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1. Assouad–Nagata dimension of minor‐closed metrics

   https://doi.org/10.1112/plms.70032  

   Liu, Chun‐Hung (March 2025, Proceedings of the London Mathematical Society)

   Abstract Assouad–Nagata dimension addresses both large‐ and small‐scale behaviors of metric spaces and is a refinement of Gromov's asymptotic dimension. A metric space is a minor‐closed metric if there exists an (edge‐)weighted graph satisfying a fixed minor‐closed property such that the underlying space of is the vertex‐set of , and the metric of is the distance function in . Minor‐closed metrics naturally arise when removing redundant edges of the underlying graphs by using edge‐deletion and edge‐contraction. In this paper, we determine the Assouad–Nagata dimension of every minor‐closed metric. Our main theorem simultaneously generalizes known results about the asymptotic dimension of ‐minor free unweighted graphs and about the Assouad–Nagata dimension of complete Riemannian surfaces with finite Euler genus (Bonamy et al., Asymptotic dimension of minor‐closed families and Assouad–Nagata dimension of surfaces,JEMS(2024)). 

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2. Bayesian Tracking of Video Graphs Using Joint Kalman Smoothing and Registration

   https://doi.org/10.1007/978-3-031-19833-5_26  

   Bal, A.B.; Mounir, R.; Aakur, S.; Sarkar, S.; Srivastava, A. (November 2022, European Conference on Computer Vision (ECCV))

   Avidan, S.; Brostow, G.; Cissé, M.; Farinella. G.M.; Hassner, T. (Ed.)

   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. 

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3. Bayesian Tracking of Video Graphs Using Joint Kalman Smoothing and Registration

   Bal, Aditi Basu; Mounir, Ramy; Aakur, Sathyanarayanan; Sarkar, Sudeep; Srivastava, Anuj (October 2022, European Conference on Computer Vision)

   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. 

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4. Bayesian Tracking of Video Graphs Using Joint Kalman Smoothing and Registration

   Aditi Basu Bal, Ramy Mounir (November 2022, European Conference on Computer Vision)

   Avidan, S. (Ed.)

   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. 

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5. Shape-Graph Matching Network (SGM-net): Registration for Statistical Shape Analysis

   https://doi.org/10.1109/ISBI56570.2024.10635203  

   Liang, Shenyuan; Srivastava, Anuj; Segundo, Mauricio Pamplona; Sarkar, Sudeep; Aakur, Sathyanarayanan N (May 2024, IEEE)

   This paper focuses on the registration problem of shape graphs, where a shape graph is a set of nodes connected by articulated curves with arbitrary shapes. This registration requires optimization over the permutation group, made challenging by differences in nodes (in terms of numbers, locations) and edges (in terms of shapes, placements, and sizes) across graphs. We tackle this registration problem using a neuralnetwork architecture with an unsupervised loss function based on the elastic shape metric for curves. This architecture results in (1) state-of-the-art matching performance and (2) an order of magnitude reduction in the computational cost relative to baseline approaches. We demonstrate the effectiveness of the proposed approach using both simulated data and real-world 2D retinal blood vessels and 3D microglia graphs. 

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