Rosanna Campagna

Vittoria Bruni, Paola Erminia Calabrese, Rosanna Campagna et al.

This paper explores the use of artificial neural networks for the stable and data-driven selection of the frequency parameter in hyperbolic polynomial penalized splines (HP-splines). This parameter defines the underlying spline space and is essential for adapting the model to exponential patterns in the data, such as those encountered in signal processing. The theoretical approximation properties of deep neural network architectures are investigated to establish a connection between classical spline-based regression and modern data-driven learning methods. Based on this analysis, a neural network is designed to predict optimal HP-spline parameters by balancing approximation accuracy, stability analysis, and complexity control, thereby producing neural architectures that are both expressive and stable. Numerical experiments confirm that the proposed approach achieves both high accuracy and stable performance, validating the theoretical findings.

Rosanna Campagna, Salvatore Mondrone, Tomas Sauer

Offset curves for planar trajectories are interesting in the generation of tool paths for numerically controlled industrial machines and in trajectory planning methods for autonomous driving systems. Theoretical offset curves may exhibit peculiar singularities, including self-intersections, which limit their use in practical applications. Existing approaches address these issue through geometric filtering techniques to detect and remove undesirable features but the computation of accurate and well-behaved offset curves remains a challenging task. We assume a first stage of functional approximation of trajectories by penalized Hermite spline regression enabling the simultaneous fitting of positions and tangents. The regularization is imposed on the second derivatives, effectively mitigating the jerk effect, which is particularly relevant in motion planning and path smoothing applications. Then, taking into account the geometrical pointwise properties of the resulting curve, we design two offset curves through the simultaneous approximation of function values and derivatives. Then, a mathematical model to obtain the so-called bi-offset as most fitting as with the original generator curve is proposed, also relating the offset range and pointwise curvature values. The adaptive reconstruction of the center line from the external boundaries is a topic of interest and is the main focus of our work. Numerical experiments confirm the reliability of our approach at every stage of the resolution process.