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This article proposes a Rao–Blackwellized particle filter (RBPF) for fully mixing state-space models that replace the Kalman filter within the RBPF method with a noise-adaptive Kalman filter. This extension aims to deal with unknown time-varying measurement variances. Consequently, a variational Bayesian (VB) adaptive Kalman filter estimates the conditionally linear states and the measurement noise variances, whereas the nonlinear (or latent) states are handled by sequential Monte Carlo sampling. Thus, by modifying the underlying mathematical framework of RBPF, we construct the Monte Carlo variational Bayesian (MCVB) filter. A stopping criterion for VB approximations is proposed by employing Tikhonov regularization. In addition, an analysis of the numerical stability of the proposed filtering mechanism is presented. The performance of the MCVB filter is illustrated in simulations and mobile robot tracking experiments in the presence of measurement model uncertainties.

The zero velocity update (ZUPT) offers an effective correction method for sensor drift in indoor positioning systems using foot-mounted inertial measurement units (IMUs). However, the heading drift is still a problem in positioning systems using IMUs. This article develops methods for positioning using two foot-mounted IMUs to solve this problem. The proposed method is based on the use of a time-varying distance constraint (VDC) and a spacing-vector constraint. Our methods are experimentally compared against other distance-constraint-based methods using a dataset that we collected for this purpose. The results show that our method has better control of the separation between the trajectories of the feet.

The automatic projection filter is a recently developed numerical method for projection filtering that leverages sparse-grid integration and automatic differentiation. However, its accuracy is highly sensitive to the accuracy of the cumulant-generating function computed via the sparse-grid integration, which in turn is also sensitive to the choice of the bijection from the canonical hypercube to the state space. In this article, we propose two new adaptive parametric bijections for the automatic projection filter. The first bijection relies on the minimization of Kullback–Leibler divergence, whereas the second method employs the sparse-grid Gauss–Hermite quadrature. The two new bijections allow the sparse-grid nodes to adaptively move within the high-density region of the state space, resulting in a substantially improved approximation while using only a small number of quadrature nodes. The practical applicability of the methodology is illustrated in three simulated nonlinear filtering problems.

Recently, joint use of thermal camera and RGB camera has emerged as a potential solution for respiration measurement. Bulky and large thermal cameras with high thermal resolution have been mostly used in thermal-imaging-based respiration measurement studies. We aim to present a method for respiration measurement using an RGB camera and a low thermal resolution thermal camera. Thermal cameras with low thermal resolution are small in size and easily portable, allowing flexible and a simple implementation of the measurement system. First, the cameras were calibrated and synchronized using calibration objects and sound analysis tools, respectively. Second, a cross-spectrum mapping method was developed which was followed by the extraction of the respiration signal. Third, the extracted and reference respiration signals were aligned. In the last step of the proposed method, a signal processing framework was applied to the estimated and reference respiration signals. The performance of the cross-spectrum mapping method was visually evaluated. The mapping method performed well when a participant had little to no head movements; however, the performance decreased when head movements were present. Our analysis showed that there is a relatively good agreement between the proposed and the reference respiration measurement methods. The number of breathing cycles from the estimated respiration signal was computed with a decent accuracy. However, the accuracy of measuring breathing rate is low compared to the previous studies. The estimated breathing patterns were measured with better accuracy than in the previous work. This study demonstrates that respiration measurement can be done by the joint use of an RGB camera and thermal camera that has a low thermal resolution, with a decent level of accuracy.

We present a continuous-time probabilistic approach for estimating the chirp signal and its instantaneous frequency function when the true forms of these functions are not accessible. Our model represents these functions by non-linearly cascaded Gaussian processes represented as non-linear stochastic differential equations. The posterior distribution of the functions is then estimated with stochastic filters and smoothers. We compute a (posterior) Cramér–Rao lower bound for the Gaussian process model, and derive a theoretical upper bound for the estimation error in the mean squared sense. The experiments show that the proposed method outperforms a number of state-of-the-art methods on a synthetic data. We also show that the method works out-of-the-box for two real-world datasets.