# A Differential Theory of Radiative Transfer

## The Need for Differentiable Rendering

Physics-based differentiable rendering is the task of estimating the derivatives of radiometric measures with respect to scene parameters. The ability to compute these derivatives is necessary for enabling gradient-based optimization in a diverse array of applications: from solving analysis-by-synthesis problems to training machine learning pipelines incorporating forward rendering processes. Unfortunately, physics-based differentiable rendering remains challenging, due to the complex and typically nonlinear relation between pixel intensities and scene parameters.

#### Differentiable

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We introduced a differential theory of radiative transfer by showing how the radiative transfer equation (RTE) can be differentiated with respect to arbitrary scene parameterizations. To this end, we derived the derivatives for individual terms of the RTE given by the transport, collision, and interfacial scattering operators.

#### Generality

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Our theory enjoys the generality to handle a large variety of radiative transfer phenomena including volumetric absorption, single and multiple scattering, anisotropic phase functions, and heterogeneity.

#### Monte Carlo Rendering

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To numerically estimate the derivatives, we presented an unbiased Monte Carlo method analogous to volumetric path tracing (VPT) to compute scene derivatives of interior and interfacial radiances.

## A Differential Theory for Radiative Transfer Equation

We introduce a differential theory of radiative transfer, which shows how individual components of the radiative transfer equation (RTE) can be differentiated with respect to arbitrary differentiable changes of a scene. Our theory encompasses the same generality as the standard RTE, allowing differentiation while accurately handling a large range of light transport phenomena such as volumetric absorption and scattering, anisotropic phase functions, and heterogeneity.

## People

#### CHENG ZHANG

#### LIFAN WU

#### CHANGXI ZHENG

#### Ioannis Gkioulekas

#### RAVI RAMAMOORTHI

#### SHUANG ZHAO

**Project Site**