ellipse to circle transformation

I took image of coin from my mobile hand set. I did the edge detection of the same image.After edge detetion i am getting elliptical shap.how to convert the elliptical shape to circle

Answer 1

[Matt J]

If you fit the ellipse boundaries with the equation,

   (x-d)'*Q*(x-d) = 1

you can then easily get the coordinate transformation needed to turn it into a circle. My ellipsefit() utility below will help.

Specifically, if the Cholesky decomposition of Q is L'*L, then the spatial transformation y=L*x will transform the above to the equation for a circle

     norm(y-L*d)^2=1

You can use IMTRANSFORM to implement the transformation.

function report=ellipsefit(XY)
  %ELLIPSEFIT - form 2D ellipse fit to given x,y data
  %
  %  report=ellipsefit(XY)
  %
  %in:
  %
  % XY: Input matrix of 2D coordinates to be fit. Each column XY(:,i) is [xi;yi]
  %
  %out: Finds the ellipse fitting the input data parametrized both as
  %     A*x^2+B*x*y C*y^2+D*x+E*y=1 and [x-x0,y-y0]*Q*[x-x0;y-y0]=1
  %
  % report: a structure output with the following fields
  %    
  %    report.Q: the matrix Q
  %    report.d: the vector [x0,y0]
  %    report.ABCDE: the vector [A,B,C,D,E]
  %    report.AxesDiams: The minor and major ellipse diameters
  %    report.theta: The counter-clockwise rotation of the ellipse.
  %
  %NOTE: The code will give errors if the data fit traces out a non-elliptic or
  %      degenerate conic section.
  %
  %See also ellipsoidfit

    X=XY(1,:).';
    Y=XY(2,:).';

    M= [X.^2, X.*Y, Y.^2, X, Y, -ones(size(X,1),1)];

    [U,S,V]=svd(M,0);
    ABCDEF=V(:,end);

    if size(ABCDEF,2)>1

        error 'Data cannot be fit with unique ellipse'
    else

        ABCDEF=num2cell(ABCDEF);
    end

    [A,B,C,D,E,F]=deal(ABCDEF{:});

    Q=[A, B/2;B/2 C];
    x0=-Q\[D;E]/2;
    dd=F+x0'*Q*x0;

    Q=Q/dd;

    [R,eigen]=eig(Q);
    eigen=eigen([1,4]);

    if ~all(eigen>=0), error 'Fit produced a non-elliptic conic section'; end

    idx=eigen>0;
    eigen(idx)=1./eigen(idx);
    AxesDiams = 2*sqrt(eigen);

    theta=atand(tand(-atan2(R(1),R(2))*180/pi));

    report.Q=Q;
    report.d=x0(:).';
    report.ABCDE=[A, B, C, D, E]/F;
    report.AxesDiams=sort(AxesDiams(:)).';
    report.theta=theta;