Objects as volumes: A stochastic geometry view of opaque solids
This work addresses the problem of accurately modeling opaque solids in 3D reconstruction for computer vision and graphics researchers, though it appears incremental as it builds on existing volumetric representations.
The paper develops a stochastic geometry theory for representing opaque solids as volumes, deriving conditions for exponential volumetric transport and expressions for attenuation coefficients from probability distributions of indicator functions. It generalizes the theory to handle isotropic/anisotropic scattering and stochastic implicit surfaces, and applies it to explain, compare, and correct previous volumetric representations while proposing extensions that improve 3D reconstruction performance.
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.