How can I get ellipse coefficient from fitEllipse

Question

I want to extract the red ball from one picture and get the detected ellipse matrix in picture.

Here is my example:

I threshold the picture, find the contour of red ball by using findContour() function and use fitEllipse() to fit an ellipse.

But what I want is to get coefficient of this ellipse. Because the fitEllipse() return a rotation rectangle (RotatedRect), so I need to re-write this function.

One Ellipse can be expressed as Ax^2 + By^2 + Cxy + Dx + Ey + F = 0; So I want to get u=(A,B,C,D,E,F) or u=(A,B,C,D,E) if F is 1 (to construct an ellipse matrix).

I read the source code of fitEllipse(), there are totally three SVD process, I think I can get the above coefficients from the results of those three SVD process. But I am quite confused what does each result (variable cv::Mat x) of each SVD process represent and why there are three SVD here?

Here is this function:

cv::RotatedRect cv::fitEllipse( InputArray _points )
{
   Mat points = _points.getMat();
   int i, n = points.checkVector(2);
   int depth = points.depth();
   CV_Assert( n >= 0 && (depth == CV_32F || depth == CV_32S));

   RotatedRect box;

   if( n < 5 )
        CV_Error( CV_StsBadSize, "There should be at least 5 points to fit the ellipse" );

    // New fitellipse algorithm, contributed by Dr. Daniel Weiss
    Point2f c(0,0);
    double gfp[5], rp[5], t;
    const double min_eps = 1e-8;
    bool is_float = depth == CV_32F;
    const Point* ptsi = points.ptr<Point>();
    const Point2f* ptsf = points.ptr<Point2f>();

    AutoBuffer<double> _Ad(n*5), _bd(n);
    double *Ad = _Ad, *bd = _bd;

    // first fit for parameters A - E
    Mat A( n, 5, CV_64F, Ad );
    Mat b( n, 1, CV_64F, bd );
    Mat x( 5, 1, CV_64F, gfp );

    for( i = 0; i < n; i++ )
    {
        Point2f p = is_float ? ptsf[i] : Point2f((float)ptsi[i].x, (float)ptsi[i].y);
        c += p;
    }
    c.x /= n;
    c.y /= n;

    for( i = 0; i < n; i++ )
    {
        Point2f p = is_float ? ptsf[i] : Point2f((float)ptsi[i].x, (float)ptsi[i].y);
        p -= c;

        bd[i] = 10000.0; // 1.0?
        Ad[i*5] = -(double)p.x * p.x; // A - C signs inverted as proposed by APP
        Ad[i*5 + 1] = -(double)p.y * p.y;
        Ad[i*5 + 2] = -(double)p.x * p.y;
        Ad[i*5 + 3] = p.x;
        Ad[i*5 + 4] = p.y;
    }

    solve(A, b, x, DECOMP_SVD);

    // now use general-form parameters A - E to find the ellipse center:
    // differentiate general form wrt x/y to get two equations for cx and cy
    A = Mat( 2, 2, CV_64F, Ad );
    b = Mat( 2, 1, CV_64F, bd );
    x = Mat( 2, 1, CV_64F, rp );
    Ad[0] = 2 * gfp[0];
    Ad[1] = Ad[2] = gfp[2];
    Ad[3] = 2 * gfp[1];
    bd[0] = gfp[3];
    bd[1] = gfp[4];
    solve( A, b, x, DECOMP_SVD );

    // re-fit for parameters A - C with those center coordinates
    A = Mat( n, 3, CV_64F, Ad );
    b = Mat( n, 1, CV_64F, bd );
    x = Mat( 3, 1, CV_64F, gfp );
    for( i = 0; i < n; i++ )
    {
        Point2f p = is_float ? ptsf[i] : Point2f((float)ptsi[i].x, (float)ptsi[i].y);
        p -= c;
        bd[i] = 1.0;
        Ad[i * 3] = (p.x - rp[0]) * (p.x - rp[0]);
        Ad[i * 3 + 1] = (p.y - rp[1]) * (p.y - rp[1]);
        Ad[i * 3 + 2] = (p.x - rp[0]) * (p.y - rp[1]);
    }
    solve(A, b, x, DECOMP_SVD);

    // store angle and radii
    rp[4] = -0.5 * atan2(gfp[2], gfp[1] - gfp[0]); // convert from APP angle usage
    if( fabs(gfp[2]) > min_eps )
        t = gfp[2]/sin(-2.0 * rp[4]);
    else // ellipse is rotated by an integer multiple of pi/2
        t = gfp[1] - gfp[0];
    rp[2] = fabs(gfp[0] + gfp[1] - t);
    if( rp[2] > min_eps )
        rp[2] = std::sqrt(2.0 / rp[2]);
    rp[3] = fabs(gfp[0] + gfp[1] + t);
    if( rp[3] > min_eps )
        rp[3] = std::sqrt(2.0 / rp[3]);

    box.center.x = (float)rp[0] + c.x;
    box.center.y = (float)rp[1] + c.y;
    box.size.width = (float)(rp[2]*2);
    box.size.height = (float)(rp[3]*2);
    if( box.size.width > box.size.height )
    {
        float tmp;
        CV_SWAP( box.size.width, box.size.height, tmp );
        box.angle = (float)(90 + rp[4]*180/CV_PI);
    }
    if( box.angle < -180 )
        box.angle += 360;
    if( box.angle > 360 )
        box.angle -= 360;

    return box;
}

The source code link: https://github.com/Itseez/opencv/blob/master/modules/imgproc/src/shapedescr.cpp

Answer 1

The function fitEllipse returns a RotatedRect that contains all the parameters of the ellipse.

An ellipse is defined by 5 parameters:

• xc : x coordinate of the center
• yc : y coordinate of the center
• a : major semi-axis
• b : minor semi-axis
• theta : rotation angle

You can obtain these parameters like:

RotatedRect e = fitEllipse(points);

float xc    = e.center.x;
float yc    = e.center.y;
float a     = e.size.width  / 2;    // width >= height
float b     = e.size.height / 2;
float theta = e.angle;              // in degrees

You can draw an ellipse with the function ellipse using the RotatedRect:

ellipse(image, e, Scalar(0,255,0)); 

or, equivalently using the ellipse parameters:

ellipse(res, Point(xc, yc), Size(a, b), theta, 0.0, 360.0, Scalar(0,255,0));

If you need the values of the coefficients of the implicit equation, you can do like (from Wikipedia):

So, you can get the parameters you need from the RotatedRect, and you don't need to change the function fitEllipse. The solve function is used to solve linear systems or least-squares problems. Using the SVD decomposition method the system can be over-defined and/or the matrix src1 can be singular.

For more details on the algorithm, you can see the paper of Fitzgibbon that proposed this fit ellipse method.