The simplest statistical functions are the height and slope distribution functions. These can be computed as non-cumulative (i.e. densities) or cumulative. These functions are computed as normalized histograms of the height or slope (obtained as dreivatives in the selected direction – horizontal or vertical) values. In other words, the quantity on the abscissa in “angle distribution” is the tangent of the angle, not the angle itself.

The normalization of the densities ρ(x) (where x is the corresponding quantity, height or slope) is such that

Evidently, the scale of the values is then independent on the number of data points and the number of histogram buckets. The cumulative distributions are integrals of the densities and they have values from interval [0, 1].

The height and slope distribution quantities belong to the first-order statistical quantities, describing only the statistical properties of the individual points. However, for the complete description of the surface properties it is necessary to study higher order functions. Usually, second-order statistical quantities observing mutual relationship of two points on the surface are employed. These functions are namely the autocorrelation function, the height-height correlation function, and the power spectral density function. A description of each of these follows:

The autocorrelation function is given by

where z₁ and z₂ are concrete values of heights at points [x₁, y₁], [x₂, y₂], τ_x =  x₁ −  x₂ and τ_y =  y₁ −  y₂. The function w(z₁, z₂, τ_x, τ_y) denotes the two-dimensional probability density of the random function ξ > (x, y) corresponding to points [x₁, y₁], [x₂, y₂] and the distance between these points τ.

From the discrete AFM data one can evaluate this function as

where m = τ_x/Δ_x, n = τ_y/Δ_y. The function can thus be evaluated in a discrete set of values of τ_x and τ_y separated by the sampling intervals Δ_x and Δ_y, respectively. The two-dimensional autocorrelation function can be calculated with Data Process → Statistics → 2D Autocorrelation.

For AFM measurements, we usually evaluate the one-dimensional autocorrelation function based only on profiles along the fast scanning axis. It can therefore be evaluated from the discrete AFM data values as

The one-dimensional autocorrelation function is often assumed to have the form of a Gaussian, i.e. it can be given by the following relation

where σ denotes the root mean square deviation of the heights and T denotes the autocorrelation length.

For the exponential autocorrelation function we have the following relation

The difference between the height-height correlation function and the autocorrelation function is very small. As with the autocorrelation function, we sum the multiplication of two different values. For the autocorrelation function, these values represented the different distances between points. For the height-height correlation function, we instead use the power of difference between the points.

For AFM measurements, we usually evaluate the one-dimensional height-height correlation function based only on profiles along the fast scanning axis. It can therefore be evaluated from the discrete AFM data values as

where m = τ_x/Δ. The function thus can be evaluated in a discrete set of values of τ separated by the sampling interval Δ.

The one-dimensional height-height correlation function is often assumed to be Gaussian, i.e. given by the following relation

where σ denotes the root mean square deviation of the heights and T denotes the autocorrelation length.

For the exponential height-height correlation function we have the following relation

In the following figure the height-height correlation function and autocorrelation function obtained for a simulated surface having Gaussian autocorrelation function are plotted. These functions are fitted by means of least-squares method using the formulae shown above. The resulting values of σ and T were practically same for both the approaches.

Figure 4.15.  Height-height correlation function and autocorrelation function obtained for simulated surface having Gaussian autocorrelation function.

The two-dimensional power spectral density function can be written in terms of the Fourier transform of the autocorrelation function

Similarly to the autocorrelation function, we also usually evaluate the one-dimensional power spectral density function which is given by the equation

This function can be evaluated by means of the Fast Fourier Transform as follows:

where P_j(K_x) is the Fourier coefficient of the j-th row, i.e.

If we have chosen the Gaussian ACF, the corresponding Gaussian relation for the PSDF is

For the surface with exponential ACF we have

In the following figure the resulting PSDF and its fit for the same surface as used in the previous figure are plotted. We can see that the function can be again fitted by Gaussian PSDF. The resulting values of σ and T were practically same as those from the HHCF and ACF fit.

Figure 4.16.  PSDF obtained for data simulated with Gaussian autocorrelation function. Points represent computed data, line represents its fit.

We can also introduce radial PSDF W_r(K), which of course contains the same information as the one-dimensional PSDF for isotropic rough surfaces:

For a surface with Gaussian ACF this function is expressed as

while for exponential ACF surface as

The Minkowski functionals are used to describe global geometric characteristics of structures. Two-dimensional discrete variants of volume V, surface S, and connectivity (Euler-Poincaré Characteristic) χ are calculated according to following formulas:

Here N denotes the total number of pixels, N_white denotes the number of “white” pixels, that is pixels above threshold (pixels below threshold are referred to as “black”). The symbol N_bound denotes the number of white-black pixel boundaries. Finally, C_white and C_black denote the number of continuous sets of white and black pixels respectively.

For an image with continuous set of values the functionals are parametrized by the height threshold value  ϑ that divides white pixels from black, that is they can be viewed as functions of this parameter. And these functions V(ϑ), S(ϑ), and χ(ϑ) are what is actually plotted.