Boundary Curvature

Boundary Curvature edit page

The curvature of a curve is defined by fitting locally a circle and taking one over its radius. Hence, a straight line will have curvature 0 and a circle of radius \(2\) will have constant curvature \(1/2\) everywhere. Hence, the unit of the curvature computed in MTEX is one over the unit of the EBSD coordinates which is usually 1/µm. Let us demonstrate boundary curvature using some artificial grain shapes

% import the artificial grain shapes
mtexdata testgrains silent

% select and smooth a few interesting grains
grains = smoothBoundary(grains('id',[2 3 9 11 15 16 18 23 31 33 38 40]),10);

Therefore, we first extract all boundary segments and colorize them according to their curvature.

% extract boundary segments
gB = grains.boundary;

% plot some dark background
plot(gB,'linewidth',10,'micronbar','off');

% colorize boundaries by curvature
hold on
plot(gB,gB.curvature,'linewidth',6);
hold off

% set a specific colormap
mtexColorMap('blue2red')
setColorRange(0.25*[-1,1])
mtexColorbar

The Sign of the Curvature

The curvature is signed, and the sign tells which of the two adjacent grains the boundary bulges into. Boundary segments are stored in walk order with the grain in the first column of gB.grainId lying to the left of the walk direction - see boundary misorientations. Accordingly a positive curvature means the boundary bulges into the grain in the second column, a negative one that it bulges into the grain in the first column.

Every grain plotted above is an island inside one large surrounding grain, grain 44, and along their outer boundaries that surrounding grain is the one stored in the first column

grains('id',2).boundary.grainId(1:5,:)

Grain 2 is a convex blob, so its boundary bulges into grain 44 everywhere - into the first column - and its curvature is negative throughout. Across the whole map this is why the convex parts come out blue while only the notches and the enclosed grains come out red. The enclosed grains are stored the other way round, with the enclosed grain in the first column, and being convex they bulge into the ring shaped grains 23 and 31 that surround them.

Which grain sits in which column is not a property of the boundary as a set of segments, it is a property of the direction each segment is walked in. The command flip reverses that direction and swaps the two columns with it, so the curvature simply changes sign. Integrating it along a closed boundary gives the total turning of the curve, which is \(\pm 2\pi\) depending on the direction it is walked in

gB2 = grains('id',2).boundary;
gB2flipped = flip(gB2);

[sum(gB2.curvature .* gB2.segLength), ...
  sum(gB2flipped.curvature .* gB2flipped.segLength)] ./ (2*pi)

ans =
   -1.0199    1.0199

Curvature With Respect to a Specific Grain

Selecting a single grain does not change the storage order - the segments of grains(k).boundary come as they are stored, and the grain we asked for may sit in either column. In order to read the curvature as convex or concave with respect to one specific grain we flip exactly those segments that have it in the second column. Then that grain lies to the left everywhere, a positive curvature marks a convex piece of it and a negative one a notch.

for k = 1:length(grains)

  gB = grains(k).boundary;

  % put the grain itself into the first column
  gB = flip(gB, gB.grainId(:,2) == grains(k).id);

  plot(gB,'linewidth',10,'micronbar','off');
  hold on
  plot(gB,gB.curvature,'linewidth',6);

end
hold off

mtexColorMap('blue2red')
setColorRange(0.25*[-1,1])
drawNow(gcm,'figSize',getMTEXpref('figSize'))

The outer boundaries are now predominantly red, i.e. convex with respect to the grain they belong to, and the boundaries of the enclosed grains have turned blue - seen from the ring shaped grains 23 and 31 they are concave.

Undefined Segments and Smoothing

The curvature of a segment is computed from the circle through its own midpoint and the midpoints of its two neighbours within the same chain. A segment at the end of an open chain has a neighbour on one side only and therefore no curvature - those segments come out NaN. All chains above are closed, so nothing is undefined there, but an arbitrary subset of segments is in general no longer closed

gB = grains.boundary;
sum(isnan(gB(1:100).curvature))

ans =
     2

These raw values are then smoothed along the chain. The number of smoothing passes is the second argument and defaults to 50. Without any smoothing the curvature follows the individual segment midpoints and is correspondingly noisy

gB = grains('id',15).boundary;

plot(gB.arcLength,gB.curvature(0),'linewidth',1)
hold on
plot(gB.arcLength,gB.curvature,'linewidth',2)
hold off

legend('no smoothing','50 smoothing passes')
xlabel(['arc length in ' gB.scanUnit])
ylabel(['curvature in 1/' gB.scanUnit])

Curvature of a Real EBSD Map

Finally we illustrate the usage of the curvature command at a real EBSD map.

% import data and reconstruct grains
mtexdata titanium silent
[grains,ebsd] = calcGrains(ebsd);
grains = smoothBoundary(grains,5);

% plot an ipf map
plot(ebsd('indexed'),ebsd('indexed').orientations)

hold on

% plot grain boundaries
plot(grains.boundary,'linewidth',4)

% colorize the grain boundary of the grain at position 596,466 according to
% curvature, with the grain itself on the left so that convex is positive
grain = grains(596,466);
gB = flip(grain.boundary, grain.boundary.grainId(:,2) == grain.id);

plot(gB,gB.curvature(5),'linewidth',6)
hold off

mtexColorMap('blue2red')
setColorRange(0.05*[-1,1])
mtexColorbar