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ParabolaFit.m
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function [BestFocus,BestDirectrix,GeneralForm,Errors,Foci,Directrices,StepSizes,GradientVectors,Penalty]=ParabolaFit(Samples,Lambda,StepSize,MaxStepSize,MinStepSize,NumSteps,T)
% Fit a parabola to data, enhanced version, with further simplifications,
% normalization of the equation of the directrix at each step and in the
% inicialization, and random restart each 1000 steps.
% Gradient descent with adaptive step size. The best solution is returned
% rather than the last one.
% Positive values of the Lambda parameters are recommended because this way
% the solutions with the focus very close to the directrix are avoided.
% Inputs:
% Samples=Matrix of size 2 x NumSamples with the training samples
% Outputs:
% BestFocus=Vector of size 2 x 1 with the focus point of the fitted
% parabola.
% BestDirectrix=Vector of size 3 x 1 with the coefficients of the
% directrix (in general form, Mx + Ny + P = 0) of the fitted parabola.
% GeneralForm=Vector of size 6 x 1 with the coefficients of the
% fitted parabola in general form, Ax^2 + Bxy + Cy^2 + Dx + Ey + F =
% 0.
% Created by Ezequiel López-Rubio and Karl Thurnhofer-Hemsi
% Last modification: 10/06/2018
[Dimension, NumSamples]=size(Samples);
% Initialize the focus to the mean of the distribution and the
% directrix to the linear regression line of the distribution
StdDev=mean(std(Samples,0,2));
% BaseFocus=mean(Samples,2)+randn(2,1); % [u v]
BaseFocus=mean(Samples,2)+0.5*std(Samples,0,2).*randn(2,1); % [u v]
%Focus=[ 8.1472; 9.0579];
LinPoly=polyfit(Samples(1,:),Samples(2,:),1);
BaseDirectrix=[LinPoly(1) -1 LinPoly(2)]'; % [a b c]
BaseDirectrix=BaseDirectrix/sqrt(BaseDirectrix(1)^2+BaseDirectrix(2)^2);
%BaseDirectrix=[1 0 -10]';
%Directrix=[1.2699; 9.1338; 6.3236];
% Initialize log variables
Errors=zeros(1,NumSteps);
Penalty=zeros(1,NumSteps);
Directrices=zeros(3,NumSteps);
Foci=zeros(2,NumSteps);
StepSizes=zeros(1,NumSteps);
StepSizes(1)=StepSize;
GradientVectors=zeros(Dimension,NumSteps);
% Main loop
BestError=1.0e300;
Gradient=zeros(5,1);
Focus=BaseFocus;
Directrix=BaseDirectrix;
for NdxStep=1:NumSteps
% Compute the distances E_f(x,y) to the current focus
DifferencesFocus=Samples-repmat(Focus,[1 NumSamples]);
DistFocus=sqrt(sum(DifferencesFocus.^2,1));
% Compute the distances E_d(x,y) to the current directrix
LineEval=Directrix'*[Samples; ones(1,NumSamples)]; % a*x_i + b*y_i + c
NormDirectrix=Directrix(1)^2+Directrix(2)^2;
DistDirectrix=sqrt((LineEval.^2)/...
NormDirectrix); % a^2+b^2
EvalDirectrix=Directrix'*[Focus;1]; % a*u+b*v+c
% Find the points inside and outside the ellipsoid
FocusPoints=DistFocus<DistDirectrix; % R_f
DirectrixPoints=~FocusPoints; % R_d
% NumInnerPoints=nnz(FocusPoints);
% NumOuterPoints=nnz(DirectrixPoints);
% Find the gradient of the parameter vector p
AuxVec=sign(LineEval).*...
(Directrix(1)*Directrix(3)+Directrix(1)*Directrix(2)*Samples(2,:) ...
-Directrix(2)^2*Samples(1,:)) ./ ...
(NormDirectrix^(3/2));
PenaltyTerm=sign(EvalDirectrix)*...
(Directrix(1)*Directrix(3)+Directrix(1)*Directrix(2)*Focus(2) ...
-Directrix(2)^2*Focus(1)) /...
((NormDirectrix^(3/2)));
Gradient(1)=(sum(AuxVec(DirectrixPoints)) ...
-sum(AuxVec(FocusPoints)))/NumSamples ...
+Lambda*PenaltyTerm;
AuxVec=sign(LineEval).*...
