WIKI de Bertrand VANDEPORTAELE

svg file for the diagram: https://bvdp.inetdoc.net/files/deflectometry/graphique_pour_daniel/dessin8.svg

display_closed_contour.m

function display_closed_contour(countour,style,origin_style)
if nargin==3
    plot([countour(1,1)],[countour(1,2)],origin_style);
end
for n=1:size(countour,1)-1
    plot([countour(n,1),countour(n+1,1)],[countour(n,2),countour(n+1,2)],style);
end
plot([countour(size(countour,1),1),countour(1,1)],[countour(size(countour,1),2),countour(1,2)],style);
end

rodrigues2R.m

function [ R] = rodrigues2R( wi )
%bvdp
% R=rodrigues2R( [1, 2,-1])
%il faut que R*R^-1= Id  et det(R)=1
theta=norm(wi,2)
w=wi/theta
 
ws=[0 -w(3) w(2) ; w(3) 0 -w(1) ; -w(2) w(1) 0]
wwt=ws*ws
if ~isa(w,'sym') & theta<eps % on ne peut pas tester theta en valeur si w est symbolique
    R=eye(3)
else
    %cette formule ne fournit pas des matrices avec vecteurs de norme 1
    %R=(cos(theta)*eye(3)) + ((1-cos(theta))*wwt) + (sin(theta)*ws)
    %la suivante est tirée de https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula
    R=eye(3)  + (sin(theta)*ws) + ((1-cos(theta))*wwt)
end
end

simu2.m

%B. Vandeportaële & D. Maestro 2017
close all
clear all
%symbolic or numeric simulation
symbolic=1
 
%size of the mirror in x
mirrorwidth=100;
%size of the mirror in y
mirrorheigth=130;
if exist('symbolic','var')
    tcx=sym('tcx','real');
    tcy=sym('tcy','real');
    tcz=sym('tcz','real');
    rcx=sym('rcx','real');
    rcy=sym('rcy','real');
    rcz=sym('rcz','real');
    rc=[rcx,rcy,rcz]';
else
    tcx= 190;
    tcy= 85;
    tcz=200;
    %rc=[0,1,0]'*110*pi/180;
    %rc=[0,1,0]'*110*pi/180;
 
    %set the rotation matrix from axis correspondances
    Rapprox=[0 0.7 -0.7;1 0 0; 0,-0.7 -0.7];
    det(Rapprox)
    [U,S,V] =svd(Rapprox)
    R=U*V'
    rr=rodrigues(R)
    rrangle=norm(rr)
    rraxis=rr/rrangle
    rc=rr
end
 
tc=[tcx,tcy,tcz]';
Tmc=[rodrigues2R(rc), tc; 0,0,0,1];
 
if exist('symbolic','var')
    fu=sym('fu','real');
    fv=sym('fv','real');
    pu=sym('pu','real');
    pv=sym('pv','real');
else
    camera_image_width=2560;
    camera_image_height=2048;
    pu=camera_image_width/2;
    pv=camera_image_height/2;
    horizontal_fov=70*pi/180;
    vertical_fov=60*pi/180;
    fu=(camera_image_width/2)/tan(horizontal_fov/2);
    fv=(camera_image_height/2)/tan(vertical_fov/2);
end
K=[fu 0 pu 0;
    0 fu pv 0;
    0 0 1 0];
 
if ~exist('symbolic','var')
    mirror_corners=[0,0;mirrorwidth,0;mirrorwidth,mirrorheigth;0,mirrorheigth];
    mirror_corners_projected=zeros(size(mirror_corners));
    for n=1:size(mirror_corners,1)
        P=[    mirror_corners(n,:),0,1]';
        PP=inv(Tmc)*P
        if (PP(3)<0)
            display('the point is behind the camera');
        end
        p=K*PP;
        p=p/p(3);
        mirror_corners_projected(n,:)=p(1:2)';
    end
end
 
 
if exist('symbolic','var')
    display_pixel_size_x=sym('display_pixel_size_x','real');
    display_pixel_size_y=sym('display_pixel_size_y','real');
else
    camera_image_boundaries=[0,0;camera_image_width,0;camera_image_width,camera_image_height;0,camera_image_height];
    display_pixel_size_x=0.205;
    display_pixel_size_y=0.205;
    display_pixel_width=2560;
    display_pixel_height=1800;
    %metric unit mm
    display_metric_width=display_pixel_size_x*display_pixel_size_x;
    display_metric_height=display_pixel_size_y*display_pixel_size_y;
end
pixel2metric=[display_pixel_size_x 0 0; 0 display_pixel_size_y 0; 0 0 1];
 
if exist('symbolic','var')
    tdx=sym('tdx','real');
    tdy=sym('tdy','real');
    tdz=sym('tdz','real');
    rdx=sym('rdx','real');
    rdy=sym('rdy','real');
    rdz=sym('rdz','real');
    rd=[rcx,rcy,rcz]';
else
    %set the rotation matrix from axis correspondances
    %close to perfect configuration
    %Rapprox2=[0 -0.7 -0.7;1 0 0; 0,-0.7 +0.7];
    %another configuration with 3 degrees of freedom rotation
    Rapprox2=[0 -0.7 -0.5;0.7 0.2 0.1; 0,-0.7 +0.9];
 
    [U,S,V] =svd(Rapprox2);
    R2=U*V';
    rr2=rodrigues(R2);
    rrangle2=norm(rr2);
    rraxis2=rr2/rrangle2;
    rd=rr2
 
    tdx=-64;
    tdy=-220;
    tdz=530;
end
 
% rd=[rdx,rdy,rdz]';
td=[tdx,tdy,tdz]';
 
