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We propose a generic spatiotemporal framework to analyze manifold-valued measurements, which allows for employing an intrinsic and computationally efficient Riemannian hierarchical model. Particularly, utilizing regression, we represent discrete trajectories in a Riemannian manifold by composite Bézier splines, propose a natural metric induced by the Sasaki metric to compare the trajectories, and estimate average trajectories as group-wise trends. We evaluate our framework in comparison to state-of-the-art methods within qualitative and quantitative experiments on hurricane tracks. Notably, our results demonstrate the superiority of spline-based approaches for an intensity classification of the tracks.
We propose two graph neural network layers for graphs with features in a Riemannian manifold. First, based on a manifold-valued graph diffusion equation, we construct a diffusion layer that can be applied to an arbitrary number of nodes and graph connectivity patterns. Second, we model a tangent multilayer perceptron by transferring ideas from the vector neuron framework to our general setting. Both layers are equivariant with respect to node permutations and isometries of the feature manifold. These properties have been shown to lead to a beneficial inductive bias in many deep learning tasks. Numerical examples on synthetic data as well as on triangle meshes of the right hippocampus to classify Alzheimer's disease demonstrate the very good performance of our layers.
For decades, de Casteljau's algorithm has been used as a fundamental building block in curve and surface design and has found a wide range of applications in fields such as scientific computing, and discrete geometry to name but a few. With increasing interest in nonlinear data science, its constructive approach has been shown to provide a principled way to generalize parametric smooth curves to manifolds. These curves have found remarkable new applications in the analysis of parameter-dependent, geometric data. This article provides a survey of the recent theoretical developments in this exciting area as well as its applications in fields such as geometric morphometrics and longitudinal data analysis in medicine, archaeology, and meteorology.
Gesture recognition is a tool to enable novel interactions with different techniques and
applications, like Mixed Reality and Virtual Reality environments. With all the recent
advancements in gesture recognition from skeletal data, it is still unclear how well state-of-
the-art techniques perform in a scenario using precise motions with two hands. This
paper presents the results of the SHREC 2024 contest organized to evaluate methods
for their recognition of highly similar hand motions using the skeletal spatial coordinate
data of both hands. The task is the recognition of 7 motion classes given their spatial
coordinates in a frame-by-frame motion. The skeletal data has been captured using
a Vicon system and pre-processed into a coordinate system using Blender and Vicon
Shogun Post. We created a small, novel dataset with a high variety of durations in
frames. This paper shows the results of the contest, showing the techniques created
by the 5 research groups on this challenging task and comparing them to our baseline
method.