There are several grain-related algorithms implemented in Gwyddion. First
of all, simple thresholding algorithms can be used (height, slope or
curvature thresholding). These procedures can be very efficient namely
within particle analysis (to mark particles located on flat surface).
Thresholding methods can be accessed within Gwyddion as
→ → .
Height, slope and curvature thresholding is implemented within this module.
The results of each individual thresholding methods can be merged together
using several operators.
Similarly, the grains can be removed from the mask using
→ → menu choice.
Maximum height and/or size thresholding methods can be used to eliminate
false grains occurred by noise or some dust particles, for example. You can
use also interactive grain removal tool for doing this manually.
For more complicated data structures the effectiveness of thresholding
algorithms can be very poor. For these data a watershed
algorithm can be used more effectively for grain or particle
marking.
The watershed algorithm is usually employed for local minima
determination and image segmentation in image processing. As the problem
of determining the grain positions can be understood as the problem of
finding local extremes on the surface this algorithm can be used also for
purposes of grain segmentation or marking. For convenience in the
following we will treat the data inverted in the
z
direction while describing the algorithm (i.e.
the grain tops are forming local minima in the following text). We
applied two stages of the grain analysis
(see [1]):
-
At each point of the inverted surface the virtual water drop was
placed. In the case that the drop was not already in a local minimum it
followed the steepest descent path to minimize its potential energy. As
soon as the drop reached any local minimum it stopped here and rested
on the surface. In this way it filled the local minimum partially by
its volume (see figure below and its caption). This process was
repeated several times. As the result a system of lakes of different
sizes filling the inverted surface depressions was obtained. Then the
area of each of the lakes was evaluated and the smallest lakes were
removed under assumption that they were formed in the local minima
originated by noise. The larger lakes were used to identify the
positions of the grains. In this way the noise in the AFM data was
eliminated.
-
The grains found in the step 1 were marked (each one by a different
number). The water drops continued in falling to the surface and
filling the local minima. As the grains were already identified and
marked after the first step, the next five situations could happen as
soon as the drop reached a local minimum.
-
The drop reached the place previously marked as a concrete grain.
In this case the drop was merged with the grain, i. e. it was
marked as a part of the same grain.
-
The drop reached the place where no grain was found but a concrete
grain was found in the closest neighbourhood of the drop. In this
case the drop was merged with the grain again.
-
The drop reached the place where no grain was found and no grain
was found even in the closest neighbourhood of the drop. In that
case the drop was not marked at all.
-
The drop reached the place where no grain was found but more than
one concrete grain was found in the closest neighbourhood (e. g.
two different grains were found in the neighbourhood). In this case
the drop was marked as the grain boundary.
-
The drop reached the place marked as grain boundary. In this case
the drop was marked as the grain boundary too.
In this way we can identify the grain positions and then determine the
volume occupied by each grain separately.
Grain properties can be studied using several functions. The simplest of
them is Grain Statistics
→ →
This function calculates the total number of marked grains, their total
(projected) area both as an absolute value and as a fraction of total
data field area, and the mean area and equivalent square size of one
grain.
Overall characteristics of the marked area can be also obtained with
Statistical Quantities tool when its Use mask
option is switched on. By inverting the mask the same information
can be obtained also for the non-grain area.
Grain Distributions
→ →
Grain Distributions is the most powerful and complex tool. It has two
basic modes of operation: graph plotting and raw data export. In
graph plotting mode selected characteristics of individual grains are
calculated, gathered and summary graphs showing their distributions
are plotted.
Raw data export is useful for experts who need for example to
correlate properties of individual grains. In this mode selected
grain characteristics are calculated and dumped to a text file table
where each row corresponds to one grain and columns correspond to
requested quantities. The order of the colums is the same as the
relative order of the quantities in the dialog; all values are
written in base SI units, as is usual in
Gwyddion.
Grain Property Correlation
→ →
Grain correlation plots a graph of one selected graph quantity as
the function of another grain quantity, visualizing correlations
between them.
Grain Measurement Tool
The grain measurement tool is the interactive method to obtain the
same information about individual grains as
Grain Distributions in raw
mode. After selecting a grain on the data window with mouse, all
the available quantities are displayed in the tool window.