(Directrix(2)*Directrix(3)+Directrix(1)*Directrix(2)*Samples(1,:) ...
-Directrix(1)^2*Samples(2,:)) ./ ...
(NormDirectrix^(3/2));
PenaltyTerm=sign(EvalDirectrix)*...
(Directrix(2)*Directrix(3)+Directrix(1)*Directrix(2)*Focus(1) ...
-Directrix(1)^2*Focus(2)) /...
((NormDirectrix^(3/2)));
Gradient(2)=(sum(AuxVec(DirectrixPoints)) ...
-sum(AuxVec(FocusPoints)))/NumSamples ...
+Lambda*PenaltyTerm;
AuxVec=sign(LineEval) ./ ...
((Directrix(1)^2+Directrix(2)^2)^(3/2));
PenaltyTerm=sign(EvalDirectrix) /...
((NormDirectrix^(3/2)));
Gradient(3)=(-sum(AuxVec(DirectrixPoints)) ...
+sum(AuxVec(FocusPoints)))/NumSamples ...
-Lambda*PenaltyTerm;
AuxVec=-DifferencesFocus(1,:)./DistFocus;
PenaltyTerm=Directrix(1)*sign(EvalDirectrix) /...
((NormDirectrix^(3/2)));
Gradient(4)=(sum(AuxVec(DirectrixPoints)) ...
-sum(AuxVec(FocusPoints)))/NumSamples ...
-Lambda*PenaltyTerm;
AuxVec=-DifferencesFocus(2,:)./DistFocus;
PenaltyTerm=Directrix(2)*sign(EvalDirectrix) /...
((NormDirectrix^(3/2)));
Gradient(5)=(sum(AuxVec(DirectrixPoints)) ...
-sum(AuxVec(FocusPoints)))/NumSamples ...
-Lambda*PenaltyTerm;
% Update the best solution found so far
Errors(NdxStep)=(sum(DistDirectrix(FocusPoints)-DistFocus(FocusPoints))+...
sum(DistFocus(DirectrixPoints)-DistDirectrix(DirectrixPoints)))/NumSamples-...
Lambda*sqrt(((Directrix'*[Focus;1])^2)/NormDirectrix);
Penalty(NdxStep)=Lambda*sqrt(((Directrix'*[Focus;1])^2)/NormDirectrix);
if Errors(NdxStep)<BestError
BestError=Errors(NdxStep);
BestFocus=Focus;
BestDirectrix=Directrix;
end
% Update the directrix and normalize it
Directrix=Directrix-StepSize*Gradient(1:3);
Directrix=Directrix/sqrt(Directrix(1)^2+Directrix(2)^2);
% Update the focus
Focus=Focus-StepSize*Gradient(4:5);
if mod(NdxStep,T)==0
Focus=BaseFocus+StdDev*0.5*randn(size(Focus));
Directrix=BaseDirectrix+StdDev*0.5*randn(size(Directrix));
end
% if mod(NdxStep,1000)==0
% Directrix=10*randn(size(Directrix));
% end
%
% if mod(NdxStep,1000)==500
% Focus=10*randn(size(Focus));
% end
% Update the logs
Directrices(:,NdxStep)=Directrix;
Foci(:,NdxStep)=Focus;
% Update the step size each 10 iterations of the main loop
if mod(NdxStep,10)==0
% Check whether the error has grown in a robust way
if median(Errors((NdxStep-4):NdxStep))<median(Errors((NdxStep-9):(NdxStep-5)))
% The error is smaller, so we increase the step size provided
% that it is not too big
if StepSize<MaxStepSize
StepSize=1.1*StepSize;
end
else
% The error is bigger, so we decrease the step size provided
% that it is not too small
if StepSize>MinStepSize
StepSize=0.9*StepSize;
end
end
end
StepSizes(NdxStep)=StepSize;
end
a=BestDirectrix(1);
b=BestDirectrix(2);
c=BestDirectrix(3);
u=BestFocus(1);
v=BestFocus(2);
GeneralForm=[ ((a^2)/(a^2+b^2))-1 ...
2*a*b/(a^2+b^2) ...
((b^2)/(a^2+b^2))-1 ...
2*u+2*a*c/(a^2+b^2) ...
2*v+2*b*c/(a^2+b^2) ...
((c^2)/(a^2+b^2))-u^2-v^2];
GeneralForm=GeneralForm/norm(GeneralForm);
if GeneralForm(1)<0
GeneralForm=-GeneralForm;
end