Comparison between Kalman and unscented Kalman filters in tracking applications of computational vision

In this paper, the problem of tracking feature points along image sequences is addressed. The establishment of correspondences between points and their tracking along image sequences is a complex problem in Computational Vision; especially, when intricate motions, erroneously detections or cases of occlusion or appearance/disappearing of features are involved. To overcome some of those difficulties, a statistical approach is frequently used in a multi-object data association and state estimation framework. Additionally, the correspondence between each measurement and predicted feature can be performed by minimizing the overall Mahalanobis distance. Under these circumstances, the estimation of the system can be accomplished using different stochastic filters. Hereby, a comparison is made between the results obtained, with the described framework, either by the Kalman Filter or the Unscented Kalman Filter, in the tracking of linear and nonlinear motions of feature points along image sequences. A central and crucial operation performed in the KF is the propagation of a Gaussian Random Variable (GRV) through the system’s dynamics. In the EKF, the state distribution is approximated by a GRV, which is then propagated analytically through the first-order linearization of the nonlinear system. This can introduce significant errors in the true posterior mean and covariance of the transformed GRV, which may lead to sub-optimal performance and sometimes divergence of the filter. Other filters have been proposed, such as the Iterated Extended Kalman Filter (IEKF), the Central Difference Filter (CDF) and the first order Divided Difference Filter. The conventional EKF and IEKF algorithms do not take the linearization errors into account, which leads to inconsistent state estimates when these errors cannot be neglected, [Lefebvre et al. (2003)]. The UKF addresses that problem by using a deterministic sampling approach. The state distribution is again approximated by a GRV, which is represented using a set of carefully chosen sample points. These sample points completely capture the true mean and covariance of the GRV, and when propagated through the true nonlinear system, also approach the posterior mean and covariance accurately to the third order (Taylor series expansion) for any nonlinearity. The EKF, in contrast, only achieves first order accuracy. Remarkably, the computational complexity of the UKF is of the same order as the EKF’s, [Wan & Van Der Merwe (2000)]. A variation of the original UKF algorithm (called the Reduced Sigma Point Filters [Julier & Uhlmann (2002)]) chooses only 1 n + sampling points. This means that the linear regression through those points is exact; therefore, it does not take the linearization errors into account. Hence, as was the case for the (I)EKF, the state estimates are generally inconsistent, [Lefebvre, Bruyninckx and De Schutter (2003)]. An arising estimation model is the Particle Filter (PF): the key idea is to represent the posterior density function by a set of random samples with associated weights and to compute estimates based on these estimates and weights. Comparing PF to KF, the former has a more robust performance in the case of non-Gaussianity and non-linearity due to the simulated posterior distribution. In a PF, a large number of particles are desirable to represent the posterior distribution, especially in situations where new measurements appear in the tail of the prior, or if the likelihood is strongly peaked. To solve the computational problem raised by the large particle numbers that can be generated, for example, in [MacCormick & Isard (2000)] a partitioned sampling is proposed, which requires that the state-space can be sliced. With the same purpose, Sullivan et al., in [Sullivan et al. (1999)], proposed layered sampling using multiscale processing of images. It turns out that these solutions significantly reduce the computational costs, but in-depth efforts are desirable for better efficiency, [Zhou et al. (2009)]. In [Merwe et al. (2000)] an Unscented Particle Filter (UPF) is introduced. The UKF allows the PF to incorporate the latest observations into a prior updating routine. In addition, the UKF generates proposal distributions that match the true posterior more closely and also has the capability of generating heavier tailed distributions than the well known EKF. Consequently, the convergence results predict that UPF should outperform standard PF, EKF and UKF. However, its computational cost can also be questioned. Other comparison studies between predictive filters have been published; for example, in [LaViola (2003)] the performance of the UKF and EKF is compared towards improving human head and hand tracking using quaternions, and in [Lefebvre, Bruyninckx and De Schutter (2003)] the process and measurement update performances are studied separately, instead of analyzing their overall behavior of some KF variants. Along the last years, numerous applications of state estimation have been reported, such as in sensor fusion and navigation, [Metzger et al. (2005)], state estimation for chemical processes, [Vachhani et al. (2005)], weather forecasting, [Carme et al. (2001)], and training of neural networks [Rios Neto (1997)]. 1.2 Adopted Tracking Framework In the applied tracking framework, the measured data is integrated using optimization upon the correspondence step. The proposed criterion minimizes the global matching cost based on the Mahalanobis distance, under the assumption that the matching cost is superior to a threshold value. To deal with the appearance and disappearance of features during the tracking process a management model is considered, [Pinho et al. (2007)].