Parameter estimation and reconstruction of digital conics in normal positions

Abstract

Reconstruction of the original curve (and the estimation of its parameters) from its digitization is a challenging problem as quantization always causes some loss of information. So we often estimate at least one (or all) continuous curve(s) which is (are) isomorphic to the original one under discretization. Some work has already been done in this respect on straight lines, circles, squares, etc. In this paper, we have attempted this problem for a specialized class of conics which are said to be in normal positions. In normal position the center of the conic is situated at a grid point and its axes are parallel to the coordinate axes. For circles and parabolas, we can directly formulate the domain, i.e., the entire set of continuous curves which produces the same digitization. For ellipses (and this can be extended to hyperbolas too), we first compute the smallest rectangle containing the domain of the given digitization and then estimate the domain itself. The major contribution of this paper lies in the development of a new method of analysis (via the iterative refinement of parameter bounds) which can be easily extended to other 1- or 2-parameter piecewise monotonic shapes such as straight lines or circles with known radius.