pylops.utils.signalprocessing.slope_estimate

pylops.utils.signalprocessing.slope_estimate(d, dz, dx, smooth=20)

Local slope estimation

Local slopes are estimated using the Structure Tensor algorithm [1]. Note that slopes are returned as \(arctan(\theta)\) where \(\theta\) is an angle defined in a RHS coordinate system with z axis pointing upward.

Parameters:

d : np.ndarray

Input dataset of size \(n_z \times n_x\)

dz : float

Sampling in z-axis

dx : float

Sampling in x-axis

smooth : float, optional

Lenght of smoothing filter to be applied to the estimated gradients

Returns:

slopes : np.ndarray

Estimated local slopes

linearity : np.ndarray

Estimated linearity

Notes

For each pixel of the input dataset \(\mathbf{d}\) the local gradients \(d \mathbf{d} / dz\) and \(g_z = d\mathbf{d} \ dx\) are computed and used to define the following three quantities: \(g_{zz} = (d\mathbf{d} / dz) ^ 2\), \(g_{xx} = (d\mathbf{d} / dx) ^ 2\), and \(g_{zx} = d\mathbf{d} / dz * d\mathbf{d} / dx\). Such quantities are spatially smoothed and at each pixel their smoothed versions are arranged in a \(2 \times 2\) matrix called the smoothed gradient-square tensor:

\[\begin{split}\mathbf{G} = \begin{bmatrix} g_{zz} & g_{zx} \\ g_{zx} & g_{xx} \end{bmatrix}\end{split}\]

Local slopes can be expressed as \(p = arctan(\frac{\lambda_{max} - g_{xx}}{g_{zx}})\).

[1]	Van Vliet, L. J., Verbeek, P. W., “Estimators for orientation and anisotropy in digitized images”, Journal ASCI Imaging Workshop. 1995.

Examples using pylops.utils.signalprocessing.slope_estimate

Seislet transform

Slope estimation via Structure Tensor algorithm