Most existing geometry processing algorithms use meshes as the default shape representation. Manipulating meshes, however, requires one to maintain high quality in the surface discretization. For example, changing the topology of a mesh usually requires additional procedures such as remeshing. This paper instead proposes the use of neural fields for geometry processing. Neural fields can compactly store complicated shapes without spatial discretization. Moreover, neural fields are infinitely differentiable, which allows them to be optimized for objectives that involve higher-order derivatives.

The Adam optimization algorithm has proven remarkably effective for optimization problems across machine learning and even traditional tasks in geometry processing. At the same time, the development of equivariant methods, which preserve their output under the action of rotation or some other transformation, has proven to be important for geometry problems across these domains.

Most existing geometry processing algorithms use meshes as the default shape representation. Manipulating meshes, however, requires one to maintain high quality in the surface discretization. For example, changing the topology of a mesh usually requires additional procedures such as remeshing. This paper instead proposes the use of neural fields for geometry processing. Neural fields can compactly store complicated shapes without spatial discretization. Moreover, neural fields are infinitely differentiable, which allows them to be optimized for objectives that involve higher-order derivatives.

The Adam optimization algorithm has proven remarkably effective for optimization problems across machine learning and even traditional tasks in geometry processing. At the same time, the development of equivariant methods, which preserve their output under the action of rotation or some other transformation, has proven to be important for geometry problems across these domains. In this work, we observe that Adam $-$ when treated as a function that maps initial conditions to optimized results $-$ is not rotation equivariant for vector-valued parameters due to per-coordinate moment updates. This leads to significant artifacts and biases in practice. We propose to resolve this deficiency with VectorAdam, a simple modification which makes Adam rotation-equivariant by accounting for the vector structure of optimization variables. We demonstrate this approach on problems in machine learning and traditional geometric optimization, showing that equivariant VectorAdam resolves the artifacts and biases of traditional Adam when applied to vector-valued data, with equivalent or even improved rates of convergence.

Abstract: "Geometric data abounds, but our algorithms for geometry processing are failing. Whether from medical imagery, free-form architecture, self-driving cars, or 3D-printed parts, geometric data is often messy, riddled with "defects" that cause algorithms to crash or behave unpredictably. The traditional philosophy assumes geometry is given with 100% certainty and that algorithms can use whatever discretization is most convenient. As a result, geometric pipelines are leaky patchworks requiring esoteric training and involving many different people. Instead, we adapt fundamental mathematics to work directly on messy geometric data. As an archetypical example, I will discuss how to generalize the classic formula for determining the inside from the outside of a curve to messy representations of a 3D surface. This work helps keep the geometry processing pipeline flowing, as validated on our large-scale geometry benchmarks. Our long term vision is to replace the current leaky geometry processing pipeline with a robust workflow where processing operates directly on real geometric data found "in the wild". To do this, we need to rethink how algorithms should gracefully degrade when confronted with imprecision and uncertainty. Our most recent work on differentiable rendering and geometry-based adversarial attacks on image classification demonstrates the potential power of this philosophy."