Denoising Orientation Maps edit page

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 indecing algorithm can be modeled as random errors. In this section we demonstrate how random errors can be significantly reduced using denoising techniques.

Simultanously while denoising orientation maps one can also fill not indexed pixels. This is explained in the section Fill Missing Data.

We shall demonstrate the denoising capabilities of MTEX at the hand of an orientation map of deformed Magnesium.

% import the data
mtexdata twins

% 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 segementation
[grains,ebsd.grainId] = calcGrains(ebsd,'angle',10*degree);

% smooth grain boundaries
grains = smooth(grains,5);

% plot the orientation map
ipfKey = ipfColorKey(ebsd.CS.properGroup);
plot(ebsd,ipfKey.orientation2color(ebsd.orientations))

% and on top the grain boundaries
hold on
plot(grains.boundary,'linewidth',2)
hold off
ebsd = EBSD  
 
 Phase  Orientations     Mineral         Color  Symmetry  Crystal reference frame
     0     46 (0.2%)  notIndexed                                                 
     1  22833 (100%)   Magnesium  LightSkyBlue     6/mmm       X||a*, Y||b, Z||c*
 
 Properties: bands, bc, bs, error, mad, x, y
 Scan unit : um

At the first glance, the orientation data look not noisy at all. However, if we look at orientation changes within the grains the noise we become clearly visible. To do so we colorize the orientation data with respect to their misorientation to the grain mean orientation

% the axisAngleColorKey colorizes misorientation according to their axis
% and angle
colorKey = axisAngleColorKey;

% we need to set the reference orientations are the mean orientation of
% each grain
colorKey.oriRef = grains(ebsd.grainId).meanOrientation;

% lets plot the result
plot(ebsd,colorKey.orientation2color(ebsd.orientations))
hold on
plot(grains.boundary,'linewidth',2)
hold off

We clearly observe some deformation gradients withing the grains which are superposed by random noise.

The Mean Filter

The simplest filter to apply to orientation data is the @meanFilter which replaces all orientations by the mean of all neighboring orientations.

% define the meanFilter
F = meanFilter;
F.numNeighbours = 1;

% smooth the data
ebsdS = smooth(ebsd,F);
ebsdS = ebsdS('indexed');

% plot the smoothed data
colorKey.oriRef = grains(ebsdS.grainId).meanOrientation;
plot(ebsdS, colorKey.orientation2color(ebsdS.orientations))

hold on
plot(grains.boundary,'linewidth',2)
hold off

We clearly see how the noise has been reduces. In order to further reduce the noise we may increase the number of neighbours that are taken into account.

F.numNeighbours = 3;

% smooth the data
ebsdS = smooth(ebsd,F);
ebsdS = ebsdS('indexed');

% plot the smoothed data
colorKey.oriRef = grains(ebsdS.grainId).meanOrientation;
plot(ebsdS,colorKey.orientation2color(ebsdS.orientations))

hold on
plot(grains.boundary,'linewidth',2)
hold off

The Median Filter

The disadvantage of the mean filter is that is smoothes away all subgrain boundaries and is quite sensitive against outliers. A more robust filter which also preserves subgrain boundaries is the median filter

F = medianFilter;

% define the size of the window to be used for finding the median
F.numNeighbours = 3; % this corresponds to a 7x7 window

% smooth the data
ebsdS = smooth(ebsd,F);
ebsdS = ebsdS('indexed');

% plot the smoothed data
colorKey.oriRef = grains(ebsdS.grainId).meanOrientation;
plot(ebsdS,colorKey.orientation2color(ebsdS.orientations))

hold on
plot(grains.boundary,'linewidth',2)
hold off

The disadvantage of the median filter is that it leads to cartoon like images which suffer from the staircase effect.

The Kuwahara Filter

Another filter that was designed to be robust against outliers and does not smooth away subgrain boundaries is the Kuwahara filter. However, in practical applications the results are often not satisfactory.

F = KuwaharaFilter;
F.numNeighbours = 5;

% smooth the data
ebsdS = smooth(ebsd,F);
ebsdS = ebsdS('indexed');

% plot the smoothed data
colorKey.oriRef = grains(ebsdS.grainId).meanOrientation;
plot(ebsdS,colorKey.orientation2color(ebsdS.orientations))

hold on
plot(grains.boundary,'linewidth',2)
hold off

The Smoothing Spline Filter

All the above filters are so called sliding windows filters which consider for the denoising operation only neighboring orientations within a certain window. The next filters belong to the class of variational filters which determine the denoised orientation map as the solution of an minimization problem that is simultanously close to the original map and "smooth". The resulting orientation map heavily depends on the specific definition of "smooth" and on the regularization parameter which controls the trade of between fitting the original data and forcing the resulting map to be smooth.

The @splineFilter uses as definition of smoothnes the curvature of the orientation map. As a cosequence, the denoised images are very "round" and similarly as for the @meanFilter subgrain boundaries will be smoothed away. On the positive side the @meanFilter is up to now the only filter that automatically calibrates the regularization parameter.

F = splineFilter;

% smooth the data
ebsdS = smooth(ebsd,F);
ebsdS = ebsdS('indexed');

% plot the smoothed data
colorKey.oriRef = grains(ebsdS.grainId).meanOrientation;
plot(ebsdS,colorKey.orientation2color(ebsdS.orientations))

hold on
plot(grains.boundary,'linewidth',2)
hold off

% the smoothing parameter determined during smoothing is
F.alpha
ans =
    4.6123

The Halfquadratic Filter

In the default setting the @halfQuadraticFilter uses definition of smoothnes the total variation functional. This functional was developed to preserve subgrain boundaries. Similarly as the @medianFilter it tends to carton like images and staircasing.

F = halfQuadraticFilter;

% smooth the data
ebsdS = smooth(ebsd,F);
ebsdS = ebsdS('indexed');

% plot the smoothed data
colorKey.oriRef = grains(ebsdS.grainId).meanOrientation;
plot(ebsdS,colorKey.orientation2color(ebsdS.orientations))

hold on
plot(grains.boundary,'linewidth',2)
hold off

The Infimal Convolution Filter

The infimal convolution filter was designed as a compromise between the @splineFilter and the @halfQuadraticFilter. It is still under deveopment and its use is not recommendet.

F = infimalConvolutionFilter;
F.lambda = 0.01; % sssmoothing parameter for the gradient
F.mu = 0.005;    % smoothing parameter for the hessian

% smooth the data
ebsdS = smooth(ebsd,F);
ebsdS = ebsdS('indexed');

% plot the smoothed data
colorKey.oriRef = grains(ebsdS.grainId).meanOrientation;
plot(ebsdS,colorKey.orientation2color(ebsdS.orientations))

hold on
plot(grains.boundary,'linewidth',2)
hold off