We develop a theory for the representation of opaque solids as volumetric models. 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 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：我们发展了一套将不透明固体表示为体素模型的理论。从将不透明固体表示为随机指示函数的随机模型出发，我们证明了此类固体可使用指数体传输建模的条件。我们还推导了体衰减系数作为底层指示函数概率分布泛函的表达式。我们将理论推广至考虑固体不同部位的各向同性与各向异性散射，以及不透明固体作为隐式曲面的表示。我们从第一性原理推导体素表示，确保其满足互易性和可逆性等物理约束。利用该理论，我们解释、比较并修正了先前的体素表示，同时提出有意义的扩展，从而在三维重建任务中实现性能提升。