%%%% Ellipse fitting from :
%%%% Ohad Gal (2022). fit_ellipse (https://www.mathworks.com/matlabcentral/fileexchange/3215-fit_ellipse), MATLAB Central File Exchange. Retrieved May 27, 2022.

function ellipse_t = fit_ellipse( x,y,axis_handle )
    %
    % fit_ellipse - finds the best fit to an ellipse for the given set of points.
    %
    % Format:   ellipse_t = fit_ellipse( x,y,axis_handle )
    %
    % Input:    x,y         - a set of points in 2 column vectors. AT LEAST 5 points are needed !
    %           axis_handle - optional. a handle to an axis, at which the estimated ellipse 
    %                         will be drawn along with it's axes
    %
    % Output:   ellipse_t - structure that defines the best fit to an ellipse
    %                       a           - sub axis (radius) of the X axis of the non-tilt ellipse
    %                       b           - sub axis (radius) of the Y axis of the non-tilt ellipse
    %                       phi         - orientation in radians of the ellipse (tilt)
    %                       X0          - center at the X axis of the non-tilt ellipse
    %                       Y0          - center at the Y axis of the non-tilt ellipse
    %                       X0_in       - center at the X axis of the tilted ellipse
    %                       Y0_in       - center at the Y axis of the tilted ellipse
    %                       long_axis   - size of the long axis of the ellipse
    %                       short_axis  - size of the short axis of the ellipse
    %                       status      - status of detection of an ellipse
    %
    % Note:     if an ellipse was not detected (but a parabola or hyperbola), then
    %           an empty structure is returned
    % =====================================================================================
    %                  Ellipse Fit using Least Squares criterion
    % =====================================================================================
    % We will try to fit the best ellipse to the given measurements. the mathematical
    % representation of use will be the CONIC Equation of the Ellipse which is:
    % 
    %    Ellipse = a*x^2 + b*x*y + c*y^2 + d*x + e*y + f = 0
    %   
    % The fit-estimation method of use is the Least Squares method (without any weights)
    % The estimator is extracted from the following equations:
    %
    %    g(x,y;A) := a*x^2 + b*x*y + c*y^2 + d*x + e*y = f
    %
    %    where:
    %       A   - is the vector of parameters to be estimated (a,b,c,d,e)
    %       x,y - is a single measurement
    %
    % We will define the cost function to be:
    %
    %   Cost(A) := (g_c(x_c,y_c;A)-f_c)'*(g_c(x_c,y_c;A)-f_c)
    %            = (X*A+f_c)'*(X*A+f_c) 
    %            = A'*X'*X*A + 2*f_c'*X*A + N*f^2
    %
    %   where:
    %       g_c(x_c,y_c;A) - vector function of ALL the measurements
    %                        each element of g_c() is g(x,y;A)
    %       X              - a matrix of the form: [x_c.^2, x_c.*y_c, y_c.^2, x_c, y_c ]
    %       f_c            - is actually defined as ones(length(f),1)*f
    %
    % Derivation of the Cost function with respect to the vector of parameters "A" yields:
    %
    %   A'*X'*X = -f_c'*X = -f*ones(1,length(f_c))*X = -f*sum(X)
    %
    % Which yields the estimator:
    %
    %       ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    %       |  A_least_squares = -f*sum(X)/(X'*X) ->(normalize by -f) = sum(X)/(X'*X)  |
    %       ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    %
    % (We will normalize the variables by (-f) since "f" is unknown and can be accounted for later on)
    %  