Tmd=[rodrigues2R(rd), td; 0,0,0,1];
%Tdm=inv(Tmd);
 
Treflection=[1 0 0 0; 0 1 0 0 ; 0 0 -1 0; 0 0 0 1];
MMM=[1 0 0; 0 1 0; 0 0 0; 0 0 1]; %form P2 to P3 in z=0 plane
if ~exist('symbolic','var')
    display_image_boundaries=[0,0;display_pixel_width,0;display_pixel_width,display_pixel_height;0,display_pixel_height];
    display_corners_projected=zeros(size(display_image_boundaries));
    for n=1:size(mirror_corners,1)
        Pd=[    display_image_boundaries(n,:),1]'
        Pdmetric=pixel2metric*Pd;
        %PdmetricP3=[Pdmetric(1:2);0;Pdmetric(3)]; %compute location in P3 for z=0 plane
        PdmetricP3=MMM*Pdmetric;
        P=Treflection*Tmd*PdmetricP3;
        PP=inv(Tmc)*P
        if (PP(3)<0)
            display('the point is behind the camera');
        end
        p=K*PP;
        p=p/p(3);
        display_corners_projected(n,:)=p(1:2)';
    end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%computation of the homography between display  and camera taking
%into account the reflexion
Hcd=K*inv(Tmc)*Treflection*Tmd*MMM*pixel2metric
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if exist('symbolic','var')
   jacobian(reshape(Hcd,9,1),[rd]) 
end
if exist('symbolic','var')
   jacobian(reshape(Tmd,16,1),[rd]) 
end
if exist('symbolic','var')
    %what happen if we use Tcm instead of Tmc=> other parameterization
    tcix=sym('tcix','real');
    tciy=sym('tciy','real');
    tciz=sym('tciz','real');
    rcix=sym('rcix','real');
    rciy=sym('rciy','real');
    rciz=sym('rciz','real');
 
    rci=[rcix,rciy,rciz]';
    tci=[tcix,tciy,tciz]';
    Tcm=[rodrigues2R(rci), tci; 0,0,0,1];
    Hcd=K*Tcm*Treflection*Tmd*MMM*pixel2metric
 
    J= jacobian(reshape(Hcd,9,1),[rd',td',rci',tci']) 
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%display
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if ~exist('symbolic','var')
    figure
    hold on
    axis equal
    axis ij
    display_closed_contour(camera_image_boundaries,'b-','bx')
    display_closed_contour(mirror_corners_projected,'r-','rx')
    display_closed_contour(display_corners_projected,'g-','gx')
 
    for ud=0:100:display_pixel_width-1
        for vd=0:100:display_pixel_height-1
            p=Hcd*[ud;vd;1];
            p=p/p(3);
            plot(p(1),p(2),'g+');
        end
    end
end

simu2D.m

%B. Vandeportaële & D. Maestro 2017
close all
clear all
%example for the inversion of P1 to P1 scale and offset
pixperm=sym('pixperm','real');
pud=sym('pud','real');
Kd=[pixperm  pud; 0 1]
a=sym('a','real');
b=sym('b','real');
c=sym('c','real');
d=sym('d','real');
Kdinv=[a b;c d ]
eqn= reshape(Kdinv*Kd-eye(2),4,1)
S=solve(eqn,a,b,c,d)
Kdinv=[S.a S.b;S.c S.d]
jacobian(reshape(Kdinv,4,1),[pixperm,pud])
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
tcx=sym('tcx','real');
tcy=sym('tcy','real');
r11=sym('r11','real');
r12=sym('r12','real');
r21=sym('r21','real');
r22=sym('r22','real');
M=[r11,r12,tcx;r21,r22,tcy ;0 0 1]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
fuc=sym('fuc','real');
puc=sym('puc','real');
K=[fuc  puc 0;
    0  1 0];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E=[1 0;0 0 ;0 1]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
H=K*M*E*Kdinv
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
tdx=sym('tdx','real');
tdy=sym('tdy','real');
thetad=sym('thetad','real');
Md=[cos(thetad),-sin(thetad), tdx;sin(thetad),cos(thetad),tdy; 0, 0, 1]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

phase.m

%@Bertrand Vandeportaele LAAS 2017
function phasefunc()
 
close all
 
symbolicComputation=false;
 
ud=218;
vd=153;
 
 
ud=289;
vd=93;
 
of=100;
af=73;
freq=6/(1920);
wt=2*pi*freq;
lum=[];
lumn=[];
n=[];
 
nbPointsPerPeriod=400
global gnoiseLevel
gnoiseLevel=3
 
%useSyntheticData=true
useSyntheticData=false
if useSyntheticData
    %polynomial for some non linear disturbance
    pol=[ 0.003; 0.35; 10]
    %polynomial for some no disturbance
    %pol=[ 0; 1; 0]
    for i=1:nbPoints*PerPeriod
        p=i*2*pi/nbPointsPerPeriod;
        deltai=ud*wt;
        l=of+af*sin(deltai+p);
        lum=[lum,l];
        %non linear disturbance
        l2=pol(3)+pol(2)*l+pol(1)*l^2;
        %add noise
        ln=l2+gnoiseLevel*randn(1);
        lumn=[lumn,ln];
        n=[n,i-1];
    end
else
    folder_path = './pos_02/0016/H/';
    %folder_path = './pos_02/0032/H/';
    %folder_path = './pos_02/0064/H/';
    %file_extension='bmp';
    %num_imgs= 11
 
