SurfaceTopography.Generic package

Submodules

SurfaceTopography.Generic.Curvature module

Scale-dependent slope

SurfaceTopography.Generic.Curvature.scale_dependent_curvature_from_profile(topography, **kwargs)

Compute the one-dimensional scale-dependent curvature.

The scale-dependent curvature is given by

\begin{equation} h_\text{rms}^{\prime\prime}(\lambda) = \left[8A(\lambda) - 2A(2\lambda)\right]^{1/2}/\lambda^2 \end{equation}

where \(A(\lambda)\) is the autocorrelation function.

Parameters:

topography (SurfaceTopography.Topography or SurfaceTopography.UniformLineScan) – Container storing the uniform topography map
**kwargs (dict) – Additional keyword parameters are passed on to autocorrelation_from_profile

Returns:

r (array) – Distances. (Units: length)
curvature (array) – Curvature. (Units: 1/length)

SurfaceTopography.Generic.Curvature.scale_dependent_curvature_from_area(topography, **kwargs)

Compute the two-dimensional, radially averaged scale-dependent curvature.

The scale-dependent curvature is given by

\begin{equation} h_\text{rms}^{\prime\prime}(\lambda) = 4\left[8A(\lambda/2) - 2A(\lambda)\right]^{1/2}/\lambda^2 \end{equation}

where \(A(\lambda)\) is the autocorrelation function.

Parameters:

topography (SurfaceTopography or UniformLineScan) – Container storing the uniform topography map
**kwargs (dict) – Additional keyword parameters are passed on to autocorrelation_from_area

Returns:

r (array) – Distances. (Units: length)
curvature (array) – Curvature. (Units: 1/length)

SurfaceTopography.Generic.Fractional module

SurfaceTopography.Generic.Fractional.container_variance_half_derivative_from_autocorrelation(container, scale_dependent=False, **kwargs)

The scale-dependence is introduced by considering only the ACF below the distance scale.

This is the definition of scale-dependent elastic energy as in Wang and Müser equation 44

see container.autocorrelation concerning further arguments

Returns:

if scale_dependent
r (array of floats)
distance scale
variance_half_derivative (array of floats)

SurfaceTopography.Generic.Moments module

Computation of moments of a discrete curve, e.g. the PSD

The example below computes the PSD of a surface topography and integrates it to obtain the RMS height.

>>> from SurfaceTopography.Generation import fourier_synthesis
>>> t = fourier_synthesis((256, 256), (256,256), hurst=0.8, rms_height=1, short_cutoff=4, long_cutoff=64)
>>> hrms = t.rms_height_from_profile()
>>> hrms_from_psd = np.sqrt(compute_1d_moment(,)
>>> assert abs(hrms_from_psd / hrms - 1)  < 0.1

However, because the averaging of the PSD before integration introduces errors, it is better to sum directly the raw spectum, see the methods integrate_psd of the topographies and the container.

SurfaceTopography.Generic.Moments.compute_1d_moment(x, y, order=1, cumulative=False)

Computes the moment of order \(\alpha\)

\[m_\alpha = \int dx y x^{\alpha}\]

using trapezoidal interpolation of the integrand

Parameters:

x: np.ndarray: sample points
y: np.ndarray: values of the function sampled at x
order: float, default=1: order of the moment
cumulative: bool, default=True: if True returns the cumulative value of this integral for each x (uses cumulative_trapezoid instead of trapezoid)

returns:

If cumulative is False (Default)
integral (float)
If cumulative is True
integral (np.ndarray of length)

SurfaceTopography.Generic.Moments.compute_iso_moment(x, y, order=1, cumulative=False)

Computes the moment of order \(\alpha\) of the 2D isotropic funtion \(y_{2D}(x)\)

\(y\) is the 1D representation of the isotropic function \(y_{2D}(x_1, x_2) = y(|\vec x|)\)

\[m_\alpha = \int dx_1 dx_2 y_{2D}(x_1, x_2) |\vec x|^{\alpha} = 2\pi \int dx y(x) x^{\alpha + 1}\]

using trapezoidal interpolation of the integrand

Parameters:

x: np.ndarray: sample points
y: np.ndarray: values of the function sampled at x
order: float, default=1: order of the moment
cumulative: bool, default=True: if True returns the cumulative value of this integral for each x (uses cumulative_trapezoid instead of trapezoid)

returns:

