Question

Suppose we have a connected component in the image as the following image illustrates:image http://dl.dropbox.com/u/92688392/ellipse.jpg.

My question is how can calculate the bounding ellipse of the connected components (the red ellipse in the image). I have checked MATLAB function regionprops, and understand how MATLAB can do that. I also notice that Opencv has similar function to do that CBlob::GetEllipse(). However, although I understand how they obtain the result by reading the code, the fundamental theory behind it is still unclear to me. I am therefore wondering whether there are some standard algorithms to do the job. Thanks!

EDIT:

Based on the comments, I reorganized my question: in image moment Wikipedia the calculation formula of the longest axis angle is

However, in the MATLAB function regionprops, the codes are as follows:

    % Calculate orientation.
    if (uyy > uxx)
        num = uyy - uxx + sqrt((uyy - uxx)^2 + 4*uxy^2);
        den = 2*uxy;
    else
        num = 2*uxy;
        den = uxx - uyy + sqrt((uxx - uyy)^2 + 4*uxy^2);
    end

This implementation is inconsistent with the formula in Wikipedia. I was wondering which one is correct.

Answer 1

I was trying to find out what’s the algorithm behind it as well so I could write my own implementation of it. I found it on a blog post of mathworks. In one of the comments, the author says:

regionprops calculates the 2nd-order moments of the object in question and then returns measurements of the ellipse with the same 2nd-order moments.

and later later:

The equations used are from Haralick and Shapiro, Computer and Robot Vision vol. 1, Appendix A, Addison-Wesley 1992. I did a sanity check by constructing an image containing an ellipse with major axis length = 100 and minor axis length = 50, and regionprops returned the correct measurements.

I don’t have that book but seems I’ll need to get a copy of it.