    folder_path = '/media/HD500GO/daniel/phases/gray/m_00/t4096/pos_02/0016/H/'
    file_extension='tif';
    num_imgs= 64
    ud=128;
    vd=1024;
    % vd=1
 
    folder_path = '/media/HD500GO/daniel/phases/gray/m_00/t40/m_00/pos_02/0040/H/'
    file_extension='tif';
    num_imgs= 128
    ud=1280;
    vd=1024;
    vd=1000
 
    vd=400
 
    %To avoid reloading the images each time.
    % clear I to force it
    global I
 
 
    %if true
    if (size(I,1)==0) || (size(I,2)==0)
        %% Load Images
        fprintf('Start loading data\n')
        tic;
 
        for indx=0:num_imgs-1
            ima_name  = sprintf('%s%s%03d.%s', folder_path,'img_',indx,file_extension);
            if indx==0
                I0 = imread(ima_name);
                [ rs, cs] = size(I0);
                % Allocate matrix
                I = zeros(rs, cs, num_imgs);
                I(:,:,1) = I0;
            else
                %ima_name = fullfile(folder_path, image_files(indx).name);
                I(:,:,indx+1) = imread(ima_name);
            end
        end
        fprintf([num2str(num_imgs), ' Images Loaded. '])
        toc
    end
end
n=(0:num_imgs-1)';
lumn=reshape(I(vd,ud,:),num_imgs,1)';
lum=lumn;
nbPointsPerPeriod=num_imgs
%polynomial for no disturbance
pol=[ 0; 1; 0]
 
 
%numberOfSamplesForMeaning=10
%mask=(1/numberOfSamplesForMeaning)*ones(1,numberOfSamplesForMeaning)
%lumn=conv(lumn,mask,'same')
 
global gpuls
gpuls=2*pi/nbPointsPerPeriod;
 
 
%%%%%%%%%%%%%%%%TEST to reduce the number of input data%%%%%%%%%%%%%%%%%%%%
% nbPointsPerPeriod=6
% n=n(1:nbPointsPerPeriod)
% lum=lum(1:nbPointsPerPeriod)
% lumn=lumn(1:nbPointsPerPeriod)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%TEST pertubate some points to check the robustness
%lumn(40)=6000
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
tic
[xLinear covarianceOutLinear]=fitSinusLinear(n,lumn,gpuls)
toc
 
tic
[xLinearOpt covarianceOutLinearOpt,err]=fitSinusLinearOptimized(n,lumn,gpuls)
toc
 
[xPerfect covarianceOutPerfect]=fitSinusLinear(n,lum,gpuls)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%non linear method
if symbolicComputation
    ofs = sym('ofs','real');
    amp = sym('amp','real');
    gpuls = sym('puls','real');
    phas = sym('phas','real');
    t = sym('t','real');
    y=ofs+amp*sin(gpuls*t+phas)
    ymes = sym('ymes','real');
    res=expand((ymes-y))
    %jacobian of res used during optimization
    J=jacobian(res,[ofs,amp,phas])'
    %jacobian of res^2, should be 0 at the end of the optimization only
    %when there is no noise on the data
    %J2=jacobian(res^2,[ofs,amp,phas])'
end
%ofs=of+10;
%amp=af*0.9;
%phas=0;
%x0=[ofs,amp,phas];
%x0=x0+[10,40,0.3]%add some error
[xNonLinear,resxNonLinear,JxNonLinear]= sinusNonLinearFit(n,lumn,xLinear);
 
%NON!!!!!!!!!!!!!!!!!
%covarianceInNonLinear=diag(resxNonLinear.*resxNonLinear);
%W=inv(covarianceInNonLinear);
 
%mean of absolute value of errr
mean (abs(resxNonLinear),1)
variance=var(resxNonLinear)
standardDeviation=variance^0.5
W=eye(size(resxNonLinear,1))*variance^-1;
 
JxNonLinear
 
InformationOutNonLinear= (JxNonLinear'*W*JxNonLinear)
covarianceOutNonLinear= inv(InformationOutNonLinear);
 
 
displayResult(xLinear,  covarianceOutLinear,'xLinear      ' )
%theese 2 values have comparable variances
displayResult(xNonLinear,  covarianceOutNonLinear,'xNonLinear   ' )
%return
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%non linear method  taking into account the non linearity of the
%camera/display
if symbolicComputation
    ofs = sym('ofs','real');
    amp = sym('amp','real');
    gpuls = sym('puls','real');
    phas = sym('phas','real');
    t = sym('t','real');
    yperfect=ofs+amp*sin(gpuls*t+phas);
 
    nla = sym('nla','real');
    nlb = sym('nlb','real');
    nlc = sym('nlc','real');
    y=nla * yperfect^2 + nlb * yperfect + nlc;
 
    ymes = sym('ymes','real');
    res=expand((ymes-y))
    %jacobian of res used during optimization
    J=jacobian(res,[ofs,amp,phas,nla,nlb,nlc])'
 
end
%ofs=of+10;
%amp=af*0.9;
%phas=0;
%x0=[ofs,amp,phas];
%x0=x0+[10,40,0.3]%add some error
x0_2=[xLinear, 0, 1, 0]
%x0_2=[1500 1.3 0.5 , 0, 1, 0]
[xNonLinear_2,resxNonLinear_2,JxNonLinear_2]= sinusNonLinearFit2(n,lumn,x0_2)
 
%covarianceOutNonLinear_2=zeros(6,6);
%covarianceOutNonLinear_2=JxNonLinear_2'*diag(resxNonLinear_2.*resxNonLinear_2)*JxNonLinear_2
%TODO: plot that non linear function and generate the corrected plot
%for sinus!
 
