Shape Parameters - Basic Properties

In this section we discuss basic geometric properties of grains. Due to the huge amount of shape parameters we split them into different categories: basic properties (this page), properties based on fitted ellipses, convex hull bases properties, projection bases properties. The table below summarizes the shape parameters discussed on this page.

We start our discussion by reconstructing the grain structure from a sample EBSD data set.

% load sample EBSD data set
mtexdata forsterite silent

% restrict it to a subregion of interest.
ebsd = ebsd(inpolygon(ebsd,[5 2 10 5]*10^3));

% remove all not indexed pixels
ebsd = ebsd('indexed');

% reconstruct grains
[grains, ebsd.grainId] = calcGrains(ebsd,'angle',5*degree);

% smooth them
grains = smooth(grains,5);

% plot the orientation data of the Forsterite phase
plot(ebsd('fo'),ebsd('fo').orientations)

% plot the grain boundary on top of it
hold on
plot(grains.boundary,'lineWidth',2)
hold off

Grain size vs. grain area and boundary size vs. perimeter

The most basic properties are grainSize and grain area. Those can be computed by

grains(9).grainSize
grains(9).area

ans =
     1
ans =
   1.9540e+03

Hereby grainSize refers to the number of pixels that belong to a certain grain while area represents the actual area measured in (µm)^2. We may analyze the distribution of grains by grain area using a histogram.

close all
histogram(grains.area)
xlabel('grain area')
ylabel('number of grains')

Note the large amount of very small grains. A more realistic histogram we obtain if we do not plot the number of grains at the y-axis but its total area. This can be achieved with the command histogram(grains) or hist(grains)

hist(grains)

ans =
    0.5294    0.8078    0.9804
ans =
    0.5608    0.7373    0.5608
ans =
    0.8549    0.6471    0.1255

histogram(grains)

Similarly as grainSize and area, the one-dimensional measures boundarySize and perimeter give the length of the grain boundary as number of segments and in µm, respectively.

grains(9).boundarySize
grains(9).perimeter

ans =
     4
ans =
  233.1556

We may compute these quantities also explicitly from the grain boundary segments. In the following code the first command returns simply the number of boundary segments while the second one gives the total length of the boundary by summing up the length of each individual boundary segment

length(grains(9).boundary)
sum(grains(9).boundary.segLength)

ans =
     4
ans =
  233.1556

Radius, diameter, equivalent radius, equivalent perimeter and shape factor

Another, one dimensional measure is the diameter which refers to the longest distance between any two boundary points and is given im µm as well

grains(9).diameter

ans =
  106.0650

The diameter is a special case of the caliper or Feret diameter of a grain that is explained in detail in the section Projection based parameters.

In contrast the equivalent radius is the radius of a circle with the same area as the grain. Naturally, the equivalent radius is always smaller than the actual radius of the grains. Similarly, the equivalent perimeter is defined as the perimeter of the circle the same area and is always smaller then the actual perimeter.

2*grains(9).equivalentRadius

grains(9).equivalentPerimeter

ans =
   49.8787
ans =
  156.6985

As a consequence, the ratio between between actual grain perimeter and the equivalent perimeter, the so called shape factor, is always larger then 1. The shapeFactor and measures how different a certain shape is from a circle.

plot(grains,grains.shapeFactor)
mtexColorbar('title','shape factor')

A second measure for the discrepancy between the actual shape and a circle is the relative difference between the perimeter and the equivalentPerimeter

plot(grains,(grains.perimeter - grains.equivalentPerimeter)./grains.perimeter)
setColorRange([0,0.5])
mtexColorbar

In this plot round shapes will have values close to zero while concave shapes will get values up to \(0.5\).

A third measure is the paris which stands for Percentile Average Relative Indented Surface and gives the relative difference between the actual perimeter and the perimeter of the convex hull. It is explained in more detail in the section convex hull parameters.

plot(grains,grains.paris)
mtexColorbar('title','paris')

Fractal Dimension

The fractal dimension of grain boundaries has been used to characterize the conditions of dynamical recrystallization. One way to define the fractal dimension of grain boundaries is to look at the slope of the perimeter as a function of the equivalent radius of the grains. More precisely, we consider this relationship in a log log plot and fit a linear model.

% consider the entire data set
mtexdata forsterite silent

% reconstruct grains
grains = calcGrains(ebsd('indexed'));

% smooth them
grains = smooth(grains,5);

close all
scatter(grains.equivalentRadius,grains.perimeter)
xlabel('equivalent radius')
ylabel('perimeter')

% set axes to logarithmic
logAxis(gca, [10,10^4],[10^2,10^5])

% fit a linear function in log space
ab = polyfit(log(grains.equivalentRadius),log(grains.perimeter),1);

% plot it
hold on
plot([10,10^4],exp(ab(2) + ab(1) * log([10,10^4])),'LineWidth',3)
hold off

The slope of the linear function is the fractal dimension.

fractalDimension = ab(1)

fractalDimension =
    1.0330

It is important to understand that the fractal dimension computed this way heavily depends on the smoothing applied to the grain boundaries.