Beside physical characteristics this tool also displays the grain
number. Grain numbers corresponds to row numbers (counting from 1)
in files exported by
Grain Distributions.
Grain Distributions
and Grain measurement tool
can calculate the following grain properties:
- Value-related properties
- Minimum,
the minimum value (height) occuring inside the grain.
- Maximum,
the maximum value (height) occuring inside the grain.
- Mean,
the mean of all values occuring inside the grain, that is the
mean grain height.
- Median
the median of all values occuring inside the grain, that is the
median grain height.
- Minimum on boundary,
the maximum value (height) occuring on the inner grain boundary.
This means within the set of pixels that lie inside the grain
but at least one of their neighbours lies outside.
- Maximum on boundary,
the maximum value (height) occuring on the inner grain boundary,
defined similarly to the minimum.
- Area-related properties
- Projected area,
the projected (flat) area of the grain.
- Equivalent square side,
the side of the square with the same projected area as the
grain.
- Equivalent disc radius,
the radius of the disc with the same projected area as the
grain.
- Surface area,
the surface area of the grain, see
statistical
quantities section for description of the surface area
estimation method.
- Boundary-related properties
- Projected boundary length,
the length of the grain boundary projected to the horizontal
plane (that is not taken on the real three-dimensional
surface). The method of boundary length estimation is
described below.
- Minimum bounding size,
the minimum dimension of the grain in the horizontal plane. It
can be visualized as the minimum width of a gap in the
horizontal plane the grain could pass through.
- Minimum bounding direction,
the direction of the gap from the previous item. If the grain
exhibits a symmetry that makes several directions to qualify,
an arbitrary direction is chosen.
- Maximum bounding size,
the maximum dimension of the grain in the horizontal plane. It
can be visualized as the maximum width of a gap in the
horizontal plane the grain could fill up.
- Maximum bounding direction,
the direction of the gap from the previous item. If the grain
exhibits a symmetry that makes several directions to qualify,
an arbitrary direction is chosen.
- Volume-related properties
- Zero basis,
the volume between grain surface and the plane
z = 0.
Values below zero form negative volumes. The zero level must be
set to a reasonable value (often
Fix Zero is sufficient) for
the results to make sense, which is also the advantage of
this method: one can use basis plane of his choice.
- Grain minimum basis,
the volume between grain surface and the plane
z = zmin,
where
zmin
is the minimum value (height) occuring in the grain. This
method accounts for grain surrounding but it typically
underestimates the volume, especially for small grains.
- Laplacian backround basis,
the volume between grain surface and the basis surface formed
by laplacian interpolation of surrounding values. In other
words, this is the volume that would disappear after using
Remove Data Under Mask
or
Grain Remover tool
with Laplacian interpolation on the grain. This is the most
sophisticated method, on the other hand it is the hardest to
develop intuition for.
- Position-related properties
- Center x position,
the horizontal coordinate of the grain centre. Since the
grain area is defined as the area covered by the
corresponding mask pixels, the centre of a single-pixel grain
has half-integer coordinates, not integer ones. Data field
origin offset (if any) is taken into account.
- Center y position,
the verical coordinate of the grain centre. See above for
the interpretation.
- Slope-related properties
- Inclination θ,
the deviation of the normal to the mean plane from the
z-axis,
see inclinations
for details.
- Inclination φ,
the azimuth of the slope, as defined in
inclinations.
The grain boundary length is estimated by summing estimated contributions
of each four-pixel configuration on the boundary. The contributions
are displayed on the following figure for each type of configuration,
where
hx
and
hy
are pixel dimension along corresponding axes and
h
is the length of the pixel diagonal:
The contributions correspond one-to-one to lenghts of segments of the
boundary of a polygon approximating the grain shape. The construction of
the equivalent polygonal shape can also be seen in the figure.
The grain volume is, after subtracting the basis, estimated as the
volume of exactly the same body whose upper surface is used for
surface area calculation.
Note for the volume between vertices this is equivalent to the classic
two-dimensional trapezoid integration method. However, we calculate the
volume under a mask centered on vertices, therefore their contribution
to the integral is distributed differently as shown in the following
figure.