# From the Rendering Equation to Stratified Light Transport Inversion

### Tian-Tsong Ng, Ramanpreet Singh Pahwa, Jiamin Bai, Kar-Han Tan and Ravi Ramamoorthi

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2010-151

December 9, 2010

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2010/EECS-2010-151.pdf

Recent advances in fast light transport acquisition have motivated new applications for forward and inverse light transport. While forward light transport enables image relighting, inverse light transport provides new possibilities for analyzing and cancelling interreflections, to enable applications like projector radiometric compensation and light bounce separation. With known scene geometry and diffuse reflectance, inverse light transport can be easily derived in closed form. However, with unknown scene geometry and reflectance properties, we must acquire and invert the scene’s light transport matrix to undo the effects of global illumination. For many photometric setups such as that of a projector-camera system, the light transport matrix often has a size of 105 × 105 or larger. Conventional inversion techniques, such as the pseudo-inverse, are accurate but impractical computationally at these resolutions.

In this work, we explore a theoretical analysis of inverse light transport, relating it to its forward counterpart, expressed in the form of the rendering equation. It is well known that forward light transport has a Neumann series that corresponds to adding bounces of light. In this paper, we show the existence of a similar inverse series, that zeroes out the corresponding physical bounces of light. We refer to this series solution as stratified light transport inversion, since truncating to a certain number of terms corresponds to cancelling the corresponding interreflection bounces. The framework of stratified inversion is general and may provide insight for other problems in light transport and beyond, that involve large-size matrix inversion. It is also efficient, requiring only sparse matrix-matrix multiplications. Our practical application is to radiometric compensation, where we seek to project patterns onto real-world surfaces, undoing the effects of global illumination. We use stratified light transport inversion to efficiently invert the acquired light transport matrix for a static scene, after which inter-reflection cancellation is a simple matrix-vector multiplication to compensate the input image for projection.

BibTeX citation:

@techreport{Ng:EECS-2010-151, Author = {Ng, Tian-Tsong and Pahwa, Ramanpreet Singh and Bai, Jiamin and Tan, Kar-Han and Ramamoorthi, Ravi}, Title = {From the Rendering Equation to Stratified Light Transport Inversion}, Institution = {EECS Department, University of California, Berkeley}, Year = {2010}, Month = {Dec}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2010/EECS-2010-151.html}, Number = {UCB/EECS-2010-151}, Abstract = {Recent advances in fast light transport acquisition have motivated new applications for forward and inverse light transport. While forward light transport enables image relighting, inverse light transport provides new possibilities for analyzing and cancelling interreflections, to enable applications like projector radiometric compensation and light bounce separation. With known scene geometry and diffuse reflectance, inverse light transport can be easily derived in closed form. However, with unknown scene geometry and reflectance properties, we must acquire and invert the scene’s light transport matrix to undo the effects of global illumination. For many photometric setups such as that of a projector-camera system, the light transport matrix often has a size of 105 × 105 or larger. Conventional inversion techniques, such as the pseudo-inverse, are accurate but impractical computationally at these resolutions. In this work, we explore a theoretical analysis of inverse light transport, relating it to its forward counterpart, expressed in the form of the rendering equation. It is well known that forward light transport has a Neumann series that corresponds to adding bounces of light. In this paper, we show the existence of a similar inverse series, that zeroes out the corresponding physical bounces of light. We refer to this series solution as stratified light transport inversion, since truncating to a certain number of terms corresponds to cancelling the corresponding interreflection bounces. The framework of stratified inversion is general and may provide insight for other problems in light transport and beyond, that involve large-size matrix inversion. It is also efficient, requiring only sparse matrix-matrix multiplications. Our practical application is to radiometric compensation, where we seek to project patterns onto real-world surfaces, undoing the effects of global illumination. We use stratified light transport inversion to efficiently invert the acquired light transport matrix for a static scene, after which inter-reflection cancellation is a simple matrix-vector multiplication to compensate the input image for projection.} }

EndNote citation:

%0 Report %A Ng, Tian-Tsong %A Pahwa, Ramanpreet Singh %A Bai, Jiamin %A Tan, Kar-Han %A Ramamoorthi, Ravi %T From the Rendering Equation to Stratified Light Transport Inversion %I EECS Department, University of California, Berkeley %D 2010 %8 December 9 %@ UCB/EECS-2010-151 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2010/EECS-2010-151.html %F Ng:EECS-2010-151