%JxNonLinear_2(:,4)=0
 
pol=xNonLinear_2(4:6)
%!!!!! NON j'ai estimé le polynome inverse!!!!
 
%xNonLinear_2=xNonLinear_2(1:3);
%covarianceOutNonLinear_2=JxNonLinear_2(:,1:3)'*diag(resxNonLinear_2.*resxNonLinear_2)*JxNonLinear_2(:,1:3)
%xNonLinear_2(4:6) =pol
%covarianceOutNonLinear_2(6,6)=0
%W=diag(resxNonLinear_2.*resxNonLinear_2);
%covarianceInNonLinear=inv(W)
%covarianceInNonLinear=eye(128);
%InformationOutNonLinear_2= (JxNonLinear_2'*covarianceInNonLinear*JxNonLinear_2)
%covarianceOutNonLinear_2= inv(InformationOutNonLinear_2)
 
resxNonLinear_2
%mean of absolute value of errr
mean (abs(resxNonLinear_2),1)
variance=var(resxNonLinear_2)
variancebis=mean(resxNonLinear_2.*resxNonLinear_2) %>give approximately the variance if the mean is close to 0  (no bias)
varianceter=(resxNonLinear_2'*resxNonLinear_2)/nbPointsPerPeriod
standardDeviation=variance^0.5
 
%
 
 
W1=eye(size(resxNonLinear_2,1))*variance^-1;
%W2=diag( ((resxNonLinear_2.*resxNonLinear_2)*nbPointsPerPeriod).^-1 )
%W2=diag( ((resxNonLinear_2.^2)*nbPointsPerPeriod).^-1 ) 
 
%the squared error is computed for each data point, with a sample size of
%one, so the variance of this 1 set is equal o val^2/1
%W2=diag( ((resxNonLinear_2.^2)).^-1 );   %<==> W2=diag( (resxNonLinear_2.^-2))   
%traceW1=trace(W1)
%MtraceW2=trace(W2)
%return
%[diag(W1),diag(W2)]'*ones(128,2)
 
%return
W=W1;
 
 
    %processing of uncertainty using the whole set of parameters lead to
    %non invertible information matrix
    InformationOutNonLinear_2= (JxNonLinear_2'*W*JxNonLinear_2)
    covarianceOutNonLinear_2= inv(InformationOutNonLinear_2) %= InformationOutNonLinear_2^-1
 
 
     [u,s,v]=svd(covarianceOutNonLinear_2)
 
covarianceOutNonLinear_2-(     u*s*v')
sinv=zeros(6,6);
for i=1:6
    s(i,i) %The sixth term is e-13 and the first is e+16, poor conditionning
    if (i~=6)
        sinv(i,i)=1/s(i,i);
    end
end
%sinv=inv(s)
sinv
inverse=v*sinv*u'
 
for i=1:6
    inverse(i,i)
end
 
rank(covarianceOutNonLinear_2)
 
inverse*covarianceOutNonLinear_2
if false
end
 
%work on a reduced version focucing on the sinusoidal function parameters,
%because the information matrix is of rank 3 only!
JxNonLinear_2red=JxNonLinear_2(:,1:3);
InformationOutNonLinear_2= (JxNonLinear_2red'*W*JxNonLinear_2red)
 
rank(InformationOutNonLinear_2)
 
covarianceOutNonLinear_2= inv(InformationOutNonLinear_2)
covarianceOutNonLinear_2(1,1)
covarianceOutNonLinear_2(2,2)
covarianceOutNonLinear_2(3,3)
 
%add 0s to the covariance matrix so it can be displayed later
covarianceOutNonLinear_2(6,6)=0;
 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
 
figure
subplot(1,3,1);
%xx=1:255;
xx=1:(2^12)-1;
yy=polyval(pol,xx);
plot(xx, yy,'r-')
subplot(1,3,2);
hold on
plot(n, lum,'r+')
displaySin(min(n)-1,max(n)+1,xPerfect,'r-');
plot(n, lumn,'b+')
displaySin(min(n)-1,max(n)+1,xLinear,'k:');
displaySin(min(n)-1,max(n)+1,xNonLinear,'g--');
 
displaySinNonLinear(min(n)-1,max(n)+1,xNonLinear_2,'b--');
 
 
legend('data without noise','function without noise','data with noise','fitted function using linear method','fitted function using non linear method','fitted function using non linear method and non linear correction')
subplot(1,3,3);
 
%resxNonLinear=randn(size(resxNonLinear,1),1);
resToPlot=resxNonLinear_2;
 
numberOfBinsToPlot=12;
maxErr=round(max(abs(resToPlot)))+1;
binranges = [-maxErr:maxErr/(numberOfBinsToPlot-2):maxErr];
%binranges = [-maxErr:maxErr/10:maxErr];
nbBins=size(binranges,1);
[nelements,ind] =histc(resToPlot,binranges);
plot(binranges,nelements*nbBins/nbPointsPerPeriod,'r+');
hold on
plot(resToPlot,zeros(size(resToPlot,1)),'g+');
meanOfError=mean(resToPlot)
varianceOfError=var(resToPlot)
standardDeviationOfError=sqrt(varianceOfError)
gaussianSamplesx=[-maxErr:0.1:maxErr];
gaussianSamplesy=(1/sqrt(2*pi*varianceOfError))*exp(-( ((gaussianSamplesx-meanOfError).^2)/(2*varianceOfError)));
plot(gaussianSamplesx, gaussianSamplesy ,'b-');
 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
displayResult(xPerfect, covarianceOutPerfect,'xPerfect     ' )
displayResult(xLinear,  covarianceOutLinear,'xLinear      ' )
displayResult(xNonLinear_2,  covarianceOutNonLinear_2,'xNonLinear_2 ' )
 
