Neural Fields as Distributions: Signal Processing Beyond Euclidean Space

Neural fields have emerged as a powerful and broadly applicable method for representing signals. However, in contrast to classical discrete digital signal processing, the portfolio of tools to process such representations is still severely limited and restricted to Euclidean domains. In this paper, we address this problem by showing how a probabilistic re-interpretation of neural fields can enable their training and inference processes to become "filter-aware". The formulation we propose not only merges training and filtering in an efficient way, but also generalizes beyond the familiar Euclidean coordinate spaces to the more general set of smooth manifolds and convolutions induced by the actions of Lie groups. We demonstrate how this framework can enable novel integrations of signal processing techniques for neural field applications on both Euclidean domains, such as images and audio, as well as non-Euclidean domains, such as rotations and rays. A noteworthy benefit of our method is its applicability. Our method can be summarized as primarily a modification of the loss function, and in most cases does not require changes to the network architecture or the inference process.