In this paper, we present an algorithm for fast calculation of the normalized cross correlation and its application to the problem of template matching. Given a template t, whose position is to be determined in an image f, the basic idea of the algorithm is to represent the template, for which the normalized cross correlation is calculated, as a sum of rectangular basis functions. Then the correlation is calculated for each basis function instead of the whole template. The result of the correlation of the template t and the image f is obtained as the weighted sum of the correlation functions of the basis functions. Depending on the approximation, the algorithm can by far outperform Fourier-transform based implementations of the normalized cross correlation algorithm and it is especially suited to problems, where many different templates are to be found in the same image f.

We present a novel tightly-coupled LiDAR-inertial odometry and mapping scheme for both solid-state and mechanical LiDARs. As frontend, a feature-based lightweight LiDAR odometry provides fast motion estimates for adaptive keyframe selection. As backend, a hierarchical keyframe-based sliding window optimization is performed through marginalization for directly fusing IMU and LiDAR measurements. For the Livox Horizon, a newly released solid-state LiDAR, a novel feature extraction method is proposed to handle its irregular scan pattern during preprocessing. LiLi-OM (Livox LiDAR-inertial odometry and mapping) is real-time capable and achieves superior accuracy over state-of-the-art systems for both LiDAR types on public data sets of mechanical LiDARs and in experiments using the Livox Horizon. Source code and recorded experimental data sets are available at https://github.com/KIT-ISAS/lili-om.

The random hypersurface model (RHM) is introduced for estimating a shape approximation of an extended object in addition to its kinematic state. An RHM represents the spatial extent by means of randomly scaled versions of the shape boundary. In doing so, the shape parameters and the measurements are related via a measurement equation that serves as the basis for a Gaussian state estimator. Specific estimators are derived for elliptic and star-convex shapes.

For many practical probability density representations such as for the widely used Gaussian mixture densities, an analytic evaluation of the differential entropy is not possible and thus, approximate calculations are inevitable. For this purpose, the first contribution of this paper deals with a novel entropy approximation method for Gaussian mixture random vectors, which is based on a component-wise Taylor-series expansion of the logarithm of a Gaussian mixture and on a splitting method of Gaussian mixture components. The employed order of the Taylor-series expansion and the number of components used for splitting allows balancing between accuracy and computational demand. The second contribution is the determination of meaningful and efficiently to calculate lower and upper bounds of the entropy, which can be also used for approximation purposes. In addition, a refinement method for the more important upper bound is proposed in order to approach the true entropy value.

We propose a principled algorithm for robust Bayesian filtering and smoothing in nonlinear stochastic dynamic systems when both the transition function and the measurement function are described by non-parametric Gaussian process (GP) models. GPs are gaining increasing importance in signal processing, machine learning, robotics, and control for representing unknown system functions by posterior probability distributions. This modern way of system identification is more robust than finding point estimates of a parametric function representation. Our principled filtering/smoothing approach for GP dynamic systems is based on analytic moment matching in the context of the forward-backward algorithm. Our numerical evaluations demonstrate the robustness of the proposed approach in situations where other state-of-the-art Gaussian filters and smoothers can fail.