PathSpace Differentiable Rendering: Supplemental Materials
Cheng Zhang^{1}, Bailey Miller^{2}, Kai Yan^{1}, Ioannis Gkioulekas^{2}, and Shuang Zhao^{1}
^{1}University of California, Irvine
^{2}Carnegie Mellon University
1. Validation and Evaluation
1.1. Validation
We validate our estimated derivatives by comparing to results computed using the finitedifference (FD) method in the following.
The derivatives and absolute differences are visualized using the color map below, and all visualizations in each row share the same colormap limits. The differences between the FD results and ours are due to FD bias and Monte Carlo noise.
Orig image 
Our deriv. 
FD (large spacing) 
Abs. diff 






FD (small spacing) 
Abs. diff 




Orig image 
Our deriv. 
FD (large spacing) 
Abs. diff 
FD (small spacing) 
Abs. diff 












1.2. PerComponent Derivative Images
In what follows, we provide percomponent visualizations for a few gradient images.
All 
Main 
Boundary 
Primary Boundary 












With nextevent estimation (NEE) and importance sampling (IS) introduced in Section 6 of the paper, the efficiency of MonteCarlo estimation of the boundary integral can be improved significantly.
We show equaltime comparisons of the boundary term estimated using different configurations below.
Boundary 
Boundary (IS) 
Boundary (NEE) 
Boundary (NEE + IS) 












1.3. Derivative Image Comparisons
We demonstrate the effectiveness of our material differential path integral formulation by providing two equaltime comparisons of percomponent gradient images between our technique and a stateoftheart method DTRT [Zhang et al. 2019], which is largely equivalent (for the surfaceonly case) to Redner [Li et al. 2018].

All 
Main 
Boundary 
Primary 
Our Method 




DTRT 




Our Method 




DTRT 




2. GradientBased Inverse Rendering
We show inverse rendering examples using gradients estimated with our approach as well as a few stateoftheart techniques.
The parameter RMSE plots show rootmeansquare error of the optimized parameters. This information is not used by the optimizations.
The table below summarizes the performance statistics and optimization configurations for the inverse rendering examples.
The reported Time is measured per iteration on a workstation with 8core intel i77820X CPU and Titan RTX graphics card.
Scene 
# Param. 
# Iter. 
Time (Ours) 
Time (Reparam.) 
Time (DTRT) 
Time (Redner) 
Guiding Resol. 
Branches 
1 
140 
0.5 s 
0.3 s 
5.66 s 
5.58 s 
40000 x 1 x1 
Puffer ball 
1 
160 
4.5 s 
1.5 s 
28.55 s 

100000 x 1 x 1 
Veach egg 
3 
200 
19.70 s 
153 s 
101.33 s 

5000 x 5 x 5 
Mug 
3 
180 
29.81 s 



N/A 
Ring 
100 
160 
69.25 s 



N/A 
Inverse Rendering Comparison
Left click the images below to start/pause; right click to reset the animations.
Branches
 This example contains a few twisted pipes lit by a small area source and is rendered with direct illumination only.
 We search for the rotation angle of the tree (around the vertical axis) by looking at its shadow.

All comparisons are equalsample.
Initial state 
Final state 


Target rotation = 0.2 radian:
Initial state 
Final state 


Target rotation = 0.6 radian:
Puffer ball
 This example contains a highlydetailed pufferball mesh with over one million faces illuminated by four area lights of red, green and blue colors, creating the colored shadows on the ground.
 For each light, we use a single parameter to control its size and intensity such that the total power remains constant.

All comparisons are equalsample.
Initial state 
Final state 


Veach Egg
 Modeled after a scene created by Eric Veach.
 We search for (i) the refractive index of the glass egg, and (ii) the position of emitter attached to the side wall.
 All comparisons equaltime.
Initial state 
Final state 


Glass Mug
 This example contains a glass mug with a small area light inside.
 We search for (i) the rotation of the glass mug (along a fixed axis), (ii) the vertical placement of the emitter, and (iii) the roughness of the glass mug.
 All comparisons are equaltime.
Initial state 
Final state 


Generic Inverse Rendering
Ring Caustics
 This example contains a silver ring illuminated by four area lights with different colors.
 We optimize the crosssectional shape of this ring parameterized using 100 variables.
Initial state 
Final state 

