Path-Space Differentiable Rendering: Supplemental Materials

Cheng Zhang¹, Bailey Miller², Kai Yan¹, Ioannis Gkioulekas², and Shuang Zhao^{1 1}University of California, Irvine          ²Carnegie Mellon University

We validate our estimated derivatives by comparing to results computed using the finite-difference (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 color-map 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

In what follows, we provide per-component visualizations for a few gradient images.

All	Main	Boundary	Primary Boundary

With next-event estimation (NEE) and importance sampling (IS) introduced in Section 6 of the paper, the efficiency of Monte-Carlo estimation of the boundary integral can be improved significantly. We show equal-time comparisons of the boundary term estimated using different configurations below.

Boundary	Boundary (IS)	Boundary (NEE)	Boundary (NEE + IS)

We demonstrate the effectiveness of our material differential path integral formulation by providing two equal-time comparisons of per-component gradient images between our technique and a state-of-the-art method DTRT [Zhang et al. 2019], which is largely equivalent (for the surface-only case) to Redner [Li et al. 2018].

	All	Main	Boundary	Primary
Our Method
DTRT
Our Method
DTRT

We show inverse rendering examples using gradients estimated with our approach as well as a few state-of-the-art techniques. The parameter RMSE plots show root-mean-square 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 8-core intel i7-7820X 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

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 equal-sample.

Initial state	Final state

Target rotation = 0.2 radian:

Initial state	Final state

Target rotation = 0.6 radian:

Puffer ball

• This example contains a highly-detailed 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 equal-sample.

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 equal-time.

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 equal-time.

Initial state	Final state

Ring Caustics

• This example contains a silver ring illuminated by four area lights with different colors.
• We optimize the cross-sectional shape of this ring parameterized using 100 variables.

Initial state	Final state