Toward a statistically optimal method for estimating geometric relations from noisy data: cases of linear relations

In many problems of computer vision we have to estimate parameters in the presence of nuisance parameters increasing with the amount of data. It is known that, unlike in the cases without nuisance parameters, maximum likelihood estimation (MLE) is not optimal in the presence of nuisance parameters. By optimal we mean that the resulting estimate is unbiased and its variance attains the theoretical lower bound in an asymptotic sense. Thus, naive application of MLE to computer vision has a potential problem. This applies to a wide range of problems from conic fitting to bundle adjustment. For this nuisance parameter problem, studies have been conducted in statistics for a long time, whereas they have hardly been studied in the computer vision community. We cast light on the methods developed in statistics for obtaining optimal estimates and explore the possibility of applying them to computer vision problems. In this paper we focus on the cases where data and nuisance parameters are linearly connected. As examples, optical flow estimation and affine structure and motion problems are considered. Through experiments, we show that the estimation accuracy is improved in several cases.