Consider the biquadratic representation of an ellipse:
Given n noisy points ( ), we want to estimate the coefficients of the ellipse: . Due to the homogeneity, we set .
For each point , we thus have one scalar equation:
Hence, can be estimated by minimizing the following objective function (weighted least-squares optimization)
where 's are positive weights.
Assume that each point has the same error distribution with mean zero and covariance . The covariance of is then given by
Thus we have
The weights can then be chosen to the inverse proportion of the variances. Since multiplication by a constant does not affect the result of the estimation, we set
The objective function (16) can be rewritten as
which is a quadratic form in unit vector . Let
The solution is the eigenvector of associated to the smallest eigenvalue.
If each point is perturbed by noise of with
the matrix is perturbed accordingly: , where is the unperturbed matrix. If , then the estimate is statistically unbiased; otherwise, it is statistically biased, because following the perturbation theorem the bias of , i.e., .
Let , then . We have
If we carry out the Taylor development and ignore quantities of order higher than 2, it can be shown that the expectation of is given by
It is clear that if we define
then is unbiased, i.e., , and hence the unit eigenvector of associated to the smallest eigenvalue is an unbiased estimate of the exact solution .
Ideally, the constant c should be chosen so that , but this is impossible unless image noise characteristics are known. On the other hand, if , we have
because takes its absolute minimum 0 for the exact solution in the absence of noise. This suggests that we require that at each iteration. If for the current c and , , we can update c by such that
To summarize, the renormalization procedure can be described as:
associated to the smallest eigenvalue, which is denoted by .
and recompute using the new .
Remark 1: This implementation is different from that described in the paper of Kanatani . This is because in his implementation, he uses the N-vectors to represent the 2-D points. In the derivation of the bias, he assumes that the perturbation in each N-vector, i.e., in his notations, has the same magnitude . This is an unrealistic assumption. In fact, to the first order, thus . Hence, , where we assume the perturbation in image plane is the same for each point (with mean zero and standard deviation ).
Remark 2: This method is optimal only in the sense of unbiasness. Another criterion of optimality, namely the minimum variance of estimation, is not addressed in this method.