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## Copyright (C) 2024 David Legland
## All rights reserved.
##
## Redistribution and use in source and binary forms, with or without
## modification, are permitted provided that the following conditions are met:
##
## 1 Redistributions of source code must retain the above copyright notice,
## this list of conditions and the following disclaimer.
## 2 Redistributions in binary form must reproduce the above copyright
## notice, this list of conditions and the following disclaimer in the
## documentation and/or other materials provided with the distribution.
##
## THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ''AS IS''
## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
## IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
## ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR
## ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
## DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
## SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
## CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
## OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
##
## The views and conclusions contained in the software and documentation are
## those of the authors and should not be interpreted as representing official
## policies, either expressed or implied, of the copyright holders.
function elli = fitEllipse(varargin)
%FITELLIPSE Fit an ellipse to a set of 2D points.
%
% ELLI = fitEllipse(PTS)
%
% Example
% elli = [50 40 30 10 20];
% pts = ellipseToPolygon(elli, 60) + randn(60,2);
% figure; hold on; drawPoint(pts, 'ko');
% axis equal; axis([0 100 0 100]);
% ellFit = fitEllipse(pts);
% drawEllipse(ellFit, 'b')
%
% This is a rewrite of an original function from the authors cited below.
% Changes from original submission include:
% * convert angle of result ellipse to degrees (to comply with MatGeom
% convention)
% * update comments
%
% Authors:
% Andrew Fitzgibbon, Maurizio Pilu, Bob Fisher
% Reference: "Direct Least Squares Fitting of Ellipses", IEEE T-PAMI, 1999
%
% https://fr.mathworks.com/matlabcentral/fileexchange/3215-fit_ellipse
%
% @Article{Fitzgibbon99,
% author = "Fitzgibbon, A.~W.and Pilu, M. and Fisher, R.~B.",
% title = "Direct least-squares fitting of ellipses",
% journal = pami,
% year = 1999,
% volume = 21,
% number = 5,
% month = may,
% pages = "476--480"
% }
%
% See also
% geom2d, ellipses2d, createEllipse, equivalentEllipse, fitLine
%
% ------
% Author: David Legland
% E-mail: david.legland@inrae.fr
% Created: 2022-07-16, using Matlab 9.12.0.1884302 (R2022a)
% Copyright 2022-2023 INRAE - BIA Research Unit - BIBS Platform (Nantes)
%% Process input arguments
if nargin==1
var = varargin{1};
X = var(:,1);
Y = var(:,2);
else
X = varargin{1};
Y = varargin{2};
end
%% Normalize data
% recenter and reduce range
mx = mean(X);
my = mean(Y);
sx = (max(X) - min(X)) / 2;
sy = (max(Y) - min(Y)) / 2;
x = (X-mx) / sx;
y = (Y-my) / sy;
% Force to column vectors
x = x(:);
y = y(:);
%% Main processing
% Build design matrix
D = [ x.*x x.*y y.*y x y ones(size(x)) ];
% Build scatter matrix
S = D' * D;
% Build 6x6 constraint matrix
C(6,6) = 0;
C(1,3) = -2;
C(2,2) = 1;
C(3,1) = -2;
% Solve eigensystem
% Break into blocks
tmpA = S(1:3,1:3);
tmpB = S(1:3,4:6);
tmpC = S(4:6,4:6);
tmpD = C(1:3,1:3);
tmpE = tmpC \ tmpB';
[evec_x, eval_x] = eig(tmpD \ (tmpA - tmpB*tmpE));
% Find the positive (as det(tmpD) < 0) eigenvalue
I = real(diag(eval_x)) < 1e-8 & ~isinf(diag(eval_x));
% Extract eigenvector corresponding to negative eigenvalue
A = real(evec_x(:,I));
% Recover the bottom half...
evec_y = -tmpE * A;
A = [A; evec_y];
% re-calibrate
par = [
A(1)*sy*sy, ...
A(2)*sx*sy, ...
A(3)*sx*sx, ...
-2*A(1)*sy*sy*mx - A(2)*sx*sy*my + A(4)*sx*sy*sy, ...
-A(2)*sx*sy*mx - 2*A(3)*sx*sx*my + A(5)*sx*sx*sy, ...
A(1)*sy*sy*mx*mx + A(2)*sx*sy*mx*my + A(3)*sx*sx*my*my ...
- A(4)*sx*sy*sy*mx - A(5)*sx*sx*sy*my ...
+ A(6)*sx*sx*sy*sy ...
]';
%% Identify parameters
% rotation angle
theta = 0.5 * atan2(par(2), par(1) - par(3));
% pre-comptute trigonometrics
cost = cos(theta);
sint = sin(theta);
sin2 = sint .* sint;
cos2 = cost .* cost;
cos_sin = sint .* cost;
%
Ao = par(6);
Au = par(4) .* cost + par(5) .* sint;
Av = -par(4) .* sint + par(5) .* cost;
Auu = par(1) .* cos2 + par(3) .* sin2 + par(2) .* cos_sin;
Avv = par(1) .* sin2 + par(3) .* cos2 - par(2) .* cos_sin;
% ROTATED = [Ao Au Av Auu Avv]
tuCentre = - Au./(2.*Auu);
tvCentre = - Av./(2.*Avv);
wCentre = Ao - Auu.*tuCentre.*tuCentre - Avv.*tvCentre.*tvCentre;
uCentre = tuCentre .* cost - tvCentre .* sint;
vCentre = tuCentre .* sint + tvCentre .* cost;
Ru = -wCentre ./ Auu;
Rv = -wCentre ./ Avv;
Ru = sqrt(abs(Ru)) .* sign(Ru);
Rv = sqrt(abs(Rv)) .* sign(Rv);
% create row vector representing ellipse
elli = [uCentre, vCentre, Ru, Rv, rad2deg(theta)];
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