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275 lines
11 KiB
275 lines
11 KiB
function [RealignedAcc,RotationMatrixT,Flags] = RealignSensorSignalHRAmp(Acc, FS)
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% History
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% 2013/05/31 SR solve problem of ML-AP rotation of appr. 90 degrees ("Maximum number of function evaluations has been exceeded")
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% 2013/08/15 SR use literature HR (i.e. amplitude not power, and harmonic frequencies not sin^4 windows)
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% 2013/08/16 SR correct: use abs(X) instead of sqrt(X.*X)
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% 2013/09/11 SR use only relevant F (containing odd or even harmonics) and speed up determination of harmonic frequencies
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%% Define VT direction (RVT) as direction of mean acceleration
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MeanAcc = mean(Acc);
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RVT = MeanAcc'/norm(MeanAcc);
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RMLGuess = cross([0;0;1],RVT); RMLGuess = RMLGuess/norm(RMLGuess);
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RAPGuess = cross(RVT,RMLGuess); RAPGuess = RAPGuess/norm(RAPGuess);
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%% Estimate stride frequency
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StrideFrequency = StrideFrequencyFrom3dAcc(Acc, FS);
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AccDetrend = detrend(Acc,'constant');
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N = size(AccDetrend,1);
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%% calculate complex DFT
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F = FS*(0:(N-1))'/N;
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dF = FS/N;
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FHarmRange = [(1:20)'-0.1, (1:20)'+0.1]*StrideFrequency;
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EvenRanges = FHarmRange(2:2:end,:);
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OddRanges = FHarmRange(1:2:end,:);
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EvenHarmonics = zeros(size(F));
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for j=1:size(EvenRanges,1)
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IXRange = [EvenRanges(j,1)/dF, EvenRanges(j,2)/dF]+1;
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IXRangeRound = min(N,round(IXRange));
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EvenHarmonics(IXRangeRound(1):IXRangeRound(2)) = EvenHarmonics(IXRangeRound(1):IXRangeRound(2)) + 1;
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EvenHarmonics(IXRangeRound(1)) = EvenHarmonics(IXRangeRound(1)) - (IXRange(1)-IXRangeRound(1)+0.5);
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EvenHarmonics(IXRangeRound(2)) = EvenHarmonics(IXRangeRound(2)) - (IXRangeRound(2)-IXRange(2)+0.5);
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end
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OddHarmonics = zeros(size(F));
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for j=1:size(OddRanges,1)
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IXRange = [OddRanges(j,1)/dF, OddRanges(j,2)/dF]+1;
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IXRangeRound = min(N,round(IXRange));
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OddHarmonics(IXRangeRound(1):IXRangeRound(2)) = OddHarmonics(IXRangeRound(1):IXRangeRound(2)) + 1;
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OddHarmonics(IXRangeRound(1)) = OddHarmonics(IXRangeRound(1)) - (IXRange(1)-IXRangeRound(1)+0.5);
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OddHarmonics(IXRangeRound(2)) = OddHarmonics(IXRangeRound(2)) - (IXRangeRound(2)-IXRange(2)+0.5);
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end
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%% select only frequencies that contain odd or even harmonics
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RelevantF = OddHarmonics~=0 | EvenHarmonics~=0;
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OddHarmonics = OddHarmonics(RelevantF);
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EvenHarmonics = EvenHarmonics(RelevantF);
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DFT = fft(AccDetrend,N,1);
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DFT = DFT(RelevantF,:);
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DFTMLG = DFT*RMLGuess;
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DFTAPG = DFT*RAPGuess;
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%% Initial estimation of ML-AP realignment (educated guess)
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% We start of by finding the directions ML = MLguess + x*APguess for which
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% the 'harmonic ratio' for the ML directions is maximal or minimal, and we
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% do this by varying x in still the power variant of HR instead of amplitude:
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HRML = @(x) real(((OddHarmonics'.*(DFTMLG'+x*DFTAPG'))*(DFTMLG+x*DFTAPG))...
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/((EvenHarmonics'.*(DFTMLG'+x*DFTAPG'))*(DFTMLG+x*DFTAPG)));
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% or by substititutions
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a = real((OddHarmonics.*DFTAPG)'*DFTAPG);
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b = real((OddHarmonics.*DFTMLG)'*DFTAPG + (OddHarmonics.*DFTAPG)'*DFTMLG);
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c = real((OddHarmonics.*DFTMLG)'*DFTMLG);
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d = real((EvenHarmonics.*DFTAPG)'*DFTAPG);
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e = real((EvenHarmonics.*DFTMLG)'*DFTAPG + (EvenHarmonics.*DFTAPG)'*DFTMLG);
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f = real((EvenHarmonics.*DFTMLG)'*DFTMLG);
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% and by setting d/dx(HRML(x))==0 we get the quadratic formula
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aq = a*e-b*d;
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bq = 2*a*f-2*c*d;
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cq = b*f-c*e;
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% DT = bq^2-4*aq*cq = 4*(a*f-*c*d)*(a*f-c*d) - 4*(a*e-b*d)*(b*f-c*e)
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% = 4aaff +4ccdd -8acdf -4abef -4bcde +4acee + 4bbdf
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% = 4aaff +4ccdd -4abef -4bcde +4ac(ee-df) +4(bb-ac)df
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x = (-bq +[1 -1]*sqrt(bq.^2-4*aq*cq))/(2*aq);
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% with hopefully two solutions for x.
