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.
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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.
Figure 4.18. Image of grain-like surface structure (a) and corresponding results of height thresholding (b), curvature thresholding (c), and watershed (d) algorithm. Within watershed algorithm it is possible to segment image even further.
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 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 correlation plots a graph of one selected graph quantity as the function of another grain quantity, visualizing correlations between them.
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
- Maximum, the maximum value (height) occuring inside the grain.
- Minimum, the minimum 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.
- 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 pixel corner participating in the boundary. The contributions are displayed on the following figure for each type of corner configuration, where hx and hy are pixel dimension along corresponding axes and q is the length of the pixel diagonal.
Figure 4.20. Contributions of pixel configurations to the estimated boundary length. Grey squares represent pixels inside the grain, white squares represent outside pixels. The estimated contribution of each configuration is: (a) q/2, (b) q, (c) hy, (c) hx, (e) q/2.
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.
