diff --git a/RunExample.m b/RunExample.m
index d375dc9c7695ca60801af1ab597f44edec66e13d..f3c58976cd2c8ac7eb09d7b1041669329e205173 100755
--- a/RunExample.m
+++ b/RunExample.m
@@ -148,9 +148,13 @@ for i = 1 : n_objects
[ area, uncorr_area ] = area3d( o );
[ perim, uncorr_perim ] = perimeter3d( o );
+ [ f, E ] = fit_ellipse( o );
+ objects( i ).id = i;
objects( i ).area = area;
objects( i ).perimeter = perim;
+ objects( i ).euler_angles = E;
+ objects( i ).ellipse_fit = f;
uncorr = struct();
uncorr.area = uncorr_area;
@@ -185,11 +189,12 @@ for i = 1 : n_objects
o = objects( i );
P = o.boundary;
- err = o.perimeter / o.uncorr.perimeter - 1;
+% err = o.perimeter / o.uncorr.perimeter - 1;
+ err = abs( o.euler_angles( 2 ) );
patch( P(:,1), P(:,2), P(:,3), err, ...
'LineWidth', 2 );
- text( o.center(1), o.center(2), o.center(3) + 0.5, num2str( i ), ...
+ text( o.center(1), o.center(2), o.center(3) + 0.5, num2str( o.id ), ...
'HorizontalAlignment', 'center', ...
'VerticalAlignment', 'middle' )
diff --git a/src/fit_ellipse.m b/src/fit_ellipse.m
new file mode 100644
index 0000000000000000000000000000000000000000..07704433a2d3016c27de70a5c9d9c69c82ae30b0
--- /dev/null
+++ b/src/fit_ellipse.m
@@ -0,0 +1,156 @@
+function [ f, E ] = fit_ellipse( o )
+%FIT_ELLIPSE Fit a 2D ellipse to the 3D points.
+% The fit requires the Euler angles of the plane fitted through the
+% opints, so that we can project them on this plane. We then make a 2D
+% ellipse fit on the projected points. This turns to be much more robust
+% than a 3D fit, and also closely match our configuration.
+
+% Greatly inspired from
+% https://stackoverflow.com/questions/29051168/data-fitting-an-ellipse-in-3d-space
+
+ p = double( centered_points( o ) );
+
+ % Fit a plane to these points.
+ [ ~, ~, v ] = svd( p );
+ E = to_euler( v );
+
+ % Rotate the points into the principal axes frame.
+ p = p * v;
+
+ % Direct ellipse fit.
+% A = direct_ellipse_fit( p( :, 1:2 ) );
+ A = taubin_ellipse_fit( p( :, 1:2 ) );
+ f = quadratic_to_cartesian( A );
+
+ %% Subfunctions
+
+ function E = to_euler( R )
+ if R( 1, 3 ) == 1 || R( 1, 3 ) == -1
+ E3 = 0;
+ dlta = atan2(R(1,2),R(1,3));
+ if R(1,3) == -1
+ E2 = pi/2;
+ E1 = E3 + dlta;
+ else
+ E2 = -pi/2;
+ E1 = -E3 + dlta;
+ end
+ else
+ E2 = - asin( R( 1, 3 ) );
+ E1 = atan2( R( 2, 3 ) / cos( E2 ), R( 3, 3 ) / cos( E2 ) );
+ E3 = atan2( R( 1, 2 ) / cos( E2 ), R( 1, 1 ) / cos( E2 ) );
+ end
+ E = [ E1, E2, E3 ];
+
+ end
+
+ function f = quadratic_to_cartesian( A )
+ % Equations taken from Wolfram website.
+
+ a = A(1);
+ b = A(2);
+ c = A(3);
+ d = A(4);
+ f = A(5);
+ g = A(6);
+
+ x0 = ( c * d - b * f ) / ( b^2 - a * c );
+ y0 = ( a * f - b * d ) / ( b^2 - a * c );
+
+ l1 = sqrt( 2*(a*f^2+c*d^2+g*b^2-2*b*d*f-a*c*g) / ((b^2-a*c)*(sqrt((a-c)^2+4*b^2)-(a+c))));
+ l2 = sqrt( 2*(a*f^2+c*d^2+g*b^2-2*b*d*f-a*c*g) / ((b^2-a*c)*(-sqrt((a-c)^2+4*b^2)-(a+c))));
+
+ if b == 0 && a < c
+ phi = 0;
+ elseif b == 0 && a > c
+ phi = 0.5*pi;
+ elseif b ~= 0 && a < c
+ phi = 0.5* acot((a-c)/(2*b));
+ else
+ phi = 0.5*pi + 0.5* acot((a-c)/(2*b));
+ end
+
+ f = [ x0 y0 l1 l2 phi ];
+
+ end
+
+ function A = direct_ellipse_fit(XY) %#ok<DEFNU>
+ % Direct ellipse fit, proposed in article
+ % A. W. Fitzgibbon, M. Pilu, R. B. Fisher
+ % "Direct Least Squares Fitting of Ellipses"
+ % IEEE Trans. PAMI, Vol. 21, pages 476-480 (1999)
+ %
+ % Adapted from https://fr.mathworks.com/matlabcentral/fileexchange/22684-ellipse-fit-direct-method
+
+ centroid = mean(XY); % the centroid of the data set
+ D1 = [(XY(:,1)-centroid(1)).^2, (XY(:,1)-centroid(1)).*(XY(:,2)-centroid(2)),...