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% from here on use amplitude HR:
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HRML = @(x) real((OddHarmonics'*abs(DFTMLG+x*DFTAPG))...
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/(EvenHarmonics'*abs(DFTMLG+x*DFTAPG)));
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% now we take the x for which the HR is maximal, and the x rotated 90
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% degrees fo which it is minimal, and take the mean angle between them
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HR1 = HRML(x(1));
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HR2 = HRML(x(2));
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if HR1 > HR2
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thetas = atan([x(1),-1/x(2)]);
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else
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thetas = atan([-1/x(1),x(2)]);
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end
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theta = mean(thetas);
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xeg = tan(theta);
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if abs(diff(thetas)) > pi/2 % thetas are more than 90 degrees apart, and the 'mean of the two axes' should be rotated 90 degrees
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xeg = -1/xeg;
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end
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%% Numerical search for best harmonic ratios product
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% Now we want to improve the solution, since the two above directions are
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% not necessarily orthogonal. Thus we want to find the orthogonal
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% directions ML = MLguess + x*APguess, and AP = APguess - x*MLguess,
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% for which the product of 'harmonic ratios' for the ML and AP directions
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% is maximal, and we do this by varying x, starting with the educated guess
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% above using power instead of amplitude:
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% HRML = ((OddHarmonics'.*(DFTMLG'+x*DFTAPG'))*(DFTMLG+x*DFTAPG))...
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% /((EvenHarmonics'.*(DFTMLG'+x*DFTAPG'))*(DFTMLG+x*DFTAPG));
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% HRAP = ((EvenHarmonics'.*(-x*DFTMLG'+DFTAPG'))*(-x*DFTMLG+DFTAPG))...
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% /((OddHarmonics'.*(-x*DFTMLG'+DFTAPG'))*(-x*DFTMLG+DFTAPG));
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% and
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% HR_Prod = HRML*HRAP = ...
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% Now use amplitude
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Minus_HR_Prod = @(x) -real(...
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(OddHarmonics'*abs(DFTMLG+x*DFTAPG))...
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/(EvenHarmonics'*abs(DFTMLG+x*DFTAPG))...
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*(EvenHarmonics'*abs(-x*DFTMLG+DFTAPG))...
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/(OddHarmonics'*abs(-x*DFTMLG+DFTAPG)));
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[xopt1,Minus_HR_Prod_opt1,exitflag1]=fminsearch(Minus_HR_Prod,xeg);
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if exitflag1 == 0 && abs(xopt1) > 100 % optimum for theta beyond +- pi/2
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[xopt1a,Minus_HR_Prod_opt1a,exitflag1a]=fminsearch(Minus_HR_Prod,-xopt1);
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if ~(exitflag1a == 0 && abs(xopt1a) > 100)
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xopt1org = xopt1;
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xopt1 = xopt1a;
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Minus_HR_Prod_opt1 = Minus_HR_Prod_opt1a;
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end
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end
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options = optimset('LargeScale','off','GradObj','off','Display','notify');
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[xopt2,Minus_HR_Prod_opt2,exitflag2]=fminunc(Minus_HR_Prod,xopt1,options);
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if exitflag2 == 0 && abs(xopt2) > 100 % optimum for theta beyond +- pi/2
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[xopt2a,Minus_HR_Prod_opt2a,exitflag2a]=fminsearch(Minus_HR_Prod,-xopt2);
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if ~(exitflag2a == 0 && abs(xopt2a) > 100)
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xopt2 = xopt2a;
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Minus_HR_Prod_opt2 = Minus_HR_Prod_opt2a;
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end
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end
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RML = (RMLGuess+xopt2*RAPGuess)/sqrt(1+xopt2^2);
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RAP = cross(RVT,RML); RAP = RAP/norm(RAP);
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RT = [RVT,RML,RAP];
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if nargout > 1
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RotationMatrixT = RT;
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end
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RealignedAcc = Acc*RT;
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if nargout > 2
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Flags = [exitflag1,exitflag2];
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end
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if nargin > 2 && plotje
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Thetas = pi*(-0.49:0.01:0.49)';
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Xs = tan(Thetas);
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HR_Prod = nan(size(Xs));
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HRMLs = nan(size(Xs));
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for i=1:numel(Xs),
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HR_Prod(i,1) = -Minus_HR_Prod(Xs(i));
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HRMLs(i,1) = HRML(Xs(i));
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end
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figure();
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semilogy(Thetas,[HR_Prod,HRMLs.^2]);
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hold on;
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semilogy(atan(x),[HR1,HR2].^2,'r.');
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semilogy(atan([xopt1;xopt2]),-[Minus_HR_Prod_opt1;Minus_HR_Prod_opt2],'g.');
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semilogy(atan(xeg),-Minus_HR_Prod(xeg),'kx');
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end
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%% attempt to solve it analytically:
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% % We can rewrite HR_Prod as:
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% % HR_Prod =
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% % ( (x.^2)*((OddHarmonics.*DFTAPG)'*DFTAPG) ...
