High-order adaptive methods for computing invariant manifolds of maps

The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps

Kinematic interpolation of movement data

Mobile tracking technologies are facilitating the collection of increasingly large and detailed data sets on object movement. Movement data are collected by recording an object’s location at discrete time intervals. Often, of interest is to estimate the unknown position of the object at unrecorded time points to increase the temporal resolution of the data, to correct erroneous or missing data points, or to match the recorded times between multiple data sets. Estimating an object’s unknown location between known locations is termed path interpolation. This paper introduces a new method for path interpolation termed kinematic interpolation. Kinematic interpolation incorporates object kinematics (i.e. velocity and acceleration) into the interpolation process. Six empirical data sets (two types of correlated random walks, caribou, cyclist, hurricane and athlete tracking data) are used to compare kinematic interpolation to other interpolation algorithms. Results showed kinematic interpolation to be a suitable interpolation method with fast-moving objects (e.g. the cyclist, hurricane and athlete tracking data), while other algorithms performed best with the correlated random walk and caribou data. Several issues associated with path interpolation tasks are discussed along with potential applications where kinematic interpolation can be useful. Finally, code for performing path interpolation is provided (for each method compared within) using the statistical software R.PostprintPeer reviewe

Polynomial-based non-uniform interpolatory subdivision with features control

Starting from a well-known construction of polynomial-based interpolatory 4-point schemes, in this paper we present an original affine combination of quadratic polynomial samples that leads to a non-uniform 4-point scheme with edge parameters. This blending-type formulation is then further generalized to provide a powerful subdivision algorithm that combines the fairing curve of a non-uniform refinement with the advantages of a shape-controlled interpolation method and an arbitrary point insertion rule. The result is a non-uniform interpolatory 4-point scheme that is unique in combining a number of distinctive properties. In fact it generates visually-pleasing limit curves where special features ranging from cusps and flat edges to point/edge tension effects may be included without creating undesired undulations. Moreover such a scheme is capable of inserting new points at any positions of existing intervals, so that the most convenient parameter values may be chosen as well as the intervals for insertion. Such a fully flexible curve scheme is a fundamental step towards the construction of high-quality interpolatory subdivision surfaces with features control

Online Monocular Lane Mapping Using Catmull-Rom Spline

In this study, we introduce an online monocular lane mapping approach that solely relies on a single camera and odometry for generating spline-based maps. Our proposed technique models the lane association process as an assignment issue utilizing a bipartite graph, and assigns weights to the edges by incorporating Chamfer distance, pose uncertainty, and lateral sequence consistency. Furthermore, we meticulously design control point initialization, spline parameterization, and optimization to progressively create, expand, and refine splines. In contrast to prior research that assessed performance using self-constructed datasets, our experiments are conducted on the openly accessible OpenLane dataset. The experimental outcomes reveal that our suggested approach enhances lane association and odometry precision, as well as overall lane map quality. We have open-sourced our code1 for this project.Comment: Accepted by IROS202