%convert the 3sigma uncertainty in phase to pixel position
pixelPeriod=16
phase3sigma=3*sqrt(covarianceOutLinear(3,3))
pixelLocation3Sigma=phase3sigma*pixelPeriod/(2*pi)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%compare the distribution of the error with the theroretic one using the
%Cumulative distribution function
figure
resxNonLinear_2sorted=sort(resxNonLinear_2);
plot(resxNonLinear_2sorted,n/nbPointsPerPeriod,'r-');
%fonction de répartition: https://en.wikipedia.org/wiki/Normal_distribution
variance=var(resxNonLinear_2sorted);
x=[-5*sqrt(variance):5*sqrt(variance)]';
for i=1:size(x,1)
y(i)=(1/2)*(1+erf( (x(i)-mean(resxNonLinear_2sorted))/(sqrt(variance)*sqrt(2))));
end
hold on 
plot(x,y,'b-');
legend('observed','ideal')
title('Cumulative distribution function')
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [x,resx,Jx]= sinusNonLinearFit(datax,datay,x0)
global gdatax
global gdatay
gdatax=datax;
gdatay=datay;
[res0] = myfun2(x0);
options = optimset('Jacobian','on','Display','iter', 'MaxFunEvals',200, 'Algorithm'  ,{'levenberg-marquardt',.005});
[x,resnorm,residual,exitflag] = lsqnonlin(@myfun2,x0,-Inf,+Inf,options);    % Invoke optimizer
%[resxPerfect] = myfun2(xPerfect);
%norm(resxPerfect,2)
[resx,Jx] = myfun2(x);
norm(resx,2)
end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [res,J] = myfun2(x)
global gpuls
global gdatax
global gdatay
ofs=x(1);
amp=x(2);
phas=x(3);
nbdata=max(size(gdatax));
res=zeros(nbdata,1);
J=zeros(nbdata,3);
%J2=zeros(nbdata,3);
for i=1:nbdata
    t=gdatax(i);
    ymes=gdatay(i);
    y=ofs+amp*sin(gpuls*t+phas);
    res(i)=ymes-y;
    %jacobian for residuals
    J(i,:)=[                                         -1,
        - cos(gpuls*t)*sin(phas) - sin(gpuls*t)*cos(phas),
        amp*sin(gpuls*t)*sin(phas) - amp*cos(gpuls*t)*cos(phas)];
    %jacobian for residuals^2
    %J2(i,:)=[                                                        2*ofs - 2*ymes + 2*amp*cos(puls*t)*sin(phas) + 2*amp*sin(puls*t)*cos(phas),
    %    2*(cos(puls*t)*sin(phas) + sin(puls*t)*cos(phas))*(ofs - ymes + amp*cos(puls*t)*sin(phas) + amp*sin(puls*t)*cos(phas)),
    %    2*(amp*cos(puls*t)*cos(phas) - amp*sin(puls*t)*sin(phas))*(ofs - ymes + amp*cos(puls*t)*sin(phas) + amp*sin(puls*t)*cos(phas))];
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [res] = displaySin(tmin,tmax,x,style)
global gpuls
ofs=x(1);
amp=x(2);
phas=x(3);
t=tmin:0.1:tmax;
recons=ofs+amp*sin(gpuls.*t+phas);
plot(t,recons,style)
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [x,resx,Jx]= sinusNonLinearFit2(datax,datay,x0)
global gdatax
global gdatay
gdatax=datax;
gdatay=datay;
%[res0] = myfun3(x0);
options = optimset('Jacobian','on','Display','iter', 'MaxFunEvals',200, 'Algorithm'  ,{'levenberg-marquardt',.005});
%options = optimset('Jacobian','off','Display','iter', 'MaxFunEvals',200, 'Algorithm'  ,{'levenberg-marquardt',.005});
[x,resnorm,resx,exitflag,output,lambda,Jx] = lsqnonlin(@myfun3,x0,-Inf,+Inf,options);    % Invoke optimizer
 
%perturbate to add bias
%resx=resx+100*ones(size(resx,2));
 
%[resxPerfect] = myfun2(xPerfect);
%norm(resxPerfect,2)
%[resx,Jx] = myfun3(x);
norm(resx,2)
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [res,J] = myfun3(x)
global gpuls
global gdatax
global gdatay
ofs=x(1);
amp=x(2);
phas=x(3);
nla=x(4);
nlb=x(5);
nlc=x(6);
 
