Green's function solution to subsurface light transport for BRDF computation

Charly Collin – Ke Chen – Ajit Hakke-PatilSumanta Pattanaik – Kadi Bouatouch

Painted materials:

Painted materials:

Painted materials:

Painted materials:

Our goal:

𝑓𝑟 (𝜔𝑖𝑛 ,𝜔𝑜𝑢𝑡 )=𝐼 (𝜔𝑜𝑢𝑡)𝐸 (𝜔 𝑖𝑛)

• Base layer• Binder thickness• Particle type and distribution

Compute the diffuse BRDF from physical properties:

BRDF Computation

Several methods exist to compute the diffuse component:

• Approximate methods:– Kubelka-Munk– Dipole model

Lam bertian model Real-world material

Several methods exist to compute the diffuse component:

• Approximate methods:– Kubelka-Munk– Dipole model

• Accurate methods:– Photon mapping– Monte Carlo– Adding-Doubling Method– Discrete Ordinate Method

BRDF Computation

Stochastic methods

Deterministic methods

BRDF Computation

𝐼 (𝜔𝑜𝑢𝑡) 𝐼 (𝜔𝑜𝑢𝑡)

Our computation makes several assumptions on the material:• Plane parallel medium

BRDF Computation

Our computation makes several assumptions on the material:• Plane parallel medium• Randomly oriented particles

𝑓𝑟 (𝜇𝑖𝑛 ,𝜑𝑖𝑛 ,𝜇𝑜𝑢𝑡 ,𝜑𝑜𝑢𝑡 )= 𝑓𝑟 (𝜇𝑖𝑛 ,𝜇𝑜𝑢𝑡 ,𝜑𝑖𝑛−𝜑 𝑜𝑢𝑡 )

𝜔=(𝜇 ,𝜑 )

BRDF Computation

Our computation makes several assumptions on the material:• Plane parallel medium• Randomly oriented particles• Homogeneous layers

𝐼 (𝜔𝑜𝑢𝑡) 𝐼 (𝜔𝑜𝑢𝑡)

BRDF Computation

BRDF computation requires computing the radiance field at the top of the material

The radiance field is modeled as a solution to the Radiative Transfer Equation

𝐼 𝑖𝑛𝑐

𝐼 (𝜔𝑜𝑢𝑡)

Radiative Transfer Equation

It has 3 components:• the radiance• corresponding to the light scattering inside the material

RTE expresses the change of radiance along optical depth .

𝐽 (𝜏 ,𝜔 )= 𝛼4 𝜋∫

− 1

1

𝑝 (𝜏 ,𝜔 ,𝜔 ′ ) 𝐼 (𝜏 ,𝜔 ′ )𝑑𝜔 ′

𝜏

𝐽 (𝜏 ,𝜔 )𝐼 (𝜏 ,𝜔 ′ )

𝜇𝜕𝜕𝜏

𝐼 (𝜏 ,𝜔 )=− 𝐼 (𝜏 ,𝜔 )+ 𝐽 (𝜏 ,𝜔 )+𝑄 (𝜏 ,𝜔 ,𝜔 𝑖𝑛)

Radiative Transfer Equation

It has 3 components:• the radiance• corresponding to the light scattering inside the material• accounting for attenuated incident radiance

RTE expresses the change of radiance along optical depth .

𝜏

𝐼 𝑖𝑛𝑐

𝜇𝜕𝜕𝜏

𝐼 (𝜏 ,𝜔 )=− 𝐼 (𝜏 ,𝜔 )+ 𝐽 (𝜏 ,𝜔 )+𝑄 (𝜏 ,𝜔 ,𝜔 𝑖𝑛)

Radiative Transfer Equation

It has 3 components:• the radiance• corresponding to the light scattering inside the material• accounting for attenuated incident radiance

RTE expresses the change of radiance along optical depth .

