High-Order Similarity Relations in Radiative Transfer Shuang Zhao 1, Ravi Ramamoorthi 2, and Kavita Bala 1 1 Cornell University 2 University of California, San Diego

Translucency is everywhere Food Skin Jewelry Architecture Slide courtesy of Ioannis Gkioulekas

Rendering translucency Radiative transfer Scattering param. Appearance

Rendering translucency Radiative transfer Scattering param. 2 Appearance 2 Radiative transfer Scattering param. 1 Appearance 1 Radiative transfer Scattering param. 1 Appearance 1 Radiative transfer Scattering param. 2 Appearance 2 ≈ ≠

First-order methods Scattering param. 1 Scattering param. 2 Scattering param. 1 Scattering param. 2 First-order approx. Approx. identical appearance Cheaper to render Limited accuracy [Frisvad et al. 2007] [Arbree et al. 2011][Wang et al. 2009] [Jensen et al. 2001]

Similarity theory Scattering param. 1 Scattering param. 2 First-order approx. First-order methods Scattering param. 1 Scattering param. 2 First-order approx. Similarity theory [Wyman et al. 1989] Scattering param. 1 Scattering param. 2 Similarity relations

Similarity theory [Wyman et al. 1989] Scattering param. 1 Scattering param. 2 Similarity relations Provide fundamental insights into the structure of material parameter space

Similarity theory [Wyman et al. 1989] Scattering param. 1 Scattering param. 2 Similarity relations Originates in applied optics [Wyman et al. 1989] Similar ideas explored in neutron transfer (Condensed History Monte Carlo) [Prinja & Franke 2005], [Bhan & Spanier 2007], …

Our contribution Introducing high-order similarity theory to computer graphics Novel algorithms benefiting forward & inverse rendering

Our contribution: forward rendering Better accuracy Our approach User-specified (balancing performance and accuracy) Approx. identical appearance Cheaper to render Scattering param ~ 200 lines of MATLAB code Scattering param. 1 Up to 10X speedup

Our contribution: inverse rendering Parameter space 1 Reparameterize Parameter space 2 Gradient descent methods perform much better

Background

Material scattering parameters Extinction coefficient Scattering coefficient Phase function Light particle Absorption coefficient Absorbed Scattered Interaction

Phase function Scattered Probability density for, parameterized as Isotropic scattering Forward Forward scattering Forward

Similarity Theory

n th Legendre moment Phase function moments Legendre polynomial For a phase function “Average cosine”

Similarity relations Low-frequency radiance Band-limited up to order-N in spherical harmonics domain … [Wyman et al. 1989]

Order-N similarity relation [Wyman et al. 1989] Similarity relations … identical appearance … Derivation in the paper Radiance low-frequency everywhere

Order-N similarity relation Similarity relations … Higher order, Better accuracy Approximately identical appearance Radiance low-frequency everywhere

Challenge Order-N similarity relation … … Original (given) Altered (unknown) ? ?

Solving for Altered Parameters

The problem Altered parameters ? … ?? Order-N similarity relation Constraints Forward Original parameters

The problem Altered parameters ? ? … Order-N similarity relation … … Forward Original parameters

The problem Altered parameters ? ? … Order-N similarity relation … Forward Original parameters

Altered phase function … Altered parameters ? … ? Forward Original parameters Remaining unknown

Altered phase function Altered parameters ? Forward Original parameters Remaining unknown Legendre moments of …

Altered phase function Altered parameters ? ? Order-1 Order-2 Order-3 Order-4 … Finding highest satisfiable order N Normalization constraint

Finding order N Given desired Legendre moments (Truncated Hausdorff moment problem) [Curto and Fialkow 1991] Phase function Hankel matrices built using are positive semi-definite exists Existence condition Does phase function exist?

Finding order N Altered parameters ? Order-1 Order-2 Order-3 Order-4 … Finding highest satisfiable order N

Altered phase function Altered parameters ? Order-3 Problem: not uniquely specified InvalidValid

Constructing altered phase function … 1 0 Need: has Legendre moments non-negative Represent as a tabulated function with pieces ? … 1 0

Constructing altered phase function Need: Represent as a tabulated function with pieces ? Const.

Constructing altered phase function Solve subject to Smoothness term (favoring “uniform” solutions) 1 0 Good 1 0 Bad

Constructing altered phase function Solve subject to Quadratic programming Standard problem Solvable with many tools/libraries MATLAB, Gurobi, CVXOPT, … Our MATLAB code is available online

Constructing altered phase function Altered parameters ? Order-3 ValidInvalidValid Our approach

Forward Altered parameters Constructing altered phase function

Summary Forward Original parameters Forward Altered parameters Forward Altered parameters Compute order N Solve optimization Compute order N Solve optimization

Application: Forward Rendering

Our contribution: forward rendering Better accuracy Our approach Approx. identical appearance Cheaper to render Scattering param. 2 Scattering param. 1 Effort-free speedups! User-specified (balancing performance and accuracy)

Application: forward rendering 0 1 No change in parameters large Better accuracy Lower speedup small Worse accuracy Greater speedup Perform test renderings to find optimal Reuse for high-resolution renderings or videos is a good start

Experimental Results

Performance vs. accuracy α = 0.05 (44 min, 8.0X) Relative error 0% 30% Reference (350 min)

Performance vs. accuracy Reference (350 min) α = 0.05 (44 min, 8.0X) Relative error α = 0.10 (63 min, 5.6X) Relative error 0% 30% 0% 30%

Performance vs. accuracy α = 0.20 (103 min, 3.4X) Relative error α = 0.30 (126 min, 2.8X) Relative error 0% 30% α = 0.10 (63 min, 5.6X) Relative error α = 0.10 (63 min, 5.6X) Relative error Visually identical

Power of high-order relations Used by first-order methods: Altered parameters (Order-1) Forward Original parameters Reduced scattering coefficient Satisfies order-1 similarity relation

Power of high-order relations Altered parameters (Order-3) Forward Original parameters Altered parameters (Order-1) Forward

Power of high-order relations Altered parameters (Order-3) Original parameters Altered parameters (Order-1) Original parameters Altered parameters (Order-1) Altered parameters (Order-3) 426 min (reference)119 min (3.6X)115 min (3.7X)

More renderings Reference 473 min Ours 178 min (2.7X) Reference 23 min Ours 20 min Equal-timeEqual-sample

Conclusion Order-N similarity relation … Introducing high-order similarity relations to graphics Proposing a practical algorithm to solve for altered parameters ? OriginalAltered

Picking automatically and adaptively Alternative versions of similarity theory Future work

Thank you! High-Order Similarity Relations in Radiative Transfer Shuang Zhao 1, Ravi Ramamoorthi 2, Kavita Bala 1 1 Cornell University, 2 University of California, San Diego Project website: (MATLAB code available!) Project website: (MATLAB code available!) Funding: NSF IIS grants , , Intel Science and Technology Center – Visual Computing Reference Ours (3.7X)

Extra Slides

Order-1 similarity relation Reduced scattering coefficient Special case (used by diffusion methods): Order-N similarity relation …

Prior work: solving for altered parameters [Wyman et al. 1989] fixed such that given by the user Discrete scattering angle [Prinja & Franke 2005] Represent as the sum of delta functions “Spiky” phase functions do not perform as well as “uniform” ones for rendering applications

Constructing altered phase function Represent as a tabulated function with pieces Quadratic programming Solve subject to Hankel matrices built using being positive semi-definite Existence condition

Performance vs. accuracy Reference (350 min)

Discarded Slides