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.

翻译：我们提出了一种将不透明固体表示为体积的理论。从将不透明固体作为随机指示函数的随机表示出发，我们证明了此类固体能够使用指数体积传输模型进行建模的条件。我们还推导了体积衰减系数作为底层指示函数概率分布泛函的表达式。我们将理论推广，以考虑固体不同部分的各向同性和各向异性散射，以及将不透明固体表示为随机隐式曲面。我们从第一性原理推导体积表示，确保其满足互易性和可逆性等物理约束。我们利用该理论解释、比较并修正了先前的体积表示方法，同时提出了有意义的扩展，从而在三维重建任务中实现了性能提升。