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% % + x*((OddHarmonics.*DFTMLG)'*DFTAPG + (OddHarmonics.*DFTAPG)'*DFTMLG)...
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% % + ((OddHarmonics.*DFTMLG)'*DFTMLG) )...
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% % /
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% % ( (x.^2)*((EvenHarmonics.*DFTAPG)'*DFTAPG) ...
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% % + x*((EvenHarmonics.*DFTMLG)'*DFTAPG + (EvenHarmonics.*DFTAPG)'*DFTMLG)...
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% % + ((EvenHarmonics.*DFTMLG)'*DFTMLG) )
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% % *
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% % ( (x.^2)*((EvenHarmonics.*DFTMLG)'*DFTMLG) ...
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% % - x*((EvenHarmonics.*DFTMLG)'*DFTAPG + (EvenHarmonics.*DFTAPG)'*DFTMLG)...
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% % + ((EvenHarmonics.*DFTAPG)'*DFTAPG) )...
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% % /
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% % ( (x.^2)*((OddHarmonics.*DFTMLG)'*DFTMLG) ...
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% % - x*((OddHarmonics.*DFTMLG)'*DFTAPG + (OddHarmonics.*DFTAPG)'*DFTMLG)...
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% % + ((OddHarmonics.*DFTAPG)'*DFTAPG) )
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% %
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% % or by making these substitutions:
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% a2 = (OddHarmonics.*DFTAPG)'*DFTAPG;
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% a1 = (OddHarmonics.*DFTMLG)'*DFTAPG + (OddHarmonics.*DFTAPG)'*DFTMLG;
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% a0 = (OddHarmonics.*DFTMLG)'*DFTMLG;
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% c2 = (EvenHarmonics.*DFTAPG)'*DFTAPG;
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% c1 = (EvenHarmonics.*DFTMLG)'*DFTAPG + (EvenHarmonics.*DFTAPG)'*DFTMLG;
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% c0 = (EvenHarmonics.*DFTMLG)'*DFTMLG;
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% b2 = c0;
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% b1 = c1;
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% b0 = c2;
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% d2 = a0;
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% d1 = a1;
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% d0 = a2;
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% % we get:
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% % HR_Prod = (x.^2*a2 + x*a1 + a0) * (x.^2*b2 + x*b1 + b0) / (x.^2*c2 + x*c1 + c0) * (x.^2*d2 + x*d1 + d0)
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% % or:
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% % HR_Prod = N/D
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% % with
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% % N = ( x.^4*(a2*b2)...
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% % +x.^3*(a2*b1+a1*b2)...
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% % +x.^2*(a2*b0+a1*b1+a0*b2)...
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% % +x *(a1*b0+a0*b1)...
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% % + a0*b0 )...
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% % and with
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% % D = ( x.^4*(c2*d2)...
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% % +x.^3*(c2*d1+c1*d2)...
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% % +x.^2*(c2*d0+c1*d1+c0*d2)...
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% % +x *(c1*d0+c0*d1)...
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% % + c0*d0 )
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% % or D = ( x.^4*(a0*b0)...
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% % +x.^3*(a1*b0+a0*b1)...
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% % +x.^2*(a2*b0+a1*b1+a0*b2)...
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% % +x *(a2*b1+a1*b2)...
