Error analysis for circle fitting algorithms

Abstract

We study the problem of fitting circles (or circular arcs) to data points observed with errors in both variables. A detailed error analysis for all popular circle fitting methods -geometric fit, Kåsa fit, Pratt fit, and Taubin fit -is presented. Our error analysis goes deeper than the traditional expansion to the leading order. We obtain higher order terms, which show exactly why and by how much circle fits differ from each other. Our analysis allows us to construct a new algebraic (non-iterative) circle fitting algorithm that outperforms all the existing methods, including the (previously regarded as unbeatable) geometric fit.

Statistical analysis of curve fitting methods in errors-in-variables models

“…In the nonlinear regression, there are several popular circle fitting methods (by Kåsa, Pratt, and Taubin; see references in [2,16]). It turns out that they all (!)…”

“…General scheme. Here we present our error analysis of curve fitting methods following [2]. For convenience we use vector notation…”

Dusty plasma (Yukawa) rings

“…The camera resolution was 0.0317 mm/pixel . To determine the geometric center of the ring we fitted the particle coordinates to a circle [22] and then averaged over all frames. (Because of variable particle spacing the geometrical center and the center of mass are not identical.)…”

Untitled

“…The expansion (7) is valid when the functionθ(x, y) satisfies standard regularity conditions (more precisely, when it has a continuous third derivative). Such conditions can be verified by the implicit value theorem; see [2] for circle estimators.…”