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