    % NOW, all that is left to do is to extract the parameters from the Conic Equation.
    % We will deal the vector A into the variables: (A,B,C,D,E) and assume F = -1;
    %
    %    Recall the conic representation of an ellipse:
    % 
    %       A*x^2 + B*x*y + C*y^2 + D*x + E*y + F = 0
    % 
    % We will check if the ellipse has a tilt (=orientation). The orientation is present
    % if the coefficient of the term "x*y" is not zero. If so, we first need to remove the
    % tilt of the ellipse.
    %
    % If the parameter "B" is not equal to zero, then we have an orientation (tilt) to the ellipse.
    % we will remove the tilt of the ellipse so as to remain with a conic representation of an 
    % ellipse without a tilt, for which the math is more simple:
    %
    % Non tilt conic rep.:  A`*x^2 + C`*y^2 + D`*x + E`*y + F` = 0
    %
    % We will remove the orientation using the following substitution:
    %   
    %   Replace x with cx+sy and y with -sx+cy such that the conic representation is:
    %   
    %   A(cx+sy)^2 + B(cx+sy)(-sx+cy) + C(-sx+cy)^2 + D(cx+sy) + E(-sx+cy) + F = 0
    %
    %   where:      c = cos(phi)    ,   s = sin(phi)
    %
    %   and simplify...
    %
    %       x^2(A*c^2 - Bcs + Cs^2) + xy(2A*cs +(c^2-s^2)B -2Ccs) + ...
    %           y^2(As^2 + Bcs + Cc^2) + x(Dc-Es) + y(Ds+Ec) + F = 0
    %
    %   The orientation is easily found by the condition of (B_new=0) which results in:
    % 
    %   2A*cs +(c^2-s^2)B -2Ccs = 0  ==> phi = 1/2 * atan( b/(c-a) )
    %   
    %   Now the constants   c=cos(phi)  and  s=sin(phi)  can be found, and from them
    %   all the other constants A`,C`,D`,E` can be found.
    %
    %   A` = A*c^2 - B*c*s + C*s^2                  D` = D*c-E*s
    %   B` = 2*A*c*s +(c^2-s^2)*B -2*C*c*s = 0      E` = D*s+E*c 
    %   C` = A*s^2 + B*c*s + C*c^2
    %
    % Next, we want the representation of the non-tilted ellipse to be as:
    %
    %       Ellipse = ( (X-X0)/a )^2 + ( (Y-Y0)/b )^2 = 1
    %
    %       where:  (X0,Y0) is the center of the ellipse
    %               a,b     are the ellipse "radiuses" (or sub-axis)
    %
    % Using a square completion method we will define:
    %       
    %       F`` = -F` + (D`^2)/(4*A`) + (E`^2)/(4*C`)
    %
    %       Such that:    a`*(X-X0)^2 = A`(X^2 + X*D`/A` + (D`/(2*A`))^2 )
    %                     c`*(Y-Y0)^2 = C`(Y^2 + Y*E`/C` + (E`/(2*C`))^2 )
    %
    %       which yields the transformations:
    %       
    %           X0  =   -D`/(2*A`)
    %           Y0  =   -E`/(2*C`)
    %           a   =   sqrt( abs( F``/A` ) )
    %           b   =   sqrt( abs( F``/C` ) )
    %
    % And finally we can define the remaining parameters:
    %
    %   long_axis   = 2 * max( a,b )
    %   short_axis  = 2 * min( a,b )
    %   Orientation = phi
    %
    %
    % initialize
    orientation_tolerance = 1e-3;
    % empty warning stack
    warning( '' );
    % prepare vectors, must be column vectors
    x = x(:);
    y = y(:);
    % remove bias of the ellipse - to make matrix inversion more accurate. (will be added later on).
    mean_x = mean(x);
    mean_y = mean(y);
    x = x-mean_x;
    y = y-mean_y;
    % the estimation for the conic equation of the ellipse
    X = [x.^2, x.*y, y.^2, x, y ];
    a = sum(X)/(X'*X);
    % check for warnings
    if ~isempty( lastwarn )
        disp( 'stopped because of a warning regarding matrix inversion' );
        ellipse_t = [];
        return
    end
    % extract parameters from the conic equation
    [a,b,c,d,e] = deal( a(1),a(2),a(3),a(4),a(5) );
    % remove the orientation from the ellipse
    if ( min(abs(b/a),abs(b/c)) > orientation_tolerance )