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% % + a2*b2 )
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% % or by substition of
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% n4 = a2*b2;
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% n3 = a2*b1+a1*b2;
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% n2 = a2*b0+a1*b1+a0*b2;
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% n1 = a1*b0+a0*b1;
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% n0 = a0*b0;
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% % d4 = n0;
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% % d3 = n1;
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% % d2 = n2;
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% % d1 = n3;
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% % d0 = n4;
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% % we get
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% % N = sum(ni*x^i) and D=sum(di*x^i)
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% %
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% % Then HR_Prod reaches a maximum or minimum when d/dx (N/D) = 0
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% % or (dN/dx)/D - N/(D^2)*(dD/dx) = 0
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% % or (dN/dx)*D - (dD/dx)*N = 0
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% % or
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% % x^7*(4*n4*d4 - 4*n4*d4)
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% % + x^6*(4*n4*d3 - 4*n3*d4 + 3*n3*d4 - 3*n4*d3)
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% % + x^5*(4*n4*d2 - 4*n2*d4 + 3*n3*d3 - 3*n3*d3 + 2*n2*d4 - 2*n4*d2)
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% % + x^4*(4*n4*d1 - 4*n1*d4 + 3*n3*d2 - 3*n2*d3 + 2*n2*d3 - 2*n3*d2 + 1*n1*d4 - 1*n4*d1)
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% % + x^3*(4*n4*d0 - 4*n0*d4 + 3*n3*d1 - 3*n1*d3 + 2*n2*d2 - 2*n2*d2 + 1*n1*d3 - 1*n3*d1)
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% % + x^2*( 3*n3*d0 - 3*n0*d3 + 2*n2*d1 - 2*n1*d2 + 1*n1*d2 - 1*n2*d1)
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% % + x^1*( 2*n2*d0 - 2*n0*d2 + 1*n1*d1 - 1*n1*d1)
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% % + x^0*( 1*n1*d0 - 1*n0*d1)
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% % = 0
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% % or
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% % x^6*(1*n4*d3 - 1*n3*d4)
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% % + x^5*(2*n4*d2 - 2*n2*d4)
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% % + x^4*(3*n4*d1 - 3*n1*d4 + 1*n3*d2 - 1*n2*d3)
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% % + x^3*(4*n4*d0 - 4*n0*d4 + 2*n3*d1 - 2*n1*d3)
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% % + x^2*( 3*n3*d0 - 3*n0*d3 + 1*n2*d1 - 1*n1*d2)
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% % + x^1*( 2*n2*d0 - 2*n0*d2)
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% % + x^0*( 1*n1*d0 - 1*n0*d1)
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% % = 0
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% % or
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% % (x^6+x^0) * (1*n4*n1 - 1*n3*n0)
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% % + (x^5+x^1) * (2*n4*n2 - 2*n2*n0)
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% % + (x^4+x^2) * (3*n4*n3 - 3*n1*n0 + 1*n3*n2 - 1*n2*n1)
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% % + x^3 * (4*n4*n4 - 4*n0*n0 + 2*n3*n3 - 2*n1*n1)
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% % = 0
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% % or
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% % (x^6+x^0) * (1*(a2*b2)*(a1*b0+a0*b1) - 1*(a2*b1+a1*b2)*(a0*b0))
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% % + (x^5+x^1) * (2*(a2*b2)*(a2*b0+a1*b1+a0*b2) - 2*(a2*b0+a1*b1+a0*b2)*(a0*b0))
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% % + (x^4+x^2) * (3*(a2*b2)*(a2*b1+a1*b2) - 3*(a1*b0+a0*b1)*(a0*b0) + 1*(a2*b1+a1*b2)*(a2*b0+a1*b1+a0*b2) - 1*(a2*b0+a1*b1+a0*b2)*(a1*b0+a0*b1))
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% % + x^3 * (4*(a2*b2)*(a2*b2) - 4*(a0*b0)*(a0*b0) + 2*(a2*b1+a1*b2)*(a2*b1+a1*b2) - 2*(a1*b0+a0*b1)*(a1*b0+a0*b1))
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% % = 0
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% % or
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% % (x^6+x^0) * (a2*b2*a1*b0 + a2*b2*b1*a0 - a2*b1*a0*b0 - b2*a1*a0*b0)
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% % + (x^5+x^1) * (2*(a2*a2*b2*b0+a2*b2*b2*a0+a2*b2*a1*b1) - 2*(a2*a0*b0*b0+b2*a0*a0*b0+a1*b1*a0*b0))
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% % + (x^4+x^2) * (3*a2*a2*b2*b1 +a2*a2*b1*b0 +3*a2*b2*b2*a1 +a2*b2*a1*b0 +a2*b2*b1*a0 +a2*a1*b1*b1 -a2*a1*b0*b0 -a2*a0*b1*b0 +b2*b2*a1*a0 +b2*a1*a1*b1 -b2*a1*a0*b0 -b2*b1*a0*a0 -a1*a1*b1*b0 -a1*b1*b1*a0 -3*a1*a0*b0*b0 -3*b1*a0*a0*b0)
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% % + x^3 * (4*a2*a2*b2*b2 +2*a2*a2*b1*b1 +4*a2*b2*a1*b1 +2*b2*b2*a1*a1 -2*a1*a1*b0*b0 -4*a1*b1*a0*b0 -2*b1*b1*a0*a0 -4*a0*a0*b0*b0)
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% % = 0
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% % which can probably not be solved analytically?.
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