Interactive Hair Rendering Under Environment Lighting

Valentin JANIAUT

Zhong Ren, Kun Zhou, Tengfei Li, Wei Hua, Baining Guo

2

Hair Rendering

● Hair fiber represented with lines primitives

● Basic shading model is not realistic at all.

Basic OpenGL illumination

Deep Opacity Map [YUK08]

1 fiber4 strands

3

Environment Lighting

● Natural Illumination● No directional light

Environment LightingSingle Light

4

Spherical Function

How to represent a spherical function?

SRBF

5

Spherical Radial Basis Function

● Useful to approximate spherical function

€

f (θ ,ϕ ) ≈ c jj =1

N

∑ R(θ ,ϕ,ξ j ,λ j )

Spherical Coordinate of the

Spherical Function

Number of SRBF to use for the

approximation

Coefficient depending of the

problem

SRBF with actually 5 parameters

Spherical Coordinate of the

center of the SRBF

Bandwidth of the center of the SRBF

● Same idea than Fourier Series.

6

SRBF Light

● A SRBF function can represent a light in graphic rendering.

€

R j (ω i,ξ j ,λ j ) = L jG(ω i,ξ j ,λ j )Expression of the SRBF light j.

Intensity of the light j.

Gaussian distribution Result on the sphere

2D 3D

€

exp(−λ j ) × exp(λ j × (ω i • ξ j ))

Gaussian distribution.

7

SRBF and Environment Lighting

● We can now represent the environment lighting as the sum of the SRBF lights, as following:

€

L(ω i) ≈ L jj =1

N

∑ G(ω i,ξ j ,λ j )

8

Outgoing Curved Intensity

€

L(ωo) = D L(ω i)T(ω i)Ω∫ S(ω i,ωo)cosθ idω i

Diameter of the hair fiber Environment Lighting Transmittance Bidirectional

scattering function

9

Transmittance or Absorbtance

Transmittance is the fraction of incident light that passes through a

sample.

€

T(x,ω i) = exp(−σ a ρ(x)dxx

∞ω∫ )

Attenuation coefficient. Density function:

• 1 if covered by hair fiber.• 0 otherwise

10

Bidirectional scattering function

● S(ωi,ωo) will be the bidirectional scattering function, similar to BRDF in surface reflectance.

● The scattering is the deviation of the straight trajectory of a ray light due to an obstacle.

● Kajiya and Kay model [1989]

€

S(ω i,ωo) = Kd + Ks

cosp (θ i +θ o)

cosθ i

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Environment Lighting Approximation

● Remember SRBF? It’s time to use it.

€

L(ω i) ≈ L jj =1

N

∑ G(ω i,ξ j ,λ j )

€

L(ωo) = D L jj =1

N

∑ G j (ω i)T(ω i)Ω∫ S(ω i,ωo)cosθ idω i

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Effective Transmittance

● Last step of our simplification● Average attenuation of the SRBF

Lighting j.

€

˜ T (ξ j ,λ j ) =G j (ω i)T(ω i)Ω

∫ S(ω i,ωo)cosθ idω i

G j (ω i)Ω∫ S(ω i,ωo)cosθ idω i

€

L(ωo) = D L jj =1

N

∑ ˜ T (ξ j ,λ j ) G j (ω i)Ω∫ S(ω i,ωo)cosθ idω i

● How to compute this equation?

13

Splitting the equation

€

L(ωo) = D L jj =1

N

∑ ˜ T (ξ j ,λ j ) G j (ω i)Ω∫ S(ω i,ωo)cosθ idω i

€

˜ T (ξ j ,λ j )

€

G j (ω i)Ω∫ S(ω i,ωo)cosθ idω i

Transmittance Convolution of SRBF and scattering function.

€

I(ωo,ξ,λ )

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€

G j (ω i)Ω∫ S(ω i,ωo)cosθ idω i

Convolving SRBF and Scattering Function

● Marschner et al. model [2003]

€

S(ω i,ωo) = M t (θ h )N t (θ d ,φ)t

∑

€

θh =(θ i +θ o)

2;

θ d =(θ i −θ o)

2;

φ = φo + φi

With:

€

IM (cosθξ ,cosθ o,cos(φξ − φo), 1λ )

 

 

 cos(ϕξ-ϕo)

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Computing Effective Transmittance

€

L(ωo) = D L jj =1

N

∑ ˜ T (ξ j ,λ j ) G j (ω i)Ω∫ S(ω i,ωo)cosθ idω i

Precomputed in a table• Sampled at the SRBF center

• Use of the Deep Opacity Map technique

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Self-shadowing

Opacity Shadow Map Deep Opacity Map

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Deep Opacity Map

z

T

z1 z2 z3

Compute the optical depth

Zo

Z1

Z2

Z3

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Multiple Scattering

€

LD (ωo) = D L j (ξ j )Tfj =1

N

∑ (ξ j ) ψ f (ξ j ,ω i,σ f )Ω

∫ SD (ω i,ωo)cosθ idω i

€

Tf (ξ j )

€

ψ f (ξ j ,ω i,σ f )Ω

∫ SD (ω i,ωo)cosθ idω i

Transmittance Convolution of SRBF and scattering function.

€

IG (ωo,ξ,σ f )

● More realistic model.

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Multiple Scattering Computation

● Voxelize Hair Model

● For each voxel store:

● ϖ : Average Fiber

Direction

● ν : Standard Deviation of

fiber direction

● ςtΤ : Perpendicular

Attenuation Coefficient

● Sample Tf and σf on a rough

grid

● Store as 3D texture

● Hardware tri-linear

interpolation

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Algorithm OverviewSingle Scattering

● Precompute● SRBF

decomposition● Single Scattering

integration table

● Runtime● Generate Deep

Opacity Depth Map (DODM)

● Construct the Summed Area Table

● Sample the effective transmittance

● Sample the single scattering integral

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Results

hair model #fibers #segments

FPSSingle

scattering

animation 10K 270K 16.2

straight 10K 160K 17.8

ponytail 20K 900K 11.1

curly 50K 3.4M 2.30

wavy 10K 687K 12.3

natural 10K 1.6M 9.20

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Limitations

● Runtime change of hair properties● precomputation is costly (~50 minutes)

● Eccentricity of hair scattering is omitted

● Additional video memory for the integral tables● 12MB for single scattering● 24MB for single + multiple scattering● no per-fiber hair property

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References

● http://www.kunzhou.net/ (Author of the main paper, some of his slides are used in this slideshow)

● http://www.cemyuksel.com/ (Author of the Deep Opacity Maps and numerous other paper about hair rendering)

● Illustration on slide 10 comes from wikipedia.

● http://www.cse.cuhk.edu.hk/~ttwong/papers/srbf/srbf.html Lecture about SRBF.

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Q/A