Gradient-Domain Volumetric Photon Density Estimation

Supplemental material

1. Introduction

In this supplemental document, we demonstrate more results of the scenes in the main paper: "Gradient-Domain Volumetric Photon Density Estimation". The primal- and gradient-domain techniques included are:

• Volumetric photon mapping (VPM and G-VPM)
• Beam radiance estimate (BRE and G-BRE)
• Photon beam (Beam 3D and G-Beam 3D)
• Photon beam with 1D kernel (Beam 1D and G-Beam 1D)
• Uni-directional path tracing (PT and G-PT)
• Bi-directional path tracing (BDPT and G-BDPT)

By default, we employ 3D kernels in photon or beam gathering unless otherwise mentioned.

2. Code and scenes

The source code (Mistuba implementation) with instruction can be found at the following address: Github repo

Scenes (Mitsuba format) and the reference: You can download each scene individually by clicking on the scene's name below. Please note that some scenes have a Copyright. Refer to the acknowledgment section inside the paper.

3. Comparison

Equal-time comparisons are shown for the below scenes. We first show interactive 4-way comparisons between our method and other methods. Each visual comparison includes:

• L2 and L1 reconstruction images (surface-volume and volume-volume transport only)
• Viridis error map: relative MSE mapped with viridis color map. Purple indicates low value; yellow indicates high value.
• Red/green error map: positive (green) or negative (red) relative error to the reference.
• Gradient images (brightened with +3 exposure value)

After the rendering in each result page, we show the convergence plot with the relative MSE metric. You can click on the technique name to show or hide a curve in the plot.

2. Experiments

a. Convergence

Scene	Equal-time comparison	Reference	Comments
Kitchen	5 min 30 min		A complex scene with diffuse and specular light transport in homogeneous media with anisotropic phase function (g = 0.3).
Staircase	2 min 30 min		A scene with only diffuse light transport and isotropic phase function.
Bathroom	5 min 30 min		A complex scene with diffuse and specular light transport in homogeneous media with anisotropic phase function.
Spotlight	10 min 30 min		An even more challenging scene with dominant specular light transport and anisotropic phase function (g = 0.3).

b. Anisotropic Phase Function

g = 0.0	g = 0.225	g = 0.45	g = 0.675	g = 0.9
10 min	10 min	10 min	10 min	10 min

c. Dense Participating Media

Scene	Equal-time comparison	Reference	Comments
Glass	5 min		A complex scene with a glass of milk and a glass of orange juice. Manifold exploration is used to handle the dielectric interfaces.

d. Bias vs. variance trade-off

Scene	Equal-time comparison	Reference	Comments
Kitchen	5 min (VPM) 5 min (BRE) 5 min (Beam)		This presents the experiments with various kernel sizes in our technique. In general, our technique is not quite sensitive with kernel radius size unless it is set too large which causes too strong bias. Gradient-domain techniques take more time per iteration for estimate gradients to reduce variance, and so it is generally slower in reducing bias in progressive rendering.