If cumulative is False (Default)
integral (float)
If cumulative is True
integral (np.ndarray of length)

SurfaceTopography.Generic.ReliabilityCutoff module

Automatic determination of the reliability of the underyling data

SurfaceTopography.Generic.ReliabilityCutoff.short_reliability_cutoff(self, other_cutoff=None)

Determine down to which distance scale the data is reliable, i.e. below which distance it may be affected by instrumental artifacts. Currently supported are tip radius artifacts and a user specified resolution.

Returns:: short_cutoff – All data below this length scale is unreliable. For tip radius analysis, this is the length of the stencil for the second derivative used to compute the reliability cutoff.
Return type:: float

SurfaceTopography.Generic.RobustStatistics module

Robust statistics (median, median absolute deviation, etc.) for topography data.

SurfaceTopography.Generic.RobustStatistics.bisect(bearing_area, target_area, rtol=1e-06)

Find the root of a function using the bisection method.

Parameters:

bearing_area (UniformBearingArea or NonuniformBearingArea) – Instance of bearing area class.
target_area (float) – Target bearing area.
rtol (float) – Relative tolerance for the root with respect to the initial bounds.

Returns:

root – Height where the bearing area is equal to target_area.

Return type:

float

SurfaceTopography.Generic.RobustStatistics.median(self)

Compute the median height of the topography.

Parameters:: self (HeightContainer) – Topography or line scan container object.
Returns:: median – Median height.
Return type:: float

SurfaceTopography.Generic.RobustStatistics.mad_height(self, percentile=np.float64(0.15865525393145707))

Compute the median-absolute-deviation of the height.

Parameters:

self (HeightContainer) – Topography or line scan container object.
percentile (float) – Fraction of the bearing area that should be considered for the deviation. (Default: One standard deviation)

Returns:

mad_height – Median absolute deviation of the height.

Return type:

float

SurfaceTopography.Generic.RobustStatistics.mad_polyfit(self, nb_coeffs)

Find the polynomial that minimizes the median absolute deviation.

Parameters:

self (HeightContainer) – Topography or line scan container object.
nb_coeffs (int) – Number of coefficients to be fitted.

SurfaceTopography.Generic.ScaleDependentStatistics module

SurfaceTopography.Generic.ScanningProbe module

Analysis functions related to scanning-probe microscopy

SurfaceTopography.Generic.Slope module

Scale-dependent slope

SurfaceTopography.Generic.Slope.scale_dependent_slope_from_profile(topography, **kwargs)

Compute the one-dimensional scale-dependent slope.

The scale-dependent slope is given by

\begin{equation} h_\text{rms}^\prime(\lambda) = \left[2A(\lambda)\right]^{1/2}/\lambda \end{equation}

where \(A(\lambda)\) is the autocorrelation function.

Parameters:

topography (SurfaceTopography.Topography or SurfaceTopography.UniformLineScan) – Container storing the uniform topography map
**kwargs (dict) – Additional keyword parameters are passed on to autocorrelation_from_profile

Returns:

r (array) – Distances. (Units: length)
slope (array) – Slope. (Units: dimensionless)

SurfaceTopography.Generic.Slope.scale_dependent_slope_from_area(topography, **kwargs)

Compute the two-dimensional, radially averaged scale-dependent slope.

The scale-dependent slope is given by

\begin{equation} h_\text{rms}^\prime(\lambda) = \left[2A(\lambda)\right]^{1/2}/\lambda \end{equation}

where \(A(\lambda)\) is the autocorrelation function.

Parameters:

topography (SurfaceTopography or UniformLineScan) – Container storing the uniform topography map
**kwargs (dict) – Additional keyword parameters are passed on to autocorrelation_from_area

Returns:

r (array) – Distances. (Units: length)
slope (array) – Slope. (Units: dimensionless)

Module contents

Module containing generic functions that work for both uniform and nonuniform topographies