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We propose a novel framework for the statistical analysis of genus-zero 4D surfaces, i.e., 3D surfaces that deform and evolve over time. This problem is particularly challenging due to the arbitrary parameterizations of these surfaces and their varying deformation speeds, necessitating effective spatiotemporal registration. Traditionally, 4D surfaces are discretized, in space and time, before computing their spatiotemporal registrations, geodesics, and statistics. However, this approach may result in suboptimal solutions and, as we demonstrate in this paper, is not necessary. In contrast, we treat 4D surfaces as continuous functions in both space and time. We introduce Dynamic Spherical Neural Surfaces (D-SNS), an efficient smooth and continuous spatiotemporal representation for genus-0 4D surfaces. We then demonstrate how to perform core 4D shape analysis tasks such as spatiotemporal registration, geodesics computation, and mean 4D shape estimation, directly on these continuous representations without upfront discretization and meshing. By integrating neural representations with classical Riemannian geometry and statistical shape analysis techniques, we provide the building blocks for enabling full functional shape analysis. We demonstrate the efficiency of the framework on 4D human and face datasets. The source code and additional results are available at https://4d-dsns.github.io/DSNS/. 

Graph-based representations are becoming increasingly popular for representing and analyzing video data, especially in object tracking and scene understanding applications. Accordingly, an essential tool in this approach is to generate statistical inferences for graphical time series associated with videos. This paper develops a Kalman-smoothing method for estimating graphs from noisy, cluttered, and incomplete data. The main challenge here is to find and preserve the registration of nodes (salient detected objects) across time frames when the data has noise and clutter due to false and missing nodes. First, we introduce a quotient-space representation of graphs that incorporates temporal registration of nodes, and we use that metric structure to impose a dynamical model on graph evolution. Then, we derive a Kalman smoother, adapted to the quotient space geometry, to estimate dense, smooth trajectories of graphs. We demonstrate this framework using simulated data and actual video graphs extracted from the Multiview Extended Video with Activities (MEVA) dataset. This framework successfully estimates graphs despite the noise, clutter, and missed detections. 

Complex analyses involving multiple, dependent random quantities often lead to graphical models—a set of nodes denoting variables of interest, and corresponding edges denoting statistical interactions between nodes. To develop statistical analyses for graphical data, especially towards generative modeling, one needs mathematical representations and metrics for matching and comparing graphs, and subsequent tools, such as geodesics, means, and covariances. This paper utilizes a quotient structure to develop efficient algorithms for computing these quantities, leading to useful statistical tools, including principal component analysis, statistical testing, and modeling. We demonstrate the efficacy of this framework using datasets taken from several problem areas, including letters, biochemical structures, and social networks. 

Abstract We present the results of a search for gravitational-wave transients associated with core-collapse supernova SN 2023ixf, which was observed in the galaxy Messier 101 via optical emission on 2023 May 19, during the LIGO–Virgo–KAGRA 15th Engineering Run. We define a five-day on-source window during which an accompanying gravitational-wave signal may have occurred. No gravitational waves have been identified in data when at least two gravitational-wave observatories were operating, which covered ∼14% of this five-day window. We report the search detection efficiency for various possible gravitational-wave emission models. Considering the distance to M101 (6.7 Mpc), we derive constraints on the gravitational-wave emission mechanism of core-collapse supernovae across a broad frequency spectrum, ranging from 50 Hz to 2 kHz, where we assume the gravitational-wave emission occurred when coincident data are available in the on-source window. Considering an ellipsoid model for a rotating proto-neutron star, our search is sensitive to gravitational-wave energy 1 × 10⁻⁴M_⊙c²and luminosity 2.6 × 10⁻⁴M_⊙c²s⁻¹for a source emitting at 82 Hz. These constraints are around an order of magnitude more stringent than those obtained so far with gravitational-wave data. The constraint on the ellipticity of the proto-neutron star that is formed is as low as 1.08, at frequencies above 1200 Hz, surpassing past results.