We introduce Explicit Neural Surfaces (ENS), an efficient surface reconstruction method that learns an explicitly defined continuous surface from multiple views. We use a series of neural deformation fields to progressively transform a continuous input surface to a target shape. By sampling meshes as discrete surface proxies, we train the deformation fields through efficient differentiable rasterization, and attain a mesh-independent and smooth surface representation. By using Laplace-Beltrami eigenfunctions as an intrinsic positional encoding alongside standard extrinsic Fourier features, our approach can capture fine surface details. ENS trains 1 to 2 orders of magnitude faster and can extract meshes of higher quality compared to implicit representations, whilst maintaining competitive surface reconstruction performance and real-time capabilities. Finally, we apply our approach to learn a collection of objects in a single model, and achieve disentangled interpolations between different shapes, their surface details, and textures.

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