Optimization of Kernel Functions #2270
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djm87
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I tried doing an optimization on components extracted with calcComponents and it works well for the dozen or so HIPPO textures I've looked at (ranging from weak to strong cubic, some with fibers). I'm curious if there is a better way to go about the optimization? %% Make base odf
CS = crystalSymmetry("432",[3.6,3.6,3.6]);
nori = 10
seed=1
rng(seed)
ori_ini = orientation.rand(nori,CS);
ori_ini = ori_ini.project2FundamentalRegion();
halfwidth_ub = 20;
halfwidth_lb = 5;
halfwidth_ini = randi([halfwidth_lb halfwidth_ub],nori,1)
uniform_weight_ini = 0.6;
vol = rand(10,1);
vol_ini = vol*(1-uniform_weight_ini)/sum(vol);
odfc = (1-sum(vol_ini))*uniformODF(CS);
for i=1:length(ori_ini)
odfc = odfc + vol_ini(i)*unimodalODF(ori_ini(i),SO3vonMisesFisherKernel('halfwidth',halfwidth_ini(i)*degree));
end
odf_ini = SO3FunHarmonic(odfc.calcFourier(),CS)
h_plot = {Miller(1,0,0,CS),Miller(1,1,0,CS),Miller(1,1,1,CS)};
hh=figure; plotPDF(odf_ini,h_plot,'upper','minmax','figsize','tiny','projection','earea','contourf')
%% Extract modes
ori_seed = equispacedSO3Grid(odf_ini.CS,odf_ini.SS,'resolution',2.5*degree)
[ori_calc, vol_calc] = calcComponents(odf_ini,'seed',ori_seed,'tolerance',1*degree);%,'exact');
nori_calc = length(ori_calc);
odfc = (1-sum(vol_calc))*uniformODF(CS);
for i=1:nori_calc
odfc = odfc + vol_calc(i)*unimodalODF(ori_calc(i),SO3vonMisesFisherKernel('halfwidth',12*degree));
end
hh=figure; plotPDF(odfc,h_plot,'upper','minmax','figsize','tiny','projection','earea','contourf')
%% Get halfwidths and volumes
vol = optimvar('vol',nori_calc,'LowerBound',zeros(nori_calc,1),'UpperBound',ones(nori_calc,1));
halfwidth = optimvar('halfwidth',nori_calc,'LowerBound',halfwidth_lb*ones(nori_calc,1),'UpperBound',halfwidth_ub*ones(nori_calc,1));
alpha = ori_calc.alpha;
beta = ori_calc.beta;
gamma = ori_calc.gamma;
%Pass the CS to func
func = @(halfwidth,vol)obj1(halfwidth,vol,alpha,beta,gamma,odf_ini);
%Setup the problem
prob = optimproblem;
prob.Objective = fcn2optimexpr(func,halfwidth,vol)
prob.Constraints.cons1 = sum(vol)<=1;
show(prob)
x0.halfwidth = (halfwidth_ub-halfwidth_lb)/2*ones(nori_calc,1);
x0.vol = vol_calc;
options = optimoptions('fmincon','Display','iter','MaxIterations',30,'StepTolerance',0.0001,'MaxFunctionEvaluations',5000)
[sol1,fval,exitflag] = solve(prob,x0,'Options',options)
odfc = (1-sum(sol1.vol))*uniformODF(CS);
for i=1:nori_calc
odfc = odfc + sol1.vol(i)*unimodalODF(ori_calc(i),SO3vonMisesFisherKernel('halfwidth',sol1.halfwidth(i)*degree));
end
hh=figure; plotPDF(odfc,h_plot,'upper','minmax','figsize','tiny','projection','earea','contourf')
%% Try also refining orientations
alpha = optimvar('alpha',nori_calc);
beta = optimvar('beta',nori_calc);
gamma = optimvar('gamma',nori_calc);
vol = optimvar('vol',nori_calc,'LowerBound',zeros(nori_calc,1),'UpperBound',ones(nori_calc,1));
halfwidth = optimvar('halfwidth',nori_calc,'LowerBound',halfwidth_lb*ones(nori_calc,1),'UpperBound',halfwidth_ub*ones(nori_calc,1));
%Pass the CS to func
func = @(alpha,beta,gamma,vol,halfwidth)obj2(alpha,beta,gamma,vol,halfwidth,odf_ini);
%Setup the problem
prob = optimproblem;
prob.Objective = fcn2optimexpr(func,alpha,beta,gamma,vol,halfwidth)
prob.Constraints.cons1 = sum(vol)<=1;
show(prob)
x0.alpha = ori_calc.alpha;
x0.beta = ori_calc.beta;
x0.gamma = ori_calc.gamma;
x0.vol = sol1.vol;
x0.halfwidth = sol1.halfwidth;
options = optimoptions('fmincon','Display','iter','MaxIterations',50,'StepTolerance',0.0001,'MaxFunctionEvaluations',5000)
[sol2,fval,exitflag] = solve(prob,x0,'Options',options)
% Plot results
odfc = (1-sum(sol2.vol))*uniformODF(CS);
ori = orientation.byEuler(sol2.alpha,sol2.beta,sol2.gamma,'ZYZ',CS);
for i=1:length(ori)
odfc = odfc + sol2.vol(i)*unimodalODF(ori(i),SO3vonMisesFisherKernel('halfwidth',sol2.halfwidth(i)*degree));
end
hh=figure; plotPDF(odfc,h_plot,'upper','minmax','figsize','tiny','projection','earea','contourf');hold on
%% Correlate extracted modes with initial modes
ind = zeros(nori,1);
for i=1:length(ori)
ind(angle(ori(i),ori_ini) < 5*degree)=i;
end
fprintf("Missed component volume:%.2f\n",sum(vol_ini(ind==0)));
for i=1:length(ori)
fprintf("Misorientation of Calculated Component: %0.2f\n",angle(ori(i),mean(ori_ini(ind==i)))/degree);
end
for i=1:length(ori)
fprintf("Deviation in Volume: %0.4f\n",sol2.vol(i)-sum(vol_ini(ind==i)));
end |
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Hi Guys,
Recently I needed to traverse kernel based orientation distributions from various softwares.
I noticed that the von Mises Fisher kernel is defined the same way as the Matthies Standard function for a Gaussian with the exception of the definition of kappa from halfwidth. Matthies defines kappa from the fwhm and you define kappa from the halfwidth at half maximum. It seems that Matthies referring to the distribution as Gaussian was not being precise on his part. Perhaps the standard functions can be added as a kernel option in MTEX for compatibility? Below is the code for what Matthies calls the gaussian portion, he also defines a lorentzian component and defines fibers (read Crystallography Tables or Standard Distributions in Texture Analysis).
These textures models are available in MAUD and we can go from MAUD to MTEX directly for the gaussian SF:
MAUD SFs:
MTEX reproduction of SF
A nice feature in MTEX would be to determine N modes of an ODF then refine the position and halfwidth of those modes as a means of representing the ODF with only a few orientations. Is this possible to do? Ideally, this optimization would operate on sparse pole figures and ODFs. It would be advantageous for initializing MAUD standard functions for extremely sharp textures. The feature would also be beneficial for initializing spatial component mapping from Bragg-edge radiography in which we us a kernel function esque way of accounting for missing intensity at a Bragg-edge.
Best,
Dan
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