Modelling the Directionality of Light Scattered in Translucent Materials

Transcription

1 Modelling the Directionality of Light Scattered in Translucent Materials Jeppe Revall Frisvad Joint work with Toshiya Hachisuka, Aarhus University Thomas Kjeldsen, The Alexandra Institute March 2014

2 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 BRDF n 1 n 2

3 Subsurface scattering Behind the rendering equation [Nicodemus et al. 1977]: n 1 dl r (x o, ω o ) dφ i (x i, ω i ) = S(x i, ω i