+ (XY(:,2)-centroid(2)).^2];
+ D2 = [XY(:,1)-centroid(1), XY(:,2)-centroid(2), ones(size(XY,1),1)];
+ S1 = D1'*D1;
+ S2 = D1'*D2;
+ S3 = D2'*D2;
+ T = -inv(S3)*S2';
+ M = S1 + S2*T;
+ M = [M(3,:)./2; -M(2,:); M(1,:)./2];
+ [evec,eval] = eig(M); %#ok<ASGLU>
+ cond = 4*evec(1,:).*evec(3,:)-evec(2,:).^2;
+ A1 = evec(:,find(cond>0)); %#ok<FNDSB>
+ A = [A1; T*A1];
+ A4 = A(4)-2*A(1)*centroid(1)-A(2)*centroid(2);
+ A5 = A(5)-2*A(3)*centroid(2)-A(2)*centroid(1);
+ A6 = A(6)+A(1)*centroid(1)^2+A(3)*centroid(2)^2+...
+ A(2)*centroid(1)*centroid(2)-A(4)*centroid(1)-A(5)*centroid(2);
+ A(4) = A4; A(5) = A5; A(6) = A6;
+ A = A/norm(A);
+ end
+
+ function A = taubin_ellipse_fit(XY)
+ % Ellipse fit by Taubin's Method published in
+ % G. Taubin, "Estimation Of Planar Curves, Surfaces And Nonplanar
+ % Space Curves Defined By Implicit Equations, With
+ % Applications To Edge And Range Image Segmentation",
+ % IEEE Trans. PAMI, Vol. 13, pages 1115-1138, (1991)
+ %
+ % Input: XY(n,2) is the array of coordinates of n points x(i)=XY(i,1), y(i)=XY(i,2)
+ %
+ % Output: A = [a b c d e f]' is the vector of algebraic
+ % parameters of the fitting ellipse:
+ % ax^2 + bxy + cy^2 +dx + ey + f = 0
+ % the vector A is normed, so that ||A||=1
+ %
+ % Among fast non-iterative ellipse fitting methods,
+ % this is perhaps the most accurate and robust
+ %
+ % Note: this method fits a quadratic curve (conic) to a set of points;
+ % if points are better approximated by a hyperbola, this fit will
+ % return a hyperbola. To fit ellipses only, use "Direct Ellipse Fit".
+
+ centroid = mean(XY); % the centroid of the data set
+ Z = [(XY(:,1)-centroid(1)).^2, (XY(:,1)-centroid(1)).*(XY(:,2)-centroid(2)),...
+ (XY(:,2)-centroid(2)).^2, XY(:,1)-centroid(1), XY(:,2)-centroid(2), ones(size(XY,1),1)];
+ M = Z'*Z/size(XY,1);
+ P = [M(1,1)-M(1,6)^2, M(1,2)-M(1,6)*M(2,6), M(1,3)-M(1,6)*M(3,6), M(1,4), M(1,5);
+ M(1,2)-M(1,6)*M(2,6), M(2,2)-M(2,6)^2, M(2,3)-M(2,6)*M(3,6), M(2,4), M(2,5);
+ M(1,3)-M(1,6)*M(3,6), M(2,3)-M(2,6)*M(3,6), M(3,3)-M(3,6)^2, M(3,4), M(3,5);
+ M(1,4), M(2,4), M(3,4), M(4,4), M(4,5);
+ M(1,5), M(2,5), M(3,5), M(4,5), M(5,5)];
+ Q = [4*M(1,6), 2*M(2,6), 0, 0, 0;
+ 2*M(2,6), M(1,6)+M(3,6), 2*M(2,6), 0, 0;
+ 0, 2*M(2,6), 4*M(3,6), 0, 0;
+ 0, 0, 0, 1, 0;
+ 0, 0, 0, 0, 1];
+ [V,D] = eig(P,Q);
+ [Dsort,ID] = sort(diag(D)); %#ok<ASGLU>
+ A = V(:,ID(1));
+ A = [A; -A(1:3)'*M(1:3,6)];
+ A4 = A(4)-2*A(1)*centroid(1)-A(2)*centroid(2);
+ A5 = A(5)-2*A(3)*centroid(2)-A(2)*centroid(1);
+ A6 = A(6)+A(1)*centroid(1)^2+A(3)*centroid(2)^2+...
+ A(2)*centroid(1)*centroid(2)-A(4)*centroid(1)-A(5)*centroid(2);
+ A(4) = A4; A(5) = A5; A(6) = A6;
+ A = A/norm(A);
+ end % Taubin
+
+
+end
+