Geometric estimation problems in vision are often solved via minimization of statistical loss functions which account for the presence of outliers in the observations. The corresponding energy landscape often has many local minima. Many approaches attempt to avoid local minima by an-nealing the scale parameter of loss functions using methods such as graduated non-convexity (GNC). However, little attention has been paid to the annealing schedule, which is often carried out in a fixed manner, resulting in a poor speed-accuracy trade-off and unreliable convergence to the global minimum. In this paper, we propose a principled approach for adaptively annealing the scale for GNC by tracking the positive-definiteness (i.e. local convexity) of the Hessian of the cost function. We illustrate our approach using the classic problem of registering 3D correspondences in the presence of noise and outliers. We also develop approximations to the Hessian that significantly speeds up our method. The effectiveness of our approach is validated by comparing its performance with state-of-the-art 3D registration approaches on a number of synthetic and real datasets. Our approach is accurate and efficient and converges to the global solution more reliably than the state-of-the-art methods.

Rotation averaging is the problem of recovering 3D camera rotations from a number of pairwise relative rotation estimates. The state-of-the-art method of [51 involves robust averaging in the Lie-algebra of 3D rotations using an $\ell_{\frac{1}{2}}$ loss function which is carried out using an iteratively reweighted least squares (IRLS) minimization. In this paper we argue that the performance of IRLS-based rotation averaging is intimately connected with two factors: a) the nature of the robust loss function used, and b) the initialization. We make two contributions. Firstly, we analyse the pitfalls associated with the unbounded weights in IRLS minimization of $\ell_{p}(0\lt p\lt 2)$ loss functions in the context of rotation averaging. We elucidate the design choices and modifications implicit to the state-of-the-art method of [5] that overcomes these problems. Secondly, we argue that the $\ell_{\frac{1}{2}}$ -based IRLS method is inflexible in adapting to the specific noise characteristics of individual datasets, leading to poorer performance. We remedy this limitation by means of a Geman-McClure loss function embedded in a graduated optimization framework. We present results on a number of large-scale real-world datasets to demonstrate that our proposed method outpetforms state-of-the-art methods in terms of both efficiency and accuracy.