nbdata=max(size(gdatax));
res=zeros(nbdata,1);
J=zeros(nbdata,6);
%J2=zeros(nbdata,3);
for i=1:nbdata
    t=gdatax(i);
    ymes=gdatay(i);
    yperfect=ofs+amp*sin(gpuls*t+phas);
    y=nla * yperfect^2 + nlb * yperfect + nlc;
    res(i)=ymes-y;
    %jacobian for residuals
    J(i,:)=[                                                                                                                                                                                                                                                          - nlb - 2*nla*ofs - 2*amp*nla*cos(gpuls*t)*sin(phas) - 2*amp*nla*sin(gpuls*t)*cos(phas)
        - nlb*cos(gpuls*t)*sin(phas) - nlb*sin(gpuls*t)*cos(phas) - 2*amp*nla*cos(gpuls*t)^2*sin(phas)^2 - 2*amp*nla*sin(gpuls*t)^2*cos(phas)^2 - 2*nla*ofs*cos(gpuls*t)*sin(phas) - 2*nla*ofs*sin(gpuls*t)*cos(phas) - 4*amp*nla*cos(gpuls*t)*sin(gpuls*t)*cos(phas)*sin(phas)
        amp*nlb*sin(gpuls*t)*sin(phas) - amp*nlb*cos(gpuls*t)*cos(phas) - 2*amp*nla*ofs*cos(gpuls*t)*cos(phas) - 2*amp^2*nla*cos(gpuls*t)*sin(gpuls*t)*cos(phas)^2 + 2*amp^2*nla*cos(gpuls*t)*sin(gpuls*t)*sin(phas)^2 + 2*amp*nla*ofs*sin(gpuls*t)*sin(phas) - 2*amp^2*nla*cos(gpuls*t)^2*cos(phas)*sin(phas) + 2*amp^2*nla*sin(gpuls*t)^2*cos(phas)*sin(phas)
        - ofs^2 - amp^2*cos(gpuls*t)^2*sin(phas)^2 - amp^2*sin(gpuls*t)^2*cos(phas)^2 - 2*amp*ofs*cos(gpuls*t)*sin(phas) - 2*amp*ofs*sin(gpuls*t)*cos(phas) - 2*amp^2*cos(gpuls*t)*sin(gpuls*t)*cos(phas)*sin(phas)
        - ofs - amp*cos(gpuls*t)*sin(phas) - amp*sin(gpuls*t)*cos(phas)
        -1]';
 
 
 
