Hyper-renormalization: Non-minimization Approach for Geometric Estimation

Abstract

Abstract:The technique of "renormalization" for geometric estimation attracted much attention when it appeared in early 1990s for having higher accuracy than any other then known methods. The key fact is that it directly specifies equations to solve, rather than minimizing some cost function. This paper expounds this "non-minimization approach" in detail and exploits this principle to modify renormalization so that it outperforms the standard reprojection error minimization. Doing a precise error analysis in the most general situation, we derive a formula that maximizes the accuracy of the solution; we call it hyper-renormalization. Applying it to ellipse fitting, fundamental matrix computation, and homography computation, we confirm its accuracy and efficiency for sufficiently small noise. Our emphasis is on the general principle, rather than on individual methods for particular problems.

A Bayesian method for type-specific quadric fitting

“…The curvature bias correction has not been explicitly represented: G 0 here is the uncorrected vector of coefficients. The results to this point are in practice very similar to those of [10], although there are two significant differences in approach. The first difference is that in this paper I have minimised the bias from each source separately: the normalisation matrix is chosen to remove normalisation bias; reweighting bias is minimised by evaluating the gradient at a point consistent with the current best-fit model; and the curvature bias is corrected as a separate, final step.…”

“…The first difference is that in this paper I have minimised the bias from each source separately: the normalisation matrix is chosen to remove normalisation bias; reweighting bias is minimised by evaluating the gradient at a point consistent with the current best-fit model; and the curvature bias is corrected as a separate, final step. In contrast, the method of [10] chooses the normalisation matrix to minimise all three biases simultaneously, including that from Sampson reweighting. One advantage of separate treatment is that the normalisation matrix may be calculated by the same method both before and after reweighting.…”

“…A geometric fit using a computationally expensive nonlinear least-squares method would produce results very similar to those obtained by omitting the curvature bias correction. The 'hyper-renormalised' method of [10] uses the Sampson weighting but modifies the normalisation matrix to correct for this: as Fig. 5 shows, this reduces the weighting bias but does not completely eliminate it.…”

“…A recent series of papers by Kanatani and coworkers [8][9][10] have used a perturbative treatment to predict the systematic errors introduced into the fitting process by noisy data, and hence to identify analytically (rather than merely numerically) the choices of normalisation and weighting that minimise bias and statistical error. These are forward statistical calculations, that is, the noisy data is described by a sampling distribution around the true values; the final result is an unbiased point estimate of the generic conic coefficients.…”

New Methods for Detecting Concentric Objects With High Accuracy

“…Because of its complexity, we also propose a simplified version of Hyper LS that produces virtually the same results for relatively large sample sizes. Following [16,17], we call this method the Semi-Hyper method.…”