  We develop a theory for the representation of opaque solids as volumes.
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 stochastic 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.