    %    [                                         -1,
    %    - cos(gpuls*t)*sin(phas) - sin(gpuls*t)*cos(phas),
    %    amp*sin(gpuls*t)*sin(phas) - amp*cos(gpuls*t)*cos(phas)];
    %jacobian for residuals^2
    %J2(i,:)=[                                                        2*ofs - 2*ymes + 2*amp*cos(puls*t)*sin(phas) + 2*amp*sin(puls*t)*cos(phas),
    %    2*(cos(puls*t)*sin(phas) + sin(puls*t)*cos(phas))*(ofs - ymes + amp*cos(puls*t)*sin(phas) + amp*sin(puls*t)*cos(phas)),
    %    2*(amp*cos(puls*t)*cos(phas) - amp*sin(puls*t)*sin(phas))*(ofs - ymes + amp*cos(puls*t)*sin(phas) + amp*sin(puls*t)*cos(phas))];
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [res] = displaySinNonLinear(tmin,tmax,x,style)
global gpuls
ofs=x(1);
amp=x(2);
phas=x(3);
nla=x(4);
nlb=x(5);
nlc=x(6);
t=tmin:0.1:tmax;
reconsperfect=ofs+amp*sin(gpuls.*t+phas);
recons=nla * reconsperfect.^2 + nlb * reconsperfect + nlc;
plot(t,recons,style)
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [X covarianceOut]=fitSinusLinear(datax,datay,puls)
%linear method for known pulsation
%https://fr.mathworks.com/matlabcentral/fileexchange/41246-sine-function-fit
global gnoiseLevel
symbolicComputation=false
nbdata=max(size(datax));
A=zeros(nbdata,3);
B=zeros(nbdata,1);
for i=1:nbdata
    t=datax(i);
    ymes=datay(i);
    B(i)=ymes;
    dx=puls*t;
    A(i,:)= [1 cos(dx) sin(dx)]    ;
end
C=(A'*A)^(-1)*A';
%solve the system by expressing it as a linear product
Xa=C*B;
%propagate uncertainty from datay to Xa
estimateVarianceFromMSE=true
if ~estimateVarianceFromMSE
    varianceIn=gnoiseLevel^2 * eye(nbdata);
else
    %in real case, the input variance should be estimated thanks to the MSE
    %varianceIn=zeros(nbdata);
    %for i=1:nbdata
    %t=datax(i);
    %ymes=datay(i);
    %dx=puls*t;
    %ypred=(Xa(1)*1+ Xa(2)* cos(dx) + Xa(3)* sin(dx));
    %err=ymes- ypred;
    %varianceIn(i,i)=err^2;
    %end
    err=A*Xa-B;
    varianceIn=diag(err.*err);
end
%covariance for the internal parameterization
covarianceOutIntern=C*varianceIn*C';
%check if there is covariance (diag matrix)
covarianceOutIntern-diag([covarianceOutIntern(1,1), covarianceOutIntern(2,2), covarianceOutIntern(3,3)]);
%this uncertainty has to be transfered to the final parameterization
if symbolicComputation
    a=sym('a','real');
    a=a;
    b1 = sym('b1','real');
    b2 = sym('b2','real');
    b=sqrt(b1*b1+b2*b2);
    %c=wrapTo2Pi(atan2(b1,b2)); %bring c between 0 and 2pi
    c=atan(b1/b2); %bring c between 0 and 2pi
    jacobian(b,[b1,b2])  %  [ b1/(b1^2 + b2^2)^(1/2), b2/(b1^2 + b2^2)^(1/2)]
    jacobian(c,[b1,b2]) %  [ 1/(b2*(b*1^2/b2^2 + 1)), -b1/(b2^2*(b1^2/b2^2 + 1))]
    jacobian([a,b,c],[a,b1,b2])
    %[1, 0 , 0;
    %0,  b1/(b1^2 + b2^2)^(1/2), b2/(b1^2 + b2^2)^(1/2);
    %0, 1/(b2*(b1^2/b2^2 + 1)), -b1/(b2^2*(b1^2/b2^2 + 1))]
end
a=Xa(1);
b1=Xa(2);
b2=Xa(3);
%compute b & c
b=sqrt(b1*b1+b2*b2);
c=wrapTo2Pi(atan2(b1,b2)); %bring c between 0 and 2pi
%compare c with
%wt*ud
JJ=[1, 0 , 0;
    0,  b1/(b1^2 + b2^2)^(1/2), b2/(b1^2 + b2^2)^(1/2);
    0, 1/(b2*(b1^2/b2^2 + 1)), -b1/(b2^2*(b1^2/b2^2 + 1))]
covarianceOut=JJ*covarianceOutIntern*JJ'
covarianceOut-diag([covarianceOut(1,1), covarianceOut(2,2), covarianceOut(3,3)]);
X=[a b c];
if symbolicComputation
    cdx1 = sym('cdx1','real');
    sdx1 = sym('sdx1','real');
    y1=    sym('y1',  'real');
    cdx2 = sym('cdx2','real');
    sdx2 = sym('sdx2','real');
    y2=    sym('y2',  'real');
    cdx3 = sym('cdx3','real');
    sdx3 = sym('sdx3','real');
    y3=    sym('y3',  'real');
    cdx4 = sym('cdx4','real');
    sdx4 = sym('sdx4','real');
    y4=    sym('y4',  'real');
    A=[1 cdx1 sdx1;
        1 cdx2 sdx2;
        1 cdx3 sdx3]
    B=[y1;y2;y3]
    solution=simplify((A'*A)^(-1)*A'*B)
    A=[1 cdx1 sdx1;
        1 cdx2 sdx2;
        1 cdx3 sdx3;
        1 cdx4 sdx4]
    B=[y1;y2;y3;y4]
    C=simplify((A'*A)^(-1)*A')
    leastSquaresSolution=simplify(C*B)
    %practically, an incremental algorithm should keep track of the
    %symetric matrix
    %(A'*A)=[                         4,                     cdx1 + cdx2 + cdx3 + cdx4,                     sdx1 + sdx2 + sdx3 + sdx4]
    %       [ cdx1 + cdx2 + cdx3 + cdx4,             cdx1^2 + cdx2^2 + cdx3^2 + cdx4^2, cdx1*sdx1 + cdx2*sdx2 + cdx3*sdx3 + cdx4*sdx4]
    %       [ sdx1 + sdx2 + sdx3 + sdx4, cdx1*sdx1 + cdx2*sdx2 + cdx3*sdx3 + cdx4*sdx4,             sdx1^2 + sdx2^2 + sdx3^2 + sdx4^2]
    %and the vector
    %(A'*B)= [y1 + y2 + y3 + y4
    %         cdx1*y1 + cdx2*y2 + cdx3*y3 + cdx4*y4
    %         sdx1*y1 + sdx2*y2 + sdx3*y3 + sdx4*y4]
    var1 = sym('var1','real');
    var2 = sym('var2','real');
    var3 = sym('var3','real');
    var4 = sym('var4','real');
    %for constant known variance on input data
    varIn=diag([var1,var1,var1,var1]);
    simplify(A'*varIn*A)
    %[                           4*var1,                     var1*(cdx1 + cdx2 + cdx3 + cdx4),                     var1*(sdx1 + sdx2 + sdx3 + sdx4)]
    %[ var1*(cdx1 + cdx2 + cdx3 + cdx4),             var1*(cdx1^2 + cdx2^2 + cdx3^2 + cdx4^2), var1*(cdx1*sdx1 + cdx2*sdx2 + cdx3*sdx3 + cdx4*sdx4)]
    %[ var1*(sdx1 + sdx2 + sdx3 + sdx4), var1*(cdx1*sdx1 + cdx2*sdx2 + cdx3*sdx3 + cdx4*sdx4),             var1*(sdx1^2 + sdx2^2 + sdx3^2 + sdx4^2)]
    %for variances obtained from MSE
    varIn=diag([var1,var2,var3,var4]);
    simplify(A'*varIn*A)
    %[                     var1 + var2 + var3 + var4,                     cdx1*var1 + cdx2*var2 + cdx3*var3 + cdx4*var4,                     sdx1*var1 + sdx2*var2 + sdx3*var3 + sdx4*var4]
    %[ cdx1*var1 + cdx2*var2 + cdx3*var3 + cdx4*var4,             var1*cdx1^2 + var2*cdx2^2 + var3*cdx3^2 + var4*cdx4^2, cdx1*sdx1*var1 + cdx2*sdx2*var2 + cdx3*sdx3*var3 + cdx4*sdx4*var4]
    %[ sdx1*var1 + sdx2*var2 + sdx3*var3 + sdx4*var4, cdx1*sdx1*var1 + cdx2*sdx2*var2 + cdx3*sdx3*var3 + cdx4*sdx4*var4,             var1*sdx1^2 + var2*sdx2^2 + var3*sdx3^2 + var4*sdx4^2]
end
end
% A.X=B
% A'.A.X=A'.B
%p 31 of http://www.stat.columbia.edu/~fwood/Teaching/w4315/Fall2009/lecture_11
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function displayResult(res,covar,name)
%display mean and 3Sigma values
nbdata=max(size(res));
s='';
s=sprintf('%s :',name);
for n=1:nbdata
    s=sprintf('%s %d +/- %d  ,',s,res(n),3*sqrt(covar(n,n)));
end
disp(s);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function test()
%computation of the inverse of a symetric 3x3 matrix
if false
    a1 = sym('ATAa1','real');
    b1 = sym('ATAb1','real');
    c1 = sym('ATAc1','real');
    d1 = sym('ATAd1','real');
    e1 = sym('ATAe1','real');
    f1 = sym('ATAf1','real');
    A1=[a1 b1 c1 ; b1 d1 e1; c1 e1 f1]
 
