Orientation maps determined by EBSD or any other technique are, as all experimental data, effected by measurement errors. Those measurement errors can be divided into systematic errors and random errors. Systematic errors mostly occur due to a bad calibration of the EBSD system and require additional knowledge to be corrected. Deviations from the true orientation due to noisy Kikuchi pattern or tolerances of the indexing algorithm can be modeled as random errors. In this section we demonstrate how random errors can be significantly reduced using denoising techniques.
We shall demonstrate the denoising capabilities of MTEX at the hand of an orientation map of deformed Magnesium
% import the data
mtexdata ferrite
% consider only indexed data
ebsd = ebsd('indexed');
% reconstruct the grain structure
[grains,ebsd.grainId,ebsd.mis2mean] = calcGrains(ebsd,'angle',10*degree);
% remove some very small grains
ebsd(grains(grains.grainSize<5)) = [];
% redo grain segmentation
[grains,ebsd.grainId] = calcGrains(ebsd,'angle',10*degree);
% smooth grain boundaries
grains = smooth(grains,5);
% plot the orientation map
plot(ebsd,ebsd.orientations)
% and on top the grain boundaries
hold on
plot(grains.boundary,'linewidth',1.5)
hold offsaving data to /home/hielscher/mtex/master/data/ferrite.mat
ebsd = EBSD
Phase Orientations Mineral Color Symmetry Crystal reference frame
0 63044 (100%) Ferrite LightSkyBlue 432
Properties: ci, fit, iq, sem_signal, x, y, oldId
Scan unit : um
A Very Sparse Measured Data Set
Although the data set has already some not indexed pixels we artificially make the situation more worse by throwing away 75 percent of all data.
ebsdSub = ebsd(discretesample(length(ebsd),round(length(ebsd)*25/100)));
% plot the reduced data
plot(ebsdSub,ebsdSub.orientations)
Our aim is now to recover the orginal orientation map. In a first step we reconstruct the grain structure from the remaining 25 percent of pixels.
% reconstruct the grain structure
[grainsSub,ebsdSub.grainId] = calcGrains(ebsdSub,'angle',10*degree);
grainsSub = smooth(grainsSub,5);
hold on
plot(grainsSub.boundary,'linewidth',1.5)
hold off
The easiest way to reconstruct missing data is to use the command fill which interpolates missing data using the method of nearest neighbor. It is very recommended to pass the grain structure grainsSub as an additional argument to the fill function. In this case the nearest neighbors are choosen within the grains.
ebsdSub_filled = fill(ebsdSub,grainsSub);
plot(ebsdSub_filled('indexed'),ebsdSub_filled('indexed').orientations);
hold on
plot(grainsSub.boundary,'linewidth',1.5)
hold off
A much more powerful method is to use any denoising method and set the option fill.
F = halfQuadraticFilter;
F.alpha = 0.25;
% interpolate the missing data
ebsdSub_filled = smooth(ebsdSub,F,'fill',grainsSub);
ebsdSub_filled = ebsdSub_filled('indexed');
plot(ebsdSub_filled('indexed'),ebsdSub_filled('indexed').orientations);
hold on
plot(grainsSub.boundary,'linewidth',1.5)
hold off
An Example from Geoscience
Data sets with many missing pixels most often appear when measuring geological samples. The following data set of forsterite contains about 25 percent missing pixels. Lets start by importing the data and reconstructing the grain structure.
close all; plotx2east
mtexdata forsterite
ebsd = ebsd(inpolygon(ebsd,[10 4 5 3]*10^3));
plot(ebsd('Fo'),ebsd('Fo').orientations)
hold on
plot(ebsd('En'),ebsd('En').orientations)
plot(ebsd('Di'),ebsd('Di').orientations)
% compute grains
[grains,ebsd.grainId] = calcGrains(ebsd('indexed'),'angle',10*degree);
% remove small grains
ebsd(grains(grains.grainSize < 3)) = [];
% and repeat the grain computation
[grains,ebsd.grainId] = calcGrains(ebsd('indexed'),'angle',10*degree);
%
grains = smooth(grains,5);
% plot the boundary of all grains
plot(grains.boundary,'linewidth',2)
hold offebsd = EBSD
Phase Orientations Mineral Color Symmetry Crystal reference frame
0 58485 (24%) notIndexed
1 152345 (62%) Forsterite LightSkyBlue mmm
2 26058 (11%) Enstatite DarkSeaGreen mmm
3 9064 (3.7%) Diopside Goldenrod 12/m1 X||a*, Y||b*, Z||c
Properties: bands, bc, bs, error, mad, x, y
Scan unit : um
Using the option fill the command smooth fills the holes inside the grains. Note that the nonindexed pixels at the grain boundaries are kept untouched. In order to allow MTEX to decide whether a pixel is inside a grain or not, the grain variable has to be passed as an additional argument.
F = halfQuadraticFilter;
F.alpha = 10;
ebsdS = smooth(ebsd('indexed'),F,'fill',grains);
plot(ebsdS('Fo'),ebsdS('Fo').orientations)
hold on
plot(ebsdS('En'),ebsdS('En').orientations)
plot(ebsdS('Di'),ebsdS('Di').orientations)
% plot the boundary of all grains
plot(grains.boundary,'linewidth',1.5)
% stop overide mode
hold off
In order to visualize the orientation gradient within the grains, we plot the misorientation to the meanorientation. We observe that the mis2mean varies smoothly also within the regions of not indexed orientations.
% plot mis2mean for all phases
ipfKey = axisAngleColorKey(ebsdS('Fo'));
ipfKey.oriRef = grains(ebsdS('fo').grainId).meanOrientation;
ipfKey.maxAngle = 2.5*degree;
color = ipfKey.orientation2color(ebsdS('Fo').orientations);
plot(ebsdS('Fo'),color,'micronbar','off')
hold on
ipfKey.oriRef = grains(ebsdS('En').grainId).meanOrientation;
plot(ebsdS('En'),ipfKey.orientation2color(ebsdS('En').orientations))
% plot boundary
plot(grains.boundary,'linewidth',4)
plot(grains('En').boundary,'lineWidth',4,'lineColor','r')
hold off
For comparison
ipfKey.oriRef = grains(ebsd('fo').grainId).meanOrientation;
ipfKey.maxAngle = 2.5*degree;
color = ipfKey.orientation2color(ebsd('Fo').orientations);
plot(ebsd('Fo'),color,'micronbar','off')
hold on
ipfKey.oriRef = grains(ebsd('En').grainId).meanOrientation;
plot(ebsd('En'),ipfKey.orientation2color(ebsd('En').orientations))
% plot boundary
plot(grains.boundary,'linewidth',4)
plot(grains('En').boundary,'lineWidth',4,'lineColor','r')
hold off