Ellipses estimation from their digitization

Abstract

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, $A = \tilde Ar^2 , B = \tilde Br^2 , C = \tilde Cr^2 .$

Since the digitization of real shapes causes an inherent loss of information about the original objects, the precision of the original shape estimation from the corresponding digital data is limited, i.e. there is no possibility that the original ellipse can be recovered from the digital ellipse. What we present here is the estimation of parameters A, B, C and center position (a, b), of the ellipse digitized as above, with relative error bounded by $\mathcal{O}\left( {\tfrac{1}{{r^{15/11 - \varepsilon } }}} \right)$ and absolute error bounded by $\mathcal{O}\left( {\tfrac{1}{{r^{4/11 - \varepsilon } }}} \right)$, (where e is an arbitrary positive number), that is, with the error tending to zero when the picture resolution increases. The obtained results imply that the half-axes of the original ellipse can be estimated with the same bounds of relative and absolute errors.