    %     a2 = sym('a2','real');
    %     b2 = sym('b2','real');
    %     c2 = sym('c2','real');
    %     d2 = sym('d2','real');
    %     e2 = sym('e2','real');
    %     f2 = sym('f2','real');
    %     A2=[a2 b2 c2 ; b2 d2 e2; c2 e2 f2]
    %     prod=A1*A2
    %     equations=reshape(prod-eye(3),9,1)
    %     solve(equations(1:6),[a2 b2 c2  d2 e2  f2])
    %     solve(equations(1:6),a2 )
 
    % http://www.tutorvista.com/content/math/inverse-of-symmetric-matrix/
    % A-1= adj(A)/det(A)
    A2tmp=adjoint(A1)
    determinant=det(A1)
    A2=A2tmp/determinant
end
end

fitSinusLinearOptimized.m

function [X covarianceOut, err]=fitSinusLinearOptimized(datax,datay,puls)
nbdata=max(size(datax));
A=zeros(nbdata,3);
B=zeros(nbdata,1);
 
err=zeros(nbdata,1);
var=zeros(nbdata,1);
 
ATA=zeros(3,3);
ATB=zeros(3,1);
ATAa=0;
ATAb=0;
ATAc=0;
ATAd=0;
ATAe=0;
ATAf=0;
ATBa=0;
ATBb=0;
ATBc=0;
 
cdx=zeros(nbdata,1);
sdx=zeros(nbdata,1);
cdx2=zeros(nbdata,1);
sdx2=zeros(nbdata,1);
cdxsdx=zeros(nbdata,1);
 
for i=1:nbdata
    x=datax(i);
    y=datay(i);
    B(i)=y;
    dx=puls*x;
    cx=cos(dx);
    sx=sin(dx);
    cx2=cx*cx;
    sx2=sx*sx;
    cxsx=cx*sx;
    cdx(i)=cx;
    sdx(i)=sx;
    cdx2(i)=cx2;
    sdx2(i)=sx2;
    cdxsdx(i)=cxsx;
 
    ATAa=ATAa+1;
    ATAb=ATAb+cx;
    ATAc=ATAc+sx;
    ATAd=ATAd+cx2;
    ATAe=ATAe+cxsx;
    ATAf=ATAf+sx2;
 
    ATBa=ATBa+y;
    ATBb=ATBb+cx*y;
    ATBc=ATBc+sx*y;
end
%complete symetric matrices
 
%ATA=[ATAa , ATAb , ATAc ; ATAb , ATAd , ATAe ; ATAc , ATAe , ATAf];
%ATB=[ATBa , ATBb , ATBc ]';
%leastSquaresSolution=inv(ATA)* ATB
%a=leastSquaresSolution(1);
%b1=leastSquaresSolution(2);
%b2=leastSquaresSolution(3);
 
determinant =- ATAf*ATAb^2 + 2*ATAb*ATAc*ATAe - ATAd*ATAc^2 - ATAa*ATAe^2 + ATAa*ATAd*ATAf;
ATAinva= (ATAd*ATAf - ATAe^2     )/determinant;
ATAinvb= (ATAc*ATAe - ATAb*ATAf  )/determinant;
ATAinvc= (ATAb*ATAe - ATAc*ATAd  )/determinant;
ATAinvd= (ATAa*ATAf - ATAc^2     )/determinant;
ATAinve= (ATAb*ATAc - ATAa*ATAe  )/determinant;
ATAinvf= (ATAa*ATAd - ATAb^2     )/determinant;
 
ATAinv=[ ATAinva ATAinvb ATAinvc; ATAinvb ATAinvd ATAinve; ATAinvc ATAinve ATAinvf];
 
a =ATAinva * ATBa + ATAinvb * ATBb + ATAinvc * ATBc;
b1=ATAinvb * ATBa + ATAinvd * ATBb + ATAinve * ATBc;
b2=ATAinvc * ATBa + ATAinve * ATBb + ATAinvf * ATBc;
Xa=[a b1 b2]
%compute b & c
b=sqrt(b1*b1+b2*b2);
c=wrapTo2Pi(atan2(b1,b2)); %bring c between 0 and 2pi
 
X=[a b c ];
 
%compute MSE and variance....
ATVarAa=0;
ATVarAb=0;
ATVarAc=0;
ATVarAd=0;
ATVarAe=0;
ATVarAf=0;
 
for i=1:nbdata
    er=  1 * a + cdx(i,1) * b1 + sdx(i,1) * b2;
    vari= er*er;
    err(i)=er;
    var(i)=vari;
    ATVarAa= ATVarAa + vari;
    ATVarAb= ATVarAb + cdx(i,1)  * vari;
    ATVarAc= ATVarAc + sdx(i,1)  * vari;
    ATVarAd= ATVarAd + cdx2(i,1) * vari;
    ATVarAe= ATVarAe + cdx(i,1)*sdx(i,1)  * vari;
    ATVarAf= ATVarAf + sdx2(i,1) * vari;
 
end
ATVarA=[ ATVarAa ATVarAb ATVarAc; ATVarAb ATVarAd ATVarAe; ATVarAc ATVarAe ATVarAf];
covarianceOutIntern=ATAinv'*ATVarA*ATAinv;
JJ=[1, 0 , 0;
    0,  b1/(b1^2 + b2^2)^(1/2), b2/(b1^2 + b2^2)^(1/2);
    0, 1/(b2*(b1^2/b2^2 + 1)), -b1/(b2^2*(b1^2/b2^2 + 1))];
covarianceOut=JJ*covarianceOutIntern*JJ';
end