A Practical Model for Computing Subsurface BRDF of Homogeneous Materials with A Thin Layer of Paint

Ke Chen Charly Collin Ajit Hakke-Patil Sumanta Pattanaik

Overview

Introduction and motivation

BRDF = surface BRDF + subsurface BRDF [Hanrahan and Krueger 1993]

Subsurface BRDFs are directionally dependent

Shape does not change

Shape changed

Previous Works and Limitations

• Kubelka-Munk [Kubelka 1954]• Single scattering approximation [Blinn 1982],

[Farrell et al. 1992]• Adding and doubling [de Haan 1987]• Discrete Ordinate Methods[Chandrasekhar

1960]

Contributions

• Subsurface BRDF for semi-infinite homogeneous materials

based on Ambartsumian’s integral equation [Chen et al. 2013]

• Adding a thin layer of paint based on invariant imbedding method [Hansen and Travis

1974]

Radiative transfer

Plane-parallel radiative transfer equation:

),,(),,(),,(

SLd

dL

[Chandrasekhar 1960]

Ambartsumian’s integral equation),()(),( 010 ii pAR

'''0

1

02 ),(),()( dRpA i

''0

'1

03 ),(),()( dRpA i

'''''''1

0

1

0

''04 ),(),(),()( dRpdRA i

Iteratively solving each ),( 0 iR

Subsurface BRDFs are directionally dependent

Titanium dioxide Aluminium oxide

Invariant imbedding

''0

'1

0

),(),(2

dRp m

im

i

)1)(,()1(),( 00

0modi

im

im

ified RR

),(4 0

0i

m

i

p

'''0

1

00

),(),(2

dRp i

mm

'''''''1

0

1

0

''0 ),(),(),( dRpdR i

mmm

(2)

(4)

(3)

(5)

(1)

Results

Conclusions• Subsurface BRDFs are directionally dependent• Using Ambartsumian’s integral equation and invariant

imbedding to compute the subsurface BRDF is fast.• Accurate

Future work

• Polarization• Multiple layered materials

Thank You

Questions?