Gegenbauer Graph Neural Networks for Time-Varying Signal Reconstruction.

Reconstructing time-varying graph signals (or graph time-series imputation) is a critical problem in machine learning and signal processing with broad applications, ranging from missing data imputation in sensor networks to time-series forecasting. Accurately capturing the spatio-temporal information inherent in these signals is crucial for effectively addressing these tasks. However, existing approaches relying on smoothness assumptions of temporal differences and simple convex optimization techniques that have inherent limitations. To address these challenges, we propose a novel approach that incorporates a learning module to enhance the accuracy of the downstream task. To this end, we introduce the Gegenbauer-based graph convolutional (GegenConv) operator, which is a generalization of the conventional Chebyshev graph convolution by leveraging the theory of Gegenbauer polynomials. By deviating from traditional convex problems, we expand the complexity of the model and offer a more accurate solution for recovering time-varying graph signals. Building upon GegenConv, we design the Gegenbauer-based time graph neural network (GegenGNN) architecture, which adopts an encoder-decoder structure. Likewise, our approach also uses a dedicated loss function that incorporates a mean squared error (MSE) component alongside Sobolev smoothness regularization. This combination enables GegenGNN to capture both the fidelity to ground truth and the underlying smoothness properties of the signals, enhancing the reconstruction performance. We conduct extensive experiments on real datasets to evaluate the effectiveness of our proposed approach. The experimental results demonstrate that GegenGNN outperforms state-of-the-art methods, showcasing its superior capability in recovering time-varying graph signals.

Graph Moving Object Segmentation.

Moving Object Segmentation (MOS) is a fundamental task in computer vision. Due to undesirable variations in the background scene, MOS becomes very challenging for static and moving camera sequences. Several deep learning methods have been proposed for MOS with impressive performance. However, these methods show performance degradation in the presence of unseen videos; and usually, deep learning models require large amounts of data to avoid overfitting. Recently, graph learning has attracted significant attention in many computer vision applications since they provide tools to exploit the geometrical structure of data. In this work, concepts of graph signal processing are introduced for MOS. First, we propose a new algorithm that is composed of segmentation, background initialization, graph construction, unseen sampling, and a semi-supervised learning method inspired by the theory of recovery of graph signals. Second, theoretical developments are introduced, showing one bound for the sample complexity in semi-supervised learning, and two bounds for the condition number of the Sobolev norm. Our algorithm has the advantage of requiring less labeled data than deep learning methods while having competitive results on both static and moving camera videos. Our algorithm is also adapted for Video Object Segmentation (VOS) tasks and is evaluated on six publicly available datasets outperforming several state-of-the-art methods in challenging conditions.