Directional Subsurface Scattering Jeppe Revall Frisvad June 2020 - PowerPoint PPT Presentation

Directional Subsurface Scattering Jeppe Revall Frisvad June 2020 Frisvad, J. R., Hachisuka, T., and Kjeldsen, T. K. Directional dipole model for subsurface scattering. ACM Transactions on Graphics 34(1), pp. 5:1-5:12, November 2014. Presented at SIGGRAPH 2015.

Materials (scattering and absorption of light) ◮ Optical properties (index of refraction, n ( λ ) = n ′ ( λ ) + i n ′′ ( λ ) ). ◮ Reflectance distribution functions, S ( x i , � ω i ; x o , � ω o ). BSSRDF n 1 x i x o n 2 ◮ The BSSRDF (Bidirectional Scattering-Surface Reflectance Distribution Function) describes surface and subsurface scattering.

An estimator for subsurface scattering ◮ The reflected radiance equation: � � L r ( x o , � ω o ) = S ( x i , � ω i ; x o , � ω o ) L i ( x i , � ω i ) cos θ i d ω i d A i . A 2 π ◮ Monte Carlo estimator: M N 1 S ( x i , p , � ω i , q ; x o , � ω o ) L i ( x i , p , � ω i , q ) cos θ i � � L r , N , M ( x o , � ω o ) = . pdf( x i , p )pdf( � ω i , q ) NM p =1 q =1 ◮ Common direction sampling pdf (cosine-weighted hemisphere): ω i , q ) = � ω i , q · � n i = cos θ i pdf( � . π π ◮ Common area sampling pdf (triangle mesh): 1 = 1 pdf( x i , p ) = pdf( △ )pdf( x i , p , △ ) = A △ . A ℓ A △ A ℓ

Sampling a cosine-weighted hemisphere (ambient occlusion) ◮ Material: ω o ) = ρ d ( x o ) S ( x i , � ω i ; x o , � δ ( x o − x i ) . π ◮ Sampler: ω i , q · � ω i , q ) = � n i = cos θ i pdf( � . π π ◮ Estimator: L r , N ( x o , � ω o ) N L i ( x o , � ω i , q ) cos θ i = 1 ρ d ( x o ) � π pdf( � ω i , q ) N q =1 M = ρ d ( x o ) 1 � L i ( x o , � ω i , q ) . N q =1

Sampling a triangle mesh (area lights, soft shadows) ◮ Material: ω o ) = ρ d ( x o ) S ( x i , � ω i ; x o , � δ ( x o − x i ) . π ◮ Sampler: x ℓ, q − x i � ω i , q = � x ℓ, q − x i � 1 pdf( x ℓ, q ) = pdf( △ )pdf( x ℓ, q , △ ) = A △ . A ℓ A △ ◮ Estimator: L r , N ( x o , � ω o ) N = ρ d ( x o ) 1 ω i , q ) V ( x ℓ, q , x o ) cos θ i cos θ ℓ � L e ( x ℓ, q , − � � x ℓ, q − x i � 2 A ℓ . π N q =1

Sampling for subsurface scattering � n i , 1 ◮ Material: d ω i , 1 S ( x i , � ω i ; x o , � ω o ) = . . . . x i , 1 A i , 1 ◮ Sampler: ω i , q ) = � ω i , q · � = cos θ i n i pdf( � . π π � ω i , 2 � n i , 2 � ω o x o pdf( x i , p ) = pdf( △ )pdf( x i , p , △ ) � n o = A △ 1 = 1 x i , 2 . A ℓ A △ A ℓ A i , 2 References - Frisvad, J. R., Hachisuka, T., and Kjeldsen, T. K. Directional dipole model for subsurface scattering. ACM Transactions on Graphics 34 (1), pp. 5:1–5:12, November 2014. Presented at SIGGRAPH 2015. - Dal Corso, A., and Frisvad, J. R. Point cloud method for rendering BSSRDFs. Technical Report, Technical University of Denmark, 2018.

Splitting up the BSSRDF ◮ Bidirectional Scattering-Surface Reflectance Distribution Function: S = S ( x i , � ω i ; x o , � ω o ) . ◮ Away from sources and boundaries, we can use diffusion. ◮ Splitting up the BSSRDF S = T 12 ( S (0) + S (1) + S d ) T 21 . where ◮ T 12 and T 21 are Fresnel transmittance terms (using � ω i , � ω o ). ◮ S (0) is the direct transmission part (using Dirac δ -functions). ◮ S (1) is the single scattering part (using all arguments). ◮ S d is the diffusive part (multiple scattering, using | x o − x i | ). ◮ We distribute the single scattering to the other terms using the delta-Eddington approximation: S = T 12 ( S δ E + S d ) T 21 , and generalize the model such that S d = S d ( x i , � ω i ; x o ).

Analytical models for subsurface scattering standard dipole directional dipole S ( x i , � ω i ; x o , � ω o ) S ( x i , � ω i ; x o , � ω o ) = T 12 ( � ω i )( S 1 + S d ( � x o − x i � )) T 21 ( � ω o ) . = T 12 ( � ω i )( S δ E + S d ( x i , � ω i ; x o )) T 21 ( � ω o ) . ◮ Directions ( � ω i , � ω o ) also require surface normals ( � n i , � n o ) to get angles ( θ i , θ o ). ◮ T 12 and T 21 are Fresnel transmittances. ◮ S 1 and S δ E are fully directional (depend on x i , � ω i , x o , � ω o , and normals).

