self.dx, self.dt = sampling
self.slopes = (self.pad * slopes.ravel()).reshape(self.dims)
self.inv = inv
self.shape = (int(np.prod(self.slopes.size)),
int(np.prod(slopes.size)))
self.dtype = np.dtype(dtype)
self.explicit = False
def _matvec(self, x):
x = self.pad.matvec(x)
x = np.reshape(x, self.dims)
y = np.zeros(
(np.sum(self.levels_size) + self.levels_size[-1], self.nt))
for ilevel in range(self.level):
odd = x[1::2]
even = x[::2]
res = odd - self.predict(even,
self.dt,
self.dx,
self.slopes,
repeat=ilevel,
backward=False)
x = even + self.predict(res,
self.dt,
self.dx,
self.slopes,
repeat=ilevel,
backward=True) / 2.
y[self.levels_cum[ilevel]:self.levels_cum[ilevel + 1]] = res
y[self.levels_cum[-1]:] = x
return y.ravel()
def _rmatvec(self, x):
if not self.inv:
x = np.reshape(x, self.dims)
y = x[self.levels_cum[-1]:]
for ilevel in range(self.level, 0, -1):
res = x[self.levels_cum[ilevel - 1]:self.levels_cum[ilevel]]
odd = res + self.predict(y,
self.dt,
self.dx,
self.slopes,
repeat=ilevel - 1,
backward=True,
adj=True) / 2.
even = y - self.predict(odd,
self.dt,
self.dx,
self.slopes,
repeat=ilevel - 1,
backward=False,
adj=True)
y = np.zeros((2 * even.shape[0], self.nt))
y[1::2] = odd
y[::2] = even
y = self.pad.rmatvec(y.ravel())
else:
y = self.inverse(x)
return y
def inverse(self, x):
x = np.reshape(x, self.dims)
y = x[self.levels_cum[-1]:]
for ilevel in range(self.level, 0, -1):
res = x[self.levels_cum[ilevel - 1]:self.levels_cum[ilevel]]
even = y - self.predict(res,
self.dt,
self.dx,
self.slopes,
repeat=ilevel - 1,
backward=True) / 2.
odd = res + self.predict(even,
self.dt,
self.dx,
self.slopes,
repeat=ilevel - 1,
backward=False)
y = np.zeros((2 * even.shape[0], self.nt))
y[1::2] = odd
y[::2] = even
y = self.pad.rmatvec(y.ravel())
return y
class DWT2D(LinearOperator):
"""Two dimensional Wavelet operator.
Apply 2D-Wavelet Transform along two directions ``dirs`` of a
multi-dimensional array of size ``dims``.
Note that the Wavelet operator is an overload of the ``pywt``
implementation of the wavelet transform. Refer to
https://pywavelets.readthedocs.io for a detailed description of the
input parameters.
Parameters
----------
dims : :obj:`tuple`
Number of samples for each dimension
dirs : :obj:`tuple`, optional
Direction along which DWT2D 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 >=0).
dtype : :obj:`str`, optional
Type of elements in input array.
Attributes
----------
shape : :obj:`tuple`
Operator shape
explicit : :obj:`bool`
Operator contains a matrix that can be solved explicitly
(``True``) or not (``False``)
Raises
------
ModuleNotFoundError
If ``pywt`` is not installed
ValueError
If ``wavelet`` does not belong to ``pywt.families``
Notes
-----
The Wavelet operator applies the 2-dimensional multilevel Discrete
Wavelet Transform (DWT2) in forward mode and the 2-dimensional multilevel
Inverse Discrete Wavelet Transform (IDWT2) in adjoint mode.
"""
def __init__(self,
dims,
dirs=(0, 1),
wavelet="haar",
level=1,
dtype="float64"):
if pywt is None:
raise ModuleNotFoundError("The wavelet operator requires "
"the pywt package t be installed. "
'Run "pip install PyWavelets" or '
'"conda install pywavelets".')
_checkwavelet(wavelet)
# define padding for length to be power of 2
ndimpow2 = [max(2**ceil(log(dims[dir], 2)), 2**level) for dir in dirs]
pad = [(0, 0)] * len(dims)
for i, dir in enumerate(dirs):
pad[dir] = (0, ndimpow2[i] - dims[dir])
self.pad = Pad(dims, pad)
self.dims = dims
self.dirs = dirs
self.dimsd = list(dims)
for i, dir in enumerate(dirs):
self.dimsd[dir] = ndimpow2[i]
# apply transform once again to find out slices
_, self.sl = pywt.coeffs_to_array(
pywt.wavedec2(
np.ones(self.dimsd),
wavelet=wavelet,
level=level,
mode="periodization",
axes=self.dirs,
),
axes=self.dirs,
)
self.wavelet = wavelet
self.waveletadj = _adjointwavelet(wavelet)
self.level = level
self.shape = (int(np.prod(self.dimsd)), int(np.prod(self.dims)))
self.dtype = np.dtype(dtype)
self.explicit = False
def _matvec(self, x):
x = self.pad.matvec(x)
x = np.reshape(x, self.dimsd)
y = pywt.coeffs_to_array(
pywt.wavedec2(
x,
wavelet=self.wavelet,
level=self.level,
mode="periodization",
axes=self.dirs,
),
axes=(self.dirs),
)[0]
return y.ravel()
def _rmatvec(self, x):
x = np.reshape(x, self.dimsd)
x = pywt.array_to_coeffs(x, self.sl, output_format="wavedec2")
y = pywt.waverec2(x,
wavelet=self.waveletadj,
mode="periodization",
axes=self.dirs)
y = self.pad.rmatvec(y.ravel())
return y