Gesture recognition is a fundamental tool to enable novel interaction paradigms in a variety of application scenarios like Mixed Reality environments, touchless public kiosks, entertainment systems, and more. Recognition of hand gestures can be nowadays performed directly from the stream of hand skeletons estimated by software provided by low-cost trackers (Ultraleap) and MR headsets (Hololens, Oculus Quest) or by video processing software modules (e.g. Google Mediapipe). Despite the recent advancements in gesture and action recognition from skeletons, it is unclear how well the current state-of-the-art techniques can perform in a real-world scenario for the recognition of a wide set of heterogeneous gestures, as many benchmarks do not test online recognition and use limited dictionaries. This motivated the proposal of the SHREC 2021: Track on Skeleton-based Hand Gesture Recognition in the Wild. For this contest, we created a novel dataset with heterogeneous gestures featuring different types and duration. These gestures have to be found inside sequences in an online recognition scenario. This paper presents the result of the contest, showing the performances of the techniques proposed by four research groups on the challenging task compared with a simple baseline method.

We consider an inpainting model proposed by A. Bertozzi et al., which is based on a Cahn--Hilliard-type equation. This equation describes the evolution of an order parameter $u \in [0,1]$ representing an approximation of the original image which occupies a bounded two-dimensional domain $\Omega$. The given image $g$ is assumed to be damaged in a fixed subdomain $D \subset \Omega$ and the equation is characterized by a linear reaction term of the form $\lambda (u - g)$. Here $\lambda = \lambda_0 \chi_{\Omega \setminus D}$ is the so-called fidelity coefficient, $\lambda_0$ being a strictly positive bounded function. The idea is that, given an initial image $u_0$, $u$ evolves towards $g$ and this process properly diffuses through the boundary of $D$ restoring the damaged image, provided that $\lambda_0$ is large enough. Here, we formulate an optimal control problem based on this fact, namely our cost functional accounts for the magnitude of $\lambda_0$. Assuming a singular potential to assure that $u$ takes its values in $[0,1]$, we first analyse the control-to-state operator and prove the existence of at least one optimal control, establishing the validity of first-order optimality conditions. Then, under suitable assumptions, we demonstrate second-order optimality conditions. All these results depend on the existence and uniqueness of a strong solution, which we obtain thanks to the strict separation property from pure phases.