How does fitEllipse work in OpenCV?

Question

I am working with opencv and I need to understand how does the function fitEllipse exactly works. I looked at the code at (https://github.com/Itseez/opencv/blob/master/modules/imgproc/src/shapedescr.cpp) and I know it uses least-squares to determine the likely ellipses. I also looked at the paper given in the documentation(Andrew W. Fitzgibbon, R.B.Fisher. A Buyer’s Guide to Conic Fitting. Proc.5th British Machine Vision Conference, Birmingham, pp. 513-522, 1995.)

But I cannot understand exactly the algorithm. For example, why does it need to solve 3 times the least square problem? why bd is initialized to 10000 before the first svd(I guess it is juste a random value for the initialization but why this value can be random?)? why does the values in Ad needs to be negative before the first svd?

Thank you!

Answer 1

Here is Matlab code.. it might help

function [Q,a]=fit_ellipse_fitzgibbon(data)
  % function [Q,a]=fit_ellipse_fitzgibbon(data)
  %
  % Ellipse specific fit, according to:
  %
  %  Direct Least Square Fitting of Ellipses,
  %  A. Fitzgibbon, M. Pilu and R. Fisher. PAMI 1996
  %
  %
  % See Also:
  %   FIT_ELLIPSE_LS
  %   FIT_ELLIPSE_HALIR

  [m,n] = size(data);
  assert((m==2||m==3)&&n>5);
  x = data(1,:)';
  y = data(2,:)';

  D = [x.^2 x.*y y.^2 x y ones(size(x))];   % design matrix
  S = D'*D;                                 % scatter matrix
  C(6,6)=0; C(1,3)=-2; C(2,2)=1; C(3,1)=-2; % constraints matrix
  % solve the generalized eigensystem
  [V,D] = eig(S, C);
  % find the only negative eigenvalue
  [n_r, n_c] = find(D<0 & ~isinf(D));
  if isempty(n_c),
    warning('Error getting the ellipse parameters, will do LS');
    [Q,a] = fit_ellipse_ls(data); %
    return;
  end
  % the parameters
  a = V(:, n_c);
  [A B C D E F] = deal(a(1),a(2),a(3),a(4),a(5),a(6)); % deal is slow!
  Q = [A B/2 D/2; B/2 C E/2; D/2 E/2 F];
end % fit_ellipse_fitzgibbon

Fitzibbon solution has some numerical stability though. See the work of Halir for a solution to this.

It is essentially least squares solution, but specifically designed so that it will produce a valid ellipse, not just any conic.