An Improved Tobit Kalman Filter with Adaptive Censoring Limits
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An Improved Tobit Kalman Filter with Adaptive Censoring Limits Kostas Loumponias1 · Nicholas Vretos2 · George Tsaklidis1 · Petros Daras2 Received: 7 February 2019 / Revised: 7 April 2020 / Accepted: 9 April 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper deals with the Tobit Kalman filtering (TKF) process when the measurements are correlated and censored. The case of interval censoring, i.e., the case of measurements which belong to some interval with given censoring limits, is considered. Two improvements of the standard TKF process are proposed, in order to estimate the hidden state vectors. Firstly, the exact covariance matrix of the censored measurements is calculated by taking into account the censoring limits. Secondly, the probability of a latent (normally distributed) measurement to belong in or out of the uncensored region is calculated by taking into account the Kalman filter residual. The designed algorithm is tested using both synthetic and real data sets. The real data set includes human skeleton joints’ coordinates captured by the Microsoft Kinect II sensor. In order to cope with certain real-life situations that cause problems in human skeleton tracking, such as (self)-occlusions, closely interacting persons, etc., adaptive censoring limits are used in the proposed TKF process. Experiments show that the proposed method outperforms other filtering processes in minimizing the overall root-mean-square error for synthetic and real data sets. Keywords Censored data · Adaptive Tobit Kalman filter · Human skeleton tracking
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George Tsaklidis [email protected] Kostas Loumponias [email protected] Nicholas Vretos [email protected] Petros Daras [email protected]
1
Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloníki, Greece
2
Information Technologies Institute, Centre for Research and Technology-Hellas, 6th km Charilaou, Thermi 57001, Thessaloníki, Greece
Circuits, Systems, and Signal Processing
1 Introduction Human skeleton motion tracking has been studied for several decades and remains a highly active research field due to its importance in several diverse domains like surveillance applications, medical applications, serious games, educational applications, high performance sports monitoring and others [27,41,44,45]. With the advent of commercial RGB-D sensors [7,36], human skeleton motion tracking has attracted a lot of attention due to the capacity of the sensors to reliably track skeletal joints. However, regardless of the significant progress that has been achieved in both sensors’ development and human skeleton motion tracking research, many applications require more accurate tracking of the human skeleton position and motion. On the sensors’ side, high performing sensors (such as the Vicon System), which are able to accurately track at high rates, are very expensive and cumbersome to deploy. On the other hand, affordable, commercial RGB-D solutions (i.e., the Microsoft Kinect, the Xtion Pro and others) often produce
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