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.

我们开发了一种关于不透明固体的体积模型表示的理论。我们从不透明固体的随机指示函数表示出发，证明了使用指数体积传输建模这种固体的条件。我们还推导了体积衰减系数的表达式，作为底层指示函数的概率分布的函数。我们将我们的理论推广到在固体的不同部分考虑各向同性和各向异性散射以及将不透明固体表示为隐式曲面。我们从第一原理派生了我们的体积表示，以确保它满足诸如互易性和可逆性等物理约束。我们使用我们的理论来解释、比较和改正以前的体积表示，并提出有意义的扩展，以提高三维重建任务的性能。