Multivariate neural directional total variation

Multi-dimensional data (e.g., videos and spatial transcriptomics) often exhibit multi-directional local smoothness, a property that is challenging to model accurately. While classical multivariate total variation (TV) methods aim to capture such multi-directional regularity, they tend to introduce discretization errors due to discrete differences or interpolation. In this work, we propose a concise yet effective multivariate neural directional TV framework (termed N-NeurDTV). Specifically, we utilize a tensor function representation parameterized by deep neural networks (DNNs) to continuously represent an N-dimensional signal. The N-NeurDTV leverages multivariate directional derivatives of the DNN outputs w.r.t. input coordinates to capture multi-directional smoothness. By leveraging the subspace orthogonal decomposition of R^N, the framework naturally generalizes to multivariate higher-order and space-variant N-NeurDTV, where the dominant direction of N-NeurDTV is adaptively refined during inference. The multivariate N-NeurDTV eliminates the need for discretization and alleviates discretization errors. We demonstrate the effectiveness of N-NeurDTV through theoretical analyses and extensive experimental validations across various data recovery tasks, including multi-dimensional video and magnetic resonance image inpainting, multispectral image denoising, and multi-slice spatial transcriptomics reconstruction, with N-NeurDTV consistently outperforming state-of-the-art methods.