Commit 763220dd authored by Jean-Yves TINEVEZ's avatar Jean-Yves TINEVEZ
Browse files

Fit 2D ellipse on 3D points, projecting on best plane through object.

Like S. Herbert did.
parent 2633fda0
...@@ -148,9 +148,13 @@ for i = 1 : n_objects ...@@ -148,9 +148,13 @@ for i = 1 : n_objects
[ area, uncorr_area ] = area3d( o ); [ area, uncorr_area ] = area3d( o );
[ perim, uncorr_perim ] = perimeter3d( o ); [ perim, uncorr_perim ] = perimeter3d( o );
[ f, E ] = fit_ellipse( o );
objects( i ).id = i;
objects( i ).area = area; objects( i ).area = area;
objects( i ).perimeter = perim; objects( i ).perimeter = perim;
objects( i ).euler_angles = E;
objects( i ).ellipse_fit = f;
uncorr = struct(); uncorr = struct();
uncorr.area = uncorr_area; uncorr.area = uncorr_area;
...@@ -185,11 +189,12 @@ for i = 1 : n_objects ...@@ -185,11 +189,12 @@ for i = 1 : n_objects
o = objects( i ); o = objects( i );
P = o.boundary; 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, ... patch( P(:,1), P(:,2), P(:,3), err, ...
'LineWidth', 2 ); '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', ... 'HorizontalAlignment', 'center', ...
'VerticalAlignment', 'middle' ) 'VerticalAlignment', 'middle' )
......
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
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