Multidimensional trimming based on projection depth

As estimators of location parameters, univariate trimmed means are well known for their robustness and efficiency. They can serve as robust alternatives to the sample mean while possessing high efficiencies at normal as well as heavy-tailed models. This paper introduces multidimensional trimmed means based on projection depth induced regions. Robustness of these depth trimmed means is investigated in terms of the influence function and finite sample breakdown point. The influence function captures the local robustness whereas the breakdown point measures the global robustness of estimators. It is found that the projection depth trimmed means are highly robust locally as well as globally. Asymptotics of the depth trimmed means are investigated via those of the directional radius of the depth induced regions. The strong consistency, asymptotic representation and limiting distribution of the depth trimmed means are obtained. Relative to the mean and other leading competitors, the depth trimmed means are highly efficient at normal or symmetric models and overwhelmingly more efficient when these models are contaminated. Simulation studies confirm the validity of the asymptotic efficiency results at finite samples.Comment: Published at http://dx.doi.org/10.1214/009053606000000713 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Do-It-Yourself Single Camera 3D Pointer Input Device

We present a new algorithm for single camera 3D reconstruction, or 3D input for human-computer interfaces, based on precise tracking of an elongated object, such as a pen, having a pattern of colored bands. To configure the system, the user provides no more than one labelled image of a handmade pointer, measurements of its colored bands, and the camera's pinhole projection matrix. Other systems are of much higher cost and complexity, requiring combinations of multiple cameras, stereocameras, and pointers with sensors and lights. Instead of relying on information from multiple devices, we examine our single view more closely, integrating geometric and appearance constraints to robustly track the pointer in the presence of occlusion and distractor objects. By probing objects of known geometry with the pointer, we demonstrate acceptable accuracy of 3D localization.Comment: 8 pages, 6 figures, 2018 15th Conference on Computer and Robot Visio