A bilinear approach to the parameter estimation of a general heteroscedastic linear system, with application to conic fitting

Abstract

In this paper, we employ low-rank matrix approximation to solve a general parameter estimation problem: where a non-linear system is linearized by treating the carrier terms as separate variables, thereby introducing heteroscedastic noise. We extend the bilinear approach to handle cases with heteroscedastic noise, in the framework of low-rank approximation. The ellipse fitting problem is investigated as a specific example of the general theory. Despite the impression given in the literature, the ellipse fitting problem is still unsolved when the data comes from a small section of the ellipse. Although there are already some good approaches to the problem of ellipse fitting, such as FNS and HEIV, convergence in these iterative approaches is not ensured, as pointed out in the literature. Another limitation of these approaches is that they cannot model the correlations among different rows of the “general measurement matrix”. Our method, of employing the bilinear approach to solve the general heteroscedastic parameter estimation problem, overcomes these limitations: it is convergent, at least to a local optimum, and can cope with a general heteroscedastic problem. Experiments show that the proposed bilinear approach performs better than other competing approaches: although it is still far short of a solution when the data comes from a very small arc of the ellipse.