Height correlation#

Structure factor#

The structure factor is defined as:

\[S(\vec{q}) = \langle \hat{u}(\vec{q}) \hat{u}(\vec{q}) \rangle\]

where \(\vec{q}\) is the angular frequency. It can be used to compute the roughness exponent \(\zeta\):

\[S(|q|) \sim |q|^{-(d + 2 \zeta)}\]

where \(d\) is the dimension of the image. An example is shown below.

_images/structure.svg _images/structure-dark.svg

Height-height correlation#

The height-height correlation corresponds to the following:

\[\mathcal{P} (\Delta \vec{x}) = \sqrt{ \sum_{i} \; \left( z (\vec{x}_i + \Delta \vec{x}) - z (\vec{x}_i) \right)^2 }\]

Example#

_images/heightheight.svg

Note

Like for the 2-point correlation, a mask can be used. Similarly, the average can be extended to that of an ensemble of images.

heightheight.py

import GooseEYE
import numpy as np

L = 2.0 * np.pi
N = 1000
h = L / N
x = np.linspace(0, L, N)
y1 = np.sin(x)
y2 = np.sin(2.0 * x)

hh1 = GooseEYE.heightheight(roi=[200], f=y1, periodic=True)
hh2 = GooseEYE.heightheight(roi=[200], f=y2, periodic=True)
dx = GooseEYE.distance(roi=[200], h=[h], dim=0)