# A Dual Theory of Inverse and Forward Light Transport

### Jiamin Bai, Manmohan Chandraker, Tian-Tsong Ng and Ravi Ramamoorthi

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2010-101

June 29, 2010

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2010/EECS-2010-101.pdf

A cornerstone of computer graphics is the solution of the rendering equation for interreflections, which allows the simulation of global illumination, given direct lighting or corresponding light source emissions. This paper lays the foundations for the inverse problem, whereby a dual theoretical framework is presented for inverting the rendering equation to undo interreflections in a real scene, thereby obtaining the direct lighting. Inverse light transport is of growing importance, enabling a variety of new applications like separation of individual bounces of the light transport, and projector radiometric compensation to display images free of global illumination artifacts in real-world environments that exhibit complex geometric and reflectance properties. However, solving the inverse problem involves the inversion of a large light transport matrix (acquired by measurement on real scenes). While straightforward matrix inversion is intractable for most realistic resolutions, there is scant prior work on either theoretical foundations or fast computational algorithms to meet the objectives of inverse light transport. In this paper, we develop a mathematical theory that exposes the duality of forward and inverse light transport. Forward rendering also formally involves a matrix or operator inversion, which is conceptually equivalent to a multi-bounce Neumann series expansion. We show the existence of an analogous series for the inverse problem. However, the convergence is oscillatory in the inverse case, with more interesting conditions on material reflectance. Importantly, we give physical meaning to this duality, by showing that each term of our inverse series cancels an interreflection bounce, just as the forward series adds them. In algorithmic terms, we develop the analog of iterative finite element methods like forward radiosity to efficiently solve light transport inversion. Our iterative inverse light transport algorithm is very fast, requiring only matrix-vector multiplications, and follows directly from the dual theoretical formulation. We also explore the connections to forward rendering in terms of Monte Carlo and wavelet-based techniques. As an initial practical application, we first acquire the light transport of a real static scene, and then demonstrate iterative inversion for radiometric compensation on high-resolution datasets, as well as rapid separation of the bounces of global illumination.

BibTeX citation:

@techreport{Bai:EECS-2010-101, Author = {Bai, Jiamin and Chandraker, Manmohan and Ng, Tian-Tsong and Ramamoorthi, Ravi}, Title = {A Dual Theory of Inverse and Forward Light Transport}, Institution = {EECS Department, University of California, Berkeley}, Year = {2010}, Month = {Jun}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2010/EECS-2010-101.html}, Number = {UCB/EECS-2010-101}, Abstract = {A cornerstone of computer graphics is the solution of the rendering equation for interreflections, which allows the simulation of global illumination, given direct lighting or corresponding light source emissions. This paper lays the foundations for the inverse problem, whereby a dual theoretical framework is presented for inverting the rendering equation to undo interreflections in a real scene, thereby obtaining the direct lighting. Inverse light transport is of growing importance, enabling a variety of new applications like separation of individual bounces of the light transport, and projector radiometric compensation to display images free of global illumination artifacts in real-world environments that exhibit complex geometric and reflectance properties. However, solving the inverse problem involves the inversion of a large light transport matrix (acquired by measurement on real scenes). While straightforward matrix inversion is intractable for most realistic resolutions, there is scant prior work on either theoretical foundations or fast computational algorithms to meet the objectives of inverse light transport. In this paper, we develop a mathematical theory that exposes the duality of forward and inverse light transport. Forward rendering also formally involves a matrix or operator inversion, which is conceptually equivalent to a multi-bounce Neumann series expansion. We show the existence of an analogous series for the inverse problem. However, the convergence is oscillatory in the inverse case, with more interesting conditions on material reflectance. Importantly, we give physical meaning to this duality, by showing that each term of our inverse series cancels an interreflection bounce, just as the forward series adds them. In algorithmic terms, we develop the analog of iterative finite element methods like forward radiosity to efficiently solve light transport inversion. Our iterative inverse light transport algorithm is very fast, requiring only matrix-vector multiplications, and follows directly from the dual theoretical formulation. We also explore the connections to forward rendering in terms of Monte Carlo and wavelet-based techniques. As an initial practical application, we first acquire the light transport of a real static scene, and then demonstrate iterative inversion for radiometric compensation on high-resolution datasets, as well as rapid separation of the bounces of global illumination.} }

EndNote citation:

%0 Report %A Bai, Jiamin %A Chandraker, Manmohan %A Ng, Tian-Tsong %A Ramamoorthi, Ravi %T A Dual Theory of Inverse and Forward Light Transport %I EECS Department, University of California, Berkeley %D 2010 %8 June 29 %@ UCB/EECS-2010-101 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2010/EECS-2010-101.html %F Bai:EECS-2010-101