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    filename: OptimalKernel.m
    path of file: ./repos/mtex/doc/GeneralConcepts
    
    the code of the file until where you have to start completion: %% Optimal Kernel Selection
%
%%
% In the section <DensityEstimation.html density estimation> we have seen
% that the correct choice of the kernel halfwidth is essential for creating a good
% match between the true density function and the reconstructed density
% function. If the halfwidth is set too small the reconstructed density
% function is usually oscillating and the indiviudual sampling points are
% visible as sharp peaks. If the halfwidth is too large the resulting
% density function is usually too smooth and does not reproduce the
% features of the original density function. 
%
% Finding an optimal kernel halfwidth is a hard problem as the optimal
% kernel halfwidth depends not only on the number of sampling points but also
% on the smoothness of the true but unknown density function. 
% MTEX offers several options set by flags during the kernel calculation operation.  A very
% conserative choice for the kernel halfwidth that takes into account only
% the number of sampling points is implemented in MTEX with the flag |'magicRule'|. The flag
% |'RuleOfThumb'| considers both the number of sampling
% points and the variance of the sampling points as an estimate of the
% smoothness of the true density function. The most advanced (and default)
% method for estimating the optimal kernel halfwidth is
% <orientation.KLCV.html Kullback Leibler cross validation>.
% This method tests different kernel halfwidths on a subset of the
% random sample and selects the halfwidth which best reproduces the
% ommited points of the random sample.
%
% In order to demonstrate this functionality let's start with the following
% orientation density function

% Define trigonal crystal symmetry using Enantiomorphic Point Group notation
cs = crystalSymmetry('32');

% Build a density function by combining a uniform texture with two pre-defined texture components
odf = 0.25*uniformODF(cs) + 0.25*unimodalODF(orientation.brass(cs)) + ...
  0.5*fibreODF(fibre.alpha(cs),'halfwidth',10*degree);

% plot the density function as six sigma sections 
plot(odf,'sections',6,'silent','sigma')
mtexColorbar

%%
% and compute $10000$ random orientations representing this density function using the command
% |<orientation.discreteSample.html discreteSample>|

ori = odf.discreteSample(10000)

%%
% Next we estimate the optimal <ODFShapes.html kernel function> using the
% command |<orientation.calcKernel
    
function and the reconstructed density
% function. If the halfwidth is set too small the reconstructed density
% function is usually oscillating and the indiviudual sampling points are
% visible as sharp peaks. If the halfwidth is too large the resulting
% density function is usually too smooth and does not reproduce the
% features of the original density function. 
%
% Finding an optimal kernel halfwidth is a hard problem as the optimal
% kernel halfwidth depends not only on the number of sampling points but also
% on the smoothness of the true but unknown density function. 
% MTEX offers several options set by flags during the kernel calculation operation.  A very
% conserative choice for the kernel halfwidth that takes into account only
% the number of sampling points is implemented in MTEX with the flag |'magicRule'|. The flag
% |'RuleOfThumb'| considers both the number of sampling
% points and the variance of the sampling points as an estimate of the
% smoothness of the true density function. The most advanced (and default)
% method for estimating the optimal kernel halfwidth is
% <orientation.KLCV.html Kullback Leibler cross validation>.
% This method tests different kernel halfwidths on a subset of the
% random sample and selects the halfwidth which best reproduces the
% ommited points of the random sample.
%
% In order to demonstrate this functionality let's start with the following
% orientation density function

% Define trigonal crystal symmetry using Enantiomorphic Point Group notation
cs = crystalSymmetry('32');

% Build a density function by combining a uniform texture with two pre-defined texture components
odf = 0.25*uniformODF(cs) + 0.25*unimodalODF(orientation.brass(cs)) + ...
  0.5*fibreODF(fibre.alpha(cs),'halfwidth',10*degree);

% plot the density function as six sigma sections 
plot(odf,'sections',6,'silent','sigma')
mtexColorbar

%%
% and compute $10000$ random orientations representing this density function using the command
% |<orientation.discreteSample.html discreteSample>|

ori = odf.discreteSample(10000)

%%
% Next we estimate the optimal <ODFShapes.html kernel function> using the
% command |<orientation.calcKernel.html calcKernel>| with the default settings.

psi  = calcKernel(ori)

%%
% This kernel can now be used to reconstruct the original ODF from the sampled points using the command
% <DensityEsimation.html density estimation>

odf_rec = calcDensity(ori,'kernel',psi)

% plot the reconstructed ODF and compare it to the plot of the original function.  The results are similar but not identical.
figure;plot(odf_rec,'sections',6,'silent','sigma')
mtexColorbar

%% Exploration of the relationship between estimation error and number of single orientations
%
% In this section we want to compare the different methods for estimating
% the optimal kernel halfwidth. To this end