Probabilistic Directed Distance Fields for Ray-Based Shape
  Representations
Tristan Aumentado-Armstrong Stavros Tsogkas Sven Dickinson Allan Jepson
In modern computer vision, the optimal representation of 3D shape continues
to be task-dependent. One fundamental operation applied to such representations
is differentiable rendering, as it enables inverse graphics approaches in
learning frameworks. Standard explicit shape representations (voxels, point
clouds, or meshes) are often easily rendered, but can suffer from limited
geometric fidelity, among other issues. 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 devise Directed Distance Fields (DDFs), a novel
neural shape representation that builds upon classical distance fields. The
fundamental operation in a DDF maps an oriented point (position and direction)
to surface visibility and depth. This enables efficient differentiable
rendering, obtaining depth with a single forward pass per pixel, as well as
differential geometric quantity extraction (e.g., surface normals), with only
additional backward passes. Using probabilistic DDFs (PDDFs), we show how to
model inherent discontinuities in the underlying field. We then apply DDFs to
several applications, including single-shape fitting, generative modelling, and
single-image 3D reconstruction, showcasing strong performance with simple
architectural components via the versatility of our representation. Finally,
since the dimensionality of DDFs permits view-dependent geometric artifacts, we
conduct a theoretical investigation of the constraints necessary for view
consistency. We find a small set of field properties that are sufficient to
guarantee a DDF is consistent, without knowing, for instance, which shape the
field is expressing.