Ellipses Estimation from their Digitization

Ellipses in general position, and problems related to their reconstruction from digital data resulting from their digitization, are considered. If the ellipse $$E:\tilde A\left( {x - p} \right)^2 + 2\tilde B\left( {x - p} \right)\left( {y - q} \right) + \tilde C\left( {y - q} \right)^2 \leqslant 1, \tilde A\tilde C - \tilde B^2 > 0,$$ is presented on digital picture of a given resolution, then the corresponding digital ellipse is: $$D\left( E \right) = \left\{ {\left( {i,j} \right) | A\left( {i - a} \right)^2 + 2B\left( {i - a} \right)\left( {j - b} \right) + C\left( {j - b} \right)^2 \leqslant r^2 , i,j are integers} \right\},$$ where r denotes the number of pixels per unit and a = pr, b = qr, \(A = \tilde Ar^2 , B = \tilde Br^2 , C = \tilde Cr^2 .\)