A Differential Theory of Radiative Transfer:
Supplemental Materials

Validation

We validate our estimated derivatives by comparing to finite-difference (FD) results. The derivatives and absolute differences are visualized using the following colormap:

All visualizations in each row share the same color-map limits. The differences between the FD results and ours are due to FD bias and Monte Carlo noise.

Orig image Ours 1 FD 1 Abs. diff Ours 2 FD 2 Abs. diff
Gradient-Based Inverse Rendering

Here we show inverse rendering examples using gradients estimated with our approach.

The images under different viewing configurations (indicated with "diff. view") are not used for gradient-based optimization but to visualize the iterative refinement of scene parameters driven by the optimization.

The table below summarizes the performance statistics and optimization configurations for the inverse rendering examples. Time is measured in CPU core minutes per iteration.

Scene # Param. # Iter. Time
Disco Ball 2 110 22
Glass 1 4 80 12.2
Camera pose 3 220 9.3
Glossy Reflector 9 200 23.6
Shape blending 1 1 100 3.0
Shape blending 2 1 50 6.4
Shadow 1 3 100 9
Shadow 2 2 60 7.6
Multilayer 3 50 31
Spotlight 1 12 200 11.2
Spotlight 2 12 110 11.2
Logo 101 100 27.2
Point light source 3 150 0.16
High-order geometry 3 150 13.2
Smooth dielectric IOR 1 50 0.96
Left click the images below to start/pause; right click to reset the animations.
Generic Inverse Rendering

Disco Ball

Glass 1

Camera pose

Glossy reflector

Spotlight 1

Shape blending 1

Shape blending 2

Imaging-Inspired Examples

Shadow 1

Shadow 2

Multilayer

Design-Inspired Examples

Spotlight 2

Logo

Generalized Theory

We now show proof-of-concept examples generated with generalized versions of our derivation from the supplemental document.

All the result images are grayscale with pixel intensities encoded in false colors using the following color map:

clrmap

Point light source

High-order geometry

Smooth dielectric IOR