        orientation_rad = 1/2 * atan( b/(c-a) );
        cos_phi = cos( orientation_rad );
        sin_phi = sin( orientation_rad );
        [a,b,c,d,e] = deal(...
            a*cos_phi^2 - b*cos_phi*sin_phi + c*sin_phi^2,...
            0,...
            a*sin_phi^2 + b*cos_phi*sin_phi + c*cos_phi^2,...
            d*cos_phi - e*sin_phi,...
            d*sin_phi + e*cos_phi );
        [mean_x,mean_y] = deal( ...
            cos_phi*mean_x - sin_phi*mean_y,...
            sin_phi*mean_x + cos_phi*mean_y );
    else
        orientation_rad = 0;
        cos_phi = cos( orientation_rad );
        sin_phi = sin( orientation_rad );
    end
    % check if conic equation represents an ellipse
    test = a*c;
    switch (1)
    case (test>0),  status = '';
    case (test==0), status = 'Parabola found';  warning( 'fit_ellipse: Did not locate an ellipse' );
    case (test<0),  status = 'Hyperbola found'; warning( 'fit_ellipse: Did not locate an ellipse' );
    end
    % if we found an ellipse return it's data
    if (test>0)

        % make sure coefficients are positive as required
        if (a<0), [a,c,d,e] = deal( -a,-c,-d,-e ); end

        % final ellipse parameters
        X0          = mean_x - d/2/a;
        Y0          = mean_y - e/2/c;
        F           = 1 + (d^2)/(4*a) + (e^2)/(4*c);
        [a,b]       = deal( sqrt( F/a ),sqrt( F/c ) );    
        long_axis   = 2*max(a,b);
        short_axis  = 2*min(a,b);
        % rotate the axes backwards to find the center point of the original TILTED ellipse
        R           = [ cos_phi sin_phi; -sin_phi cos_phi ];
        P_in        = R * [X0;Y0];
        X0_in       = P_in(1);
        Y0_in       = P_in(2);

        % pack ellipse into a structure
        ellipse_t = struct( ...
            'a',a,...
            'b',b,...
            'phi',orientation_rad,...
            'X0',X0,...
            'Y0',Y0,...
            'X0_in',X0_in,...
            'Y0_in',Y0_in,...
            'long_axis',long_axis,...
            'short_axis',short_axis,...
            'status','' );
    else
        % report an empty structure
        ellipse_t = struct( ...
            'a',[],...
            'b',[],...
            'phi',[],...
            'X0',[],...
            'Y0',[],...
            'X0_in',[],...
            'Y0_in',[],...
            'long_axis',[],...
            'short_axis',[],...
            'status',status );
    end
    % check if we need to plot an ellipse with it's axes.
    if (nargin>2) & ~isempty( axis_handle ) & (test>0)

        % rotation matrix to rotate the axes with respect to an angle phi
        R = [ cos_phi sin_phi; -sin_phi cos_phi ];

        % the axes
        ver_line        = [ [X0 X0]; Y0+b*[-1 1] ];
        horz_line       = [ X0+a*[-1 1]; [Y0 Y0] ];
        new_ver_line    = R*ver_line;
        new_horz_line   = R*horz_line;

        % the ellipse
        theta_r         = linspace(0,2*pi);
        ellipse_x_r     = X0 + a*cos( theta_r );
        ellipse_y_r     = Y0 + b*sin( theta_r );
        rotated_ellipse = R * [ellipse_x_r;ellipse_y_r];

        % draw
        hold_state = get( axis_handle,'NextPlot' );
        set( axis_handle,'NextPlot','add' );
        plot( new_ver_line(1,:),new_ver_line(2,:),'r' );
        plot( new_horz_line(1,:),new_horz_line(2,:),'r' );
        plot( rotated_ellipse(1,:),rotated_ellipse(2,:),'r' );
        set( axis_handle,'NextPlot',hold_state );
    end
end
