In practice objects exhibiting random properties are encountered. It is
often assumed that these objects exhibit the self-affine properties in
a certain range of scales. Self-affinity is a generalization of
self-similarity which is the basic property of most of the deterministic
fractals. A part of self-affine object is similar to whole object after
anisotropic scaling. Many randomly rough surfaces are assumed to belong
to the random objects that exhibit the self-affine properties and they
are treated self-affine statistical fractals. Of course, these surfaces
can be studied using atomic force microscopy (AFM). The results of the
fractal analysis of the self-affine random surfaces using AFM are often
used to classify these surfaces prepared by various technological
Within Gwyddion, there are different methods of fractal analysis
→ → .
Cube counting method
is derived directly from a definition of box-counting fractal
dimension. The algorithm is based on the following steps: a cubic
lattice with lattice constant l
is superimposed on the z-expanded surface.
Initially l is set at
(where X is length of edge of the
surface), resulting in a lattice of
2×2×2 = 8 cubes. Then
N(l) is the number of all cubes that
contain at least one pixel of the image. The lattice constant
l is then reduced stepwise by factor of
2 and the process repeated until l
equals to the distance between two adjacent pixels. The slope of
a plot of log(N(l)) versus
gives the fractal dimension Df directly.
is very similar to cube counting method and is also based directly on
the box-counting fractal dimension definition. The method works as
follows: a grid of unit dimension l
is placed on the surface. This defines the location of the vertices of
a number of triangles. When, for example,
l = X/4,
the surface is covered by 32 triangles of different areas inclined at
various angles with respect to the xy
plane. The areas of all triangles are calculated and summed to obtain
an approximation of the surface area S(l)
corresponding to l.
The grid size is then decreased by successive factor of 2, as before,
and the process continues until l
corresponds to distance between two adjacent pixel points.
The slope of a plot of log(S(l)) versus
log(1/l) then corresponds to
Df − 2.
is based on the scale dependence of the
variance of fractional Brownian motion. In practice, in the variance
method one divides the full surface into equal-sized squared boxes, and
the variance (power of RMS value of heights), is calculated for a
particular box size. Fractal dimension is
evaluated from the slope β
of a least-square regression line fit to the data points in log-log
plot of variance as
Df = 3 − β/2.
Power spectrum method
is based on the power spectrum dependence of fractional Brownian
motion. In the power spectrum method, every line height profiles that
forms the image is Fourier transformed and the power spectrum evaluated
and then all these power spectra are averaged. Fractal dimension is
evaluated from the slope
β of a least-square regression line fit
to the data points in log-log plot of power spectrum as
Df = 7/2 − β/2.
The axes in Fractal Dimension graphs always show already logarithmed
quantities, therefore the linear dependencies mentioned above correspond
to straight lines there. The measure of the axes should be treated as
Note, that results of different methods differ. This fact is caused by
systematic error of different fractal analysis approaches.
Moreover, the results of the fractal analysis can be influenced strongly by
the tip convolution. We recommend therefore to check the certainty map
before fractal analysis. In cases when the surface is influenced a lot by
tip imaging, the results of the fractal analysis can be misrepresented
Note, that algorithms that can be used within the fractal analysis module
are also used in Fractal Correction module and Fractal Correction option
of Remove Spots tool.
 C. Douketis, Z. Wang, T. L. Haslett, M. Moskovits: Fractal
character of cold-deposited silver films determined by low-temperature
scanning tunneling microscopy. Physical Review B, Volume 51, Number 16,
15 April 1995, 51
 W. Zahn, A. Zösch: The dependence of fractal dimension on measuring
conditions of scanning probe microscopy. Fresenius J Analen Chem (1999)
 A. Van Put, A. Vertes, D. Wegrzynek, B. Treiger, R. Van Grieken:
Quantitative characterization of individual particle surfaces by fractal
analysis of scanning electron microscope images. Fresenius J Analen
Chem (1994) 350: 440-447
 A. Mannelquist, N. Almquist, S. Fredriksson: Influence of tip
geometry on fractal analysis of atomic force microscopy images. Appl.
Phys. A 66,1998, 891-895
 W. Zahn, A. Zösch: Characterization of thin film surfaces by
fractal geometry. Fresenius J Anal Chem (1997) 358: 119-121