3D Reconstruction with Fast Dipole Sums

We introduce a technique for the reconstruction of high-fidelity surfaces from multi-view images. Our technique uses a new point-based representation, the dipole sum, which generalizes the winding number to allow for interpolation of arbitrary per-point attributes in point clouds with noisy or outlier points. Using dipole sums allows us to represent implicit geometry and radiance fields as per-point attributes of a point cloud, which we initialize directly from structure from motion. We additionally derive Barnes-Hut fast summation schemes for accelerated forward and reverse-mode dipole sum queries. These queries facilitate the use of ray tracing to efficiently and differentiably render images with our point-based representations, and thus update their point attributes to optimize scene geometry and appearance. We evaluate this inverse rendering framework against state-of-the-art alternatives, based on ray tracing of neural representations or rasterization of Gaussian point-based representations. Our technique significantly improves reconstruction quality at equal runtimes, while also supporting more general rendering techniques such as shadow rays for direct illumination. In the supplement, we provide interactive visualizations of our results.

A theory of volumetric representations for opaque solids

We develop a theory for the representation of opaque solids as volumetric models. Starting from a stochastic representation of opaque solids as random indicator functions, we prove the conditions under which such solids can be modeled using exponential volumetric transport. We also derive expressions for the volumetric attenuation coefficient as a functional of the probability distributions of the underlying indicator functions. We generalize our theory to account for isotropic and anisotropic scattering at different parts of the solid, and for representations of opaque solids as implicit surfaces. We derive our volumetric representation from first principles, which ensures that it satisfies physical constraints such as reciprocity and reversibility. We use our theory to explain, compare, and correct previous volumetric representations, as well as propose meaningful extensions that lead to improved performance in 3D reconstruction tasks.