HyperLS for Parameter Estimation in Geometric Fitting

Abstract

We present a general framework of a special type of least squares (LS) estimator, which we call "HyperLS," for parameter estimation that frequently arises in computer vision applications. It minimizes the algebraic distance under a special scale normalization, which is derived by a detailed error analysis in such a way that statistical bias is removed up to second order noise terms. We discuss in detail many theoretical issues involved in its derivation. By numerical experiments, we show that HyperLS is far superior to the standard LS and comparable in accuracy to maximum likelihood (ML), which is known to produce highly accurate results but may fail to converge if poorly initialized. We conclude that HyperLS is a perfect candidate for ML initialization.

Hyper-renormalization: Non-minimization Approach for Geometric Estimation

“…This implies that the reprojection error is not a good measure of ellipse fitting; we need to evaluate the error in θ. This was also pointed out by Kanatani et al [14] in relation to HyperLS. Figure 6 (a), (b) plot the bias B and the RMS error D, respectively, defined in Eq.…”

“…Usually, the images overlap very well in the part where matching points are extracted, but a large deviation may appear in the far away part with no matching points. In such applications, the reprojection error, i.e., the sum of the square distances between the points to be matched, is more or less the same among different methods as pointed out by Kanatani et al [14]. Hence, the error in θ, which expresses the coefficients of the homography equation (see Eqs.…”

“…It turns out that the initial solution with W (kl) α = δ kl coincides with what is called HyperLS [13], [14], [24], which is derived so that the bias is removed up to second order noise terms within the framework of algebraic methods without iterations. The expression of Eq.…”

“…If we iteratively solve Eq. (59), the initial solution with W (kl) α = δ kl corresponds to the least squares, the (extended) method of Taubin [26], and HyperLS [13], [14], [24], respectively. In other words, iterative reweight, renormalization, and hyper-renormalization can be viewed as an iterative improvement of least squares, the Taubin method, and HyperLS, respectively ( Table 1).…”

“…In Sections 7 and 8, we do a detailed error analysis of the problem. In Section 9, we derive the procedure of hyper-renormalization as an iterative improvement of what is called HyperLS [13], [14], [24]. In Section 10, we summarize the non-minimization approach.…”