Moment based quantities are expressed using integrals of the height distribution function with some powers of height. They include the familiar quantities:

More precisely, RMS (σ), skewness (γ₁), and kurtosis (γ₂) are computed from central moments of i-th order μ_i

where z̄ denotes the mean value. The are expressed by the following formulas:

Note that Gwyddion computes the excess kurtosis, which is zero for Gaussian data distribution. Add 3 to obtain the area texture parameter Sku.

Mean roughness S_a is similar to RMS value, the difference being that it is calculated from the sum of absolute values of data differences from the mean, instead of their squares.

It should be noted that quantities such as σ calculated using the standard definitions are biased, especially if any scan line levelling has been done. The Correlation Length tool can calculate bias-corrected quantities. See its description for more details.

Order based quantities represent values corresponding to specific ranks if all the values were ordered. They include:

Hybrid quantities combine heights and spatial relations in the surface and include:

The surface area is estimated by the following method. Let z_i for i = 1, 2, 3, 4 denote values in four neighbour points (pixel centres), and h_x and h_y pixel dimensions along corresponding axes. If an additional point is placed in the centre of the rectangle which corresponds to the common corner of the four pixels (using the mean value of the pixels), four triangles are formed and the surface area can be approximated by summing their areas. This leads to the following formulas for the area of one triangle (top) and the surface area of one pixel (bottom):

The method is now well-defined for inner pixels of the region. Each value participates on eight triangles, two with each of the four neighbour values. Half of each of these triangles lies in one pixel, the other half in the other pixel. By counting in the area that lies inside each pixel, the total area is defined also for grains and masked areas. It remains to define it for boundary pixels of the whole data field. We do this by virtually extending the data field with a copy of the border row of pixels on each side for the purpose of surface area calculation, thus making all pixels of interest inner.

Surface area calculation triangulation scheme (left). Application of the triangulation scheme to a three-pixel masked area (right), e.g. a grain. The small circles represent pixel-centre vertices z_i, thin dashed lines stand for pixel boundaries while thick lines symbolize the triangulation. The surface area estimate equals to the area covered by the mask (grey) in this scheme.

In addition, the tool calculates a few quantities that do not belong in any of the previous categories:

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 derivatives 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(p) (where p 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 the values of heights at points (x₁, y₁), (x₂, y₂); furthermore, 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/Δx, n = τ_y/Δy. The function can thus be evaluated in a discrete set of values of τ_x and τ_y separated by the sampling intervals Δx and Δy, respectively.

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

It should be noted that this estimate is biased, especially if any scan line levelling has been done. The Correlation Length too can calculate a bias-corrected ACF. See its description for more details.

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

Autocorrelation function obtained for simulated Gaussian randomly rough surface (i.e. with a Gaussian autocorrelation function) with σ ≈ 20 nm and T ≈ 300 nm.

We can also introduce the radial ACF G_r(τ), i.e. angularly averaged two-dimensional ACF, which of course contains the same information as the one-dimensional ACF for isotropic surfaces:

The interface distribution function (IDF) is the second derivative of ACF

Positions and signs of its extrema are linked to features like grains sizes and intergrain distances.

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/Δx. The function thus can be evaluated in a discrete set of values of τ_x separated by the sampling interval Δx.

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 obtained for a simulated Gaussian surface is plotted. It is fitted using the formula shown above. The resulting values of σ and T obtained by fitting the HHCF are practically the same as for the ACF.

Height-height correlation function obtained for simulated Gaussian randomly rough surface with σ ≈ 20 nm and T ≈ 300 nm.

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 choose 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 ACF and HHCF fitting 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.

PSDF obtained for data simulated with Gaussian autocorrelation function.

We can also introduce the radial PSDF W_r(K), i.e. angularly integrated two-dimensional PSDF, 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 spectral density can also be integrated radially, giving the angular spectrum. Its peaks can be used to identify the image texture direction.

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 the following formulas:

Here N denotes the total number of pixels, N_white denotes the number of “white” pixels, that is pixels above the threshold. Pixels below the threshold are referred to as “black”. 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 plotted.

The range distribution is a plot of the growth of value range depending on the lateral distance. It is always a non-decreasing function.

For each pixel in the image and each lateral distance, it is possible to calculate the minimum and maximum from values that do not lie father than the given lateral distance from the given pixel. The local range is the difference between the maximum and minimum (and in two dimensions can be visualised using the Rank presentation function). Averaging the local ranges for all image pixels gives the range curve.

The area scale graph shows the ratio of developed surface area and projected area minus one, as the function of scale at which the surface area is measured. Because of the subtraction of unity, which would be the ratio for a completely flat surface, the quantity is called “excess” area.

Similar to the correlation functions, the area excess can be in principle defined as a directional quantity. However, the function displayed in the tool assumes an isotropic surface – the horizontal or vertical orientation that can be chosen there only determines the primary direction used in its calculation.

Most of the statistical functions support masking. In other words, they can be calculated for arbitrarily shaped image regions. For some of them, such as the height or angle distribution, no further explanation is needed: only image pixels covered (or not covered) by the mask are included in the computation. However, the meaning of the function is less clear for correlation functions and spectral densities.

We will illustrate it for the ACF (the simplest case). The full description can be found in the literature [1]. The discrete ACF formula can be rewritten

where Ω_m is the set of image pixels inside the image region for which the pixel m columns to the right also lies inside the region. If we calculate the ACF for a rectangular region, for example the entire image, nothing changes. However, this formula expresses the function meaningfully for regions of any shape. The only missing part is how to perform the computation efficiently.

For this we define c_k,l as the mask of valid pixels, i.e. it is equal to 1 for pixels we want to include and 0 for pixels we want to exclude. It then follows that

Furthermore, if image data are premultiplied by c_k,l the ACF sum over the irregular region can be replaced by the usual ACF sum over the entire image. Both are efficiently computed using FFT.

It is important to note that the amount of information available in the data about the ACF for given m depends on Ω_m. Unlike for entire rectangular images, it does not have to decrease monotonically with increasing m and can vary almost arbitrarily. There can even be holes, i.e. horizontal distances m for which the region contains no pair of pixels exactly this apart. If this occurs Gwyddion replaces the missing values using linear interpolation.

The height-height correlation function is calculated in a similar manner, only the sums have to be split into parts which can be then evaluated using FFT. For PSDF, this is not possible because each Fourier coefficient depends on each data value. So, instead, the cyclic autocorrelation function is calculated in the same manner as ACF, and PSDF is then obtained as its Fourier transform.