𝜇𝜕𝜕𝜏

𝐼 (𝜏 ,𝜔 )=− 𝐼 (𝜏 ,𝜔 )+ 𝐽 (𝜏 ,𝜔 )+𝑄 (𝜏 ,𝜔 ,𝜔 𝑖𝑛)

To compute the BRDF, RTE needs to be solved for each incident and outgoing direction.

RTE Solution

• Fourier expansion of the radiance

𝐼 (𝜏 ,𝜔 )=12𝐼 0 (𝜏 ,𝜇 )+ ∑

𝑚=1

𝐿

𝐼𝑚 (𝜏 ,𝜇) cos (𝑚𝜑 )

𝜔=(𝜇 ,𝜑 )

RTE Solution

The RTE for each expansion order can be written as:

𝜇 𝜕𝜕𝜏

𝐼𝑚 (𝜏 ,𝜇)=− 𝐼𝑚 (𝜏 ,𝜇 )+𝜔2∫− 1

1

𝑝𝑚 (𝜇 ,𝜇 ′) 𝐼𝑚 (𝜏 ,𝜇′)𝑑𝜇 ′+𝑄𝑚 (𝜏 ,𝜇 ,𝜇𝑖𝑛)

That we reorganize:

𝜇 𝜕𝜕𝜏

𝐼𝑚 (𝜏 ,𝜇) +𝐼𝑚 (𝜏 ,𝜇)−𝜔2∫−1

1

𝑝𝑚 (𝜇 ,𝜇′ ) 𝐼𝑚 (𝜏 ,𝜇′ )𝑑𝜇′=𝑄𝑚 (𝜏 ,𝜇 ,𝜇𝑖𝑛 )

Components expressed using

Components independant of

RTE Solution

We introduce an integro-differential operator :

ℒ [ 𝐼𝑚 (𝜏 ,𝜇 ) ]=𝑄𝑚(𝜏 ,𝜇 ,𝜇𝑖𝑛)

Needs to be solved for each and

𝜇 𝜕𝜕𝜏

𝐼𝑚 (𝜏 ,𝜇) +𝐼𝑚 (𝜏 ,𝜇)−𝜔2∫−1

1

𝑝𝑚 (𝜇 ,𝜇′ ) 𝐼𝑚 (𝜏 ,𝜇′ )𝑑𝜇′=𝑄𝑚 (𝜏 ,𝜇 ,𝜇𝑖𝑛 )

RTE Solution

Standard solution is the combination of the homogeneous solution...

ℒ [ 𝐼 h𝑚 (𝜏 ,𝜇𝑖 ) ]=0

... and one particular solution.

ℒ [ 𝐼𝑝𝑚 (𝜏 ,𝜇𝑖 ) ]=𝑄𝑚 (𝜏 ,𝜇 ,𝜇𝑖𝑛 )

+

RTE Solution

The homogeneous solution is independant of

The particular solution depends on

It can be solved only once for each

Its computation must be repeated for each incident direction!

How to take advantage of the similarity of the computations?

Green’s function solution

Green’s function are defined as:

ℱ [𝐺(𝑥 :𝑡)]=𝛿(𝑡−𝑥)

Property of the function:

∫𝑎

𝑏

𝛿 (𝑡−𝑥 ) 𝑑𝑡=1

ℱ [ 𝑓 (𝑥 ) ]=𝑞(𝑥 )

∫𝑎

𝑏

𝛿 (𝑡−𝑥 )𝑞(𝑡)𝑑𝑡=𝑞 (𝑥 )

For a generic differential equation:

∫𝑎

𝑏

ℱ [𝐺(𝑥 :𝑡) ]𝑞 (𝑡)𝑑𝑡=𝑞 (𝑥)

Green’s function solution

ℱ [ 𝑓 (𝑥 ) ]=𝑞(𝑥 )

ℱ [ 𝑓 𝑝 (𝑥 ) ]=𝑞(𝑥)

∫𝑎

𝑏

ℱ [𝐺(𝑥 :𝑡) ]𝑞 (𝑡)𝑑𝑡=ℱ [ 𝑓 𝑝 (𝑥 ) ]