Diffusion theory ◮ Think of multiple scattering as a diffusion process. ◮ In diffusion theory, we use quantities that describe the light field in an element of x volume of the scattering medium. ◮ Total flux, or fluence, is defined by d z y d y � φ ( x ) = L ( x , � ω ) d ω . d x 4 π x z ◮ We find an expression for φ by solving the diffusion equation ( D ∇ 2 − σ a ) φ ( x ) = − q ( x ) + 3 D ∇· Q ( x ) , where σ a and D are absorption and diffusion coefficients, while q and Q are zeroth and first order source terms.

Deriving a BSSRDF ◮ Assume that emerging light is diffuse due to a large number of scattering events: S d ( x i , � ω i ; x o , � ω o ) = S d ( x i , � ω i ; x o ). ◮ Integrating emerging diffuse radiance over outgoing directions, we find S d = C φ ( η ) φ − C E ( η ) D � n o ·∇ φ , Φ 4 π C φ (1 /η ) where ◮ Φ is the flux entering the medium at x i . ◮ � n o is the surface normal at the point of emergence x o . ◮ C φ and C E depend on the relative index of refraction η and are polynomial fits of different hemispherical integrals of the Fresnel transmittance. ◮ This connects the BSSRDF and the diffusion theory. ◮ To get an analytical model, we use a special case solution for the diffusion equation (an expression for φ ). ◮ Then, “all” we need to do is to find ∇ φ (do the math) and deal with boundary conditions (build a plausible model).

Point source diffusion or ray source diffusion directional dipole standard dipole ◮ Ray source diffusion ◮ Point source diffusion [Menon et al. 2005a; 2005b] [Bothe 1941; 1942] e − σ tr r Φ e − σ tr r Φ φ ( r , θ ) = φ ( r ) = , 4 π D r 4 π D r 1 + 3 D 1+ σ tr r � � cos θ , r where r = | x o − x i | and � σ tr = σ a / D is the effective where θ is the angle between the transport coefficient. refracted ray and x o − x i .

Diffusive part of the standard dipole S d ( r ) = α ′ � z r (1 + σ tr d r ) e − σ tr d r + z v (1 + σ tr d v ) e − σ tr d v � . 4 π 2 d 3 d 3 r v z -axis ◮ Distances: z ◮ z r = Λ . r real source ◮ z v = Λ + 4 AD . dr n r 2 + z 2 ◮ d r ( r ) = � r . r 2 + z 2 � ◮ d v ( r ) = v . n 2 ◮ Optical properties ( η = n 2 / n 1 , σ s , σ a , g ): z = 0 r n 1 ◮ Reduced scattering coefficient: σ ′ s = σ s (1 − g ) . ω i ω o ◮ Reduced extinction coefficient: σ ′ t = σ ′ s + σ a . dv incident light ◮ Reduced scattering albedo: α ′ = σ ′ s /σ ′ t . observer ◮ Transport mean free path: Λ = 1 /σ ′ t . -zv virtual source ◮ Diffusion coefficient: D = Λ / 3 . ◮ Transport coefficient: σ tr = � σ a / D . ◮ Reflection parameter: A ( η ) (ratio of polynomial fits).

Directional subsurface scattering when disregarding the boundary � � r 2 � 4 π 2 e − σ tr r S ′ 1 1 d ( x , � ω 12 , r ) = C φ ( η ) D + 3(1 + σ tr r ) x · � ω 12 4 C φ (1 /η ) r 3 � � � � (1 + σ tr r ) + 3 D 3(1+ σ tr r )+( σ tr r ) 2 � − C E ( η ) 3 D (1 + σ tr r ) � ω 12 · � n o − x · � ω 12 x · � , n o r 2 where C φ ( η ) and C E ( η ) are polynomial fits. ◮ Additional dependencies: ◮ Normal: � n o . ◮ Optical properites: η , D , σ tr . d ◮ Note the exponential term: e − σ tr r . ◮ Normal incidence: � ω 12 · � n o = ± 1 . ◮ Plane (half-space): x · � n o ≈ 0 . ◮ normal incidence on plane: x · � ω 12 ≈ 0 . ◮ r → � x o − x i � for � x o − x i � → ∞ .

Dipole configuration (method of mirror images) d ◮ We place the “real” ray source at the boundary and reflect it in an extrapolated boundary to place the “virtual” ray source. ◮ Distance to the extrapolated boundary [Davison 1958]: � d e = 2 . 131 D / 1 − 3 D σ a . ◮ In case of a refractive boundary ( η 1 � = η 2 ), the distance is A = 1 − C E ( η ) Ad e with . 2 C φ ( η )

Modified tangent plane d ◮ The dipole assumes a semi-infinite medium. ◮ We assume that the boundary contains the vector x o − x i and that it is perpendicular to the plane spanned by � n i and x o − x i . ◮ The normal of the assumed boundary plane is then i = x o − x i | x o − x i | × � n i × ( x o − x i ) n ∗ n ∗ � n i × ( x o − x i ) | , or � i = � n i if x o = x i . | � and the virtual source is given by n ∗ n ∗ n ∗ x v = x i + 2 Ad e � i , d v = | x v − x i | , � ω v = � ω 12 − 2( � ω 12 · � i ) � i .

Distance to the real source (handling the singularity) z -axis z r real source n dr d n 2 z = 0 n 1 r ω i ω o incident light dv observer -zv virtual source standard dipole directional dipole r 2 + z 2 � d r = r . d r = r ? ◮ Emergent radiance is an integral over z of a Hankel transform of a Green function which is Fourier transformed in x and y . ◮ Approximate analytic evaluation is possible if r is corrected to R 2 = r 2 + ( z ′ + d e ) 2 . ◮ The resulting model for z ′ = 0 corresponds to the standard dipole where z ′ = z r and d e is replaced by the virtual source.