NeRF, meet differential geometry!

Neural radiance fields, or NeRF, represent a breakthrough in the field of novel view synthesis and 3D modeling of complex scenes from multi-view image collections. Numerous recent works have been focusing on making the models more robust, by means of regularization, so as to be able to train with possibly inconsistent and/or very sparse data. In this work, we scratch the surface of how differential geometry can provide regularization tools for robustly training NeRF-like models, which are modified so as to represent continuous and infinitely differentiable functions. In particular, we show how these tools yield a direct mathematical formalism of previously proposed NeRF variants aimed at improving the performance in challenging conditions (i.e. RegNeRF). Based on this, we show how the same formalism can be used to natively encourage the regularity of surfaces (by means of Gaussian and Mean Curvatures) making it possible, for example, to learn surfaces from a very limited number of views.