ℱ [∫𝑎

𝑏

𝐺(𝑥 :𝑡)𝑞 (𝑡)𝑑𝑡 ]=ℱ [ 𝑓 𝑝 (𝑥 ) ]

ℱ [𝐺(𝑥 :𝑡)]=𝛿(𝑡−𝑥) ∫𝑎

𝑏

ℱ [𝐺(𝑥 :𝑡) ]𝑞 (𝑡)𝑑𝑡=𝑞 (𝑥)

Considering a particular solution :

Leading to the equality:

∫𝑎

𝑏

𝐺(𝑥 :𝑡)𝑞(𝑡 )𝑑𝑡= 𝑓 𝑝 (𝑥 )

Green’s function solution

How to compute ?

ℱ [𝐺(𝑥 :𝑡)]=𝛿(𝑡−𝑥)

For any :

ℱ [𝐺(𝑥≠ 𝑡 :𝑡)]=0 Homogeneous equation!

For , use of a jump condition:

∫𝑡−𝜀

𝑡+𝜀

ℱ [𝐺(𝑥 :𝑡)]𝑑𝑥=∫𝑡−𝜀

𝑡+𝜀

𝛿 (𝑡− 𝑥 )𝑑𝑥

𝐺 (𝑡+𝜀 :𝑡 )−𝐺 (𝑡−𝜀 :𝑡 )=1

The Green’s function can be expressed using only the homogeneous solution

Back to the RTE

ℒ [ 𝐼𝑚 (𝜏 ,𝜇 ) ]=𝑄𝑚(𝜏 ,𝜇 ,𝜇𝑖𝑛)

𝜇 𝜕𝜕𝜏

𝐼𝑚 (𝜏 ,𝜇) +𝐼𝑚 (𝜏 ,𝜇)−𝜔2∫−1

1

𝑝𝑚 (𝜇 ,𝜇′ ) 𝐼𝑚 (𝜏 ,𝜇′ )𝑑𝜇′=𝑄𝑚 (𝜏 ,𝜇 ,𝜇𝑖𝑛 )

In this case the Green’s function is defined as a 4-D function:

ℒ [𝐺𝑚 (𝜏 ,𝜇 :𝑡 ,𝜇 ′ ) ]=𝛿(𝑡−𝜏)𝛿(𝜇′−𝜇)

And our particular solution can be expressed as:

∫0

𝜏𝑚𝑎𝑥

∫− 1

1

𝐺𝑚 (𝜏 ,𝜇 :𝑡 ,𝜇′ )𝑄𝑚 ( 𝑡 ,𝜇′ )=𝐼𝑝𝑚 (𝜏 ,𝜇)

Back to the RTEℒ [𝐺𝑚 (𝜏 ≠ 𝑡 ,𝜇 :𝑡 ,𝜇 ′ ) ]=0

Using the homogeneous solution, we can express:

𝐺𝑚 (𝜏<𝑡 ,𝜇 :𝑡 ,𝜇 ′ ) 𝐺𝑚 (𝜏>𝑡 ,𝜇 :𝑡 ,𝜇 ′ )and

The jump condition becomes:

𝜇 [𝐺𝑚 (𝑡+𝜀 ,𝜇 :𝑡 ,𝜇 ′ )−𝐺𝑚 (𝑡−𝜀 ,𝜇 :𝑡 ,𝜇 ′ ) ]=𝛿(𝜇 ′ −𝜇)

RTE Solution

The homogeneous solution is independant of

The particular solution is now an integration of the Green’s function

It is solved only once

The Green’s function can be expressed using 𝐼 h𝑚 (𝜏 ,𝜇𝑖 )

Is it faster?

Without Green’s function

Using Green’s function

Time

Number of incident directions

Time needed to compute the particular solution

Results

That DOM solution can be used for computing subsurface BRDF for different pigment particles types

Results

BRDF will change as well for different material thicknesses

Results

The solution handles materials with multiple layers.

Thank you