  Differentiable rendering is an essential operation in modern vision, allowing
inverse graphics approaches to 3D understanding to be utilized in modern
machine learning frameworks. Explicit shape representations (voxels, point
clouds, or meshes), while relatively easily rendered, often suffer from limited
geometric fidelity or topological constraints. On the other hand, implicit
representations (occupancy, distance, or radiance fields) preserve greater
fidelity, but suffer from complex or inefficient rendering processes, limiting
scalability. In this work, we endeavour to address both shortcomings with a
novel shape representation that allows fast differentiable rendering within an
implicit architecture. Building on implicit distance representations, we define
Directed Distance Fields (DDFs), which map an oriented point (position and
direction) to surface visibility and depth. Such a field can render a depth map
with a single forward pass per pixel, enable differential surface geometry
extraction (e.g., surface normals and curvatures) via network derivatives, be
easily composed, and permit extraction of classical unsigned distance fields.
Using probabilistic DDFs (PDDFs), we show how to model inherent discontinuities
in the underlying field. Finally, we apply our method to fitting single shapes,
unpaired 3D-aware generative image modelling, and single-image 3D
reconstruction tasks, showcasing strong performance with simple architectural
components via the versatility of our representation.
