import numpy as np
from scipy.ndimage import gaussian_filter
from pylops.utils.backend import get_array_module, get_toeplitz
[docs]def convmtx(h, n):
r"""Convolution matrix
Equivalent of `MATLAB's convmtx function
<http://www.mathworks.com/help/signal/ref/convmtx.html>`_ .
Makes a dense convolution matrix :math:`\mathbf{C}`
such that the dot product ``np.dot(C, x)`` is the convolution of
the filter :math:`h` and the input signal :math:`x`.
Parameters
----------
h : :obj:`np.ndarray`
Convolution filter (1D array)
n : :obj:`int`
Number of columns (if :math:`\text{len}(h) < n`) or rows
(if :math:`\text{len}(h) \geq n`) of convolution matrix
Returns
-------
C : :obj:`np.ndarray`
Convolution matrix of size :math:`\text{len}(h)+n-1 \times n`
(if :math:`\text{len}(h) < n`) or :math:`n \times \text{len}(h)+n-1`
(if :math:`\text{len}(h) \geq n`)
"""
ncp = get_array_module(h)
if len(h) < n:
col_1 = ncp.r_[h[0], ncp.zeros(n - 1, dtype=h.dtype)]
row_1 = ncp.r_[h, ncp.zeros(n - 1, dtype=h.dtype)]
else:
row_1 = ncp.r_[h[0], ncp.zeros(n - 1, dtype=h.dtype)]
col_1 = ncp.r_[h, ncp.zeros(n - 1, dtype=h.dtype)]
C = get_toeplitz(h)(col_1, row_1)
return C
[docs]def nonstationary_convmtx(H, n, hc=0, pad=(0, 0)):
r"""Convolution matrix from a bank of filters
Makes a dense convolution matrix :math:`\mathbf{C}`
such that the dot product ``np.dot(C, x)`` is the nonstationary
convolution of the bank of filters :math:`H=[h_1, h_2, h_n]`
and the input signal :math:`x`.
Parameters
----------
H : :obj:`np.ndarray`
Convolution filters (2D array of shape
:math:`[n_\text{filters} \times n_{h}]`
n : :obj:`int`
Number of columns of convolution matrix
hc : :obj:`np.ndarray`, optional
Index of center of first filter
pad : :obj:`np.ndarray`
Zero-padding to apply to the bank of filters before and after the
provided values (use it to avoid wrap-around or pass filters with
enough padding)
Returns
-------
C : :obj:`np.ndarray`
Convolution matrix
"""
ncp = get_array_module(H)
H = ncp.pad(H, ((0, 0), pad), mode="constant")
C = ncp.array([ncp.roll(h, ih) for ih, h in enumerate(H)])
C = C[:, pad[0] + hc : pad[0] + hc + n].T # take away edges
return C
[docs]def slope_estimate(d, dz=1.0, dx=1.0, smooth=5, eps=0):
r"""Local slope estimation
Local slopes are estimated using the *Structure Tensor* algorithm [1]_.
Slopes are returned as :math:`\tan\theta`, defined
in a RHS coordinate system with :math:`z`-axis pointing upward.
.. note:: For stability purposes, it is important to ensure that the orders
of magnitude of the samplings are similar.
Parameters
----------
d : :obj:`np.ndarray`
Input dataset of size :math:`n_z \times n_x`
dz : :obj:`float`
Sampling in :math:`z`-axis, :math:`\Delta z`
.. warning::
Since version 1.17.0, defaults to 1.0.
dx : :obj:`float`
Sampling in :math:`x`-axis, :math:`\Delta x`
.. warning::
Since version 1.17.0, defaults to 1.0.
smooth : :obj:`float`, optional
Standard deviation for Gaussian kernel. The standard deviations of the
Gaussian filter are given for each axis as a sequence, or as a single number,
in which case it is equal for all axes.
.. warning::
Default changed in version 1.17.0 to 5 from previous value of 20.
eps : :obj:`float`, optional
.. versionadded:: 1.17.0
Regularization term. All slopes where :math:`|g_{zx}| < \epsilon \max |g_{zx}|`
are set to zero. All anisotropies where :math:`\lambda_\text{max} < \epsilon`
are also set to zero. See Notes. When using with small values of ``smooth``,
start from a very small number (e.g. 1e-10) and start increasing by a power
of 10 until results are satisfactory.
Returns
-------
slopes : :obj:`np.ndarray`
Estimated local slopes. Unit is that of :math:`\Delta z/\Delta x`.
.. warning::
Prior to version 1.17.0, erroneously returned angles in radians instead of
slopes.
anisotropies : :obj:`np.ndarray`
Estimated local anisotropies: :math:`1-\lambda_\text{min}/\lambda_\text{max}`
.. note::
Since 1.17.0, changed name from ``linearity`` to ``anisotropies``.
Definition remains the same.
Notes
-----
For each pixel of the input dataset :math:`\mathbf{d}` the local gradients
:math:`g_z = \frac{\partial \mathbf{d}}{\partial z}` and
:math:`g_x = \frac{\partial \mathbf{d}}{\partial x}` are computed
and used to define the following three quantities:
.. math::
\begin{align}
g_{zz} &= \left(\frac{\partial \mathbf{d}}{\partial z}\right)^2\\
g_{xx} &= \left(\frac{\partial \mathbf{d}}{\partial x}\right)^2\\
g_{zx} &= \frac{\partial \mathbf{d}}{\partial z}\cdot\frac{\partial \mathbf{d}}{\partial x}
\end{align}
They are then spatially smoothed and at each pixel their smoothed versions are
arranged in a :math:`2 \times 2` matrix called the *smoothed
gradient-square tensor*:
.. math::
\mathbf{G} =
\begin{bmatrix}
g_{zz} & g_{zx} \\
g_{zx} & g_{xx}
\end{bmatrix}
Local slopes can be expressed as
:math:`p = \frac{\lambda_\text{max} - g_{zz}}{g_{zx}}`,
where :math:`\lambda_\text{max}` is the largest eigenvalue of :math:`\mathbf{G}`.
Moreover, we can obtain a measure of local anisotropy, defined as
.. math::
a = 1-\lambda_\text{min}/\lambda_\text{max}
where :math:`\lambda_\text{min}` is the smallest eigenvalue of :math:`\mathbf{G}`.
A value of :math:`a = 0` indicates perfect isotropy whereas :math:`a = 1`
indicates perfect anisotropy.
.. [1] Van Vliet, L. J., Verbeek, P. W., "Estimators for orientation and
anisotropy in digitized images", Journal ASCI Imaging Workshop. 1995.
"""
slopes = np.zeros_like(d)
anisos = np.zeros_like(d)
gz, gx = np.gradient(d, dz, dx)
gzz, gzx, gxx = gz * gz, gz * gx, gx * gx
# smoothing
gzz = gaussian_filter(gzz, sigma=smooth)
gzx = gaussian_filter(gzx, sigma=smooth)
gxx = gaussian_filter(gxx, sigma=smooth)
gmax = np.max(np.abs(gzx))
if gmax == 0.0:
return slopes, anisos
gzz /= gmax
gzx /= gmax
gxx /= gmax
lcommon1 = 0.5 * (gzz + gxx)
lcommon2 = 0.5 * np.sqrt((gzz - gxx) ** 2 + 4 * gzx ** 2)
l1 = lcommon1 + lcommon2
l2 = lcommon1 - lcommon2
regdata = np.abs(gzx) > eps
slopes[regdata] = (l1 - gzz)[regdata] / gzx[regdata]
regdata = np.abs(l1) > eps
anisos[regdata] = 1 - l2[regdata] / l1[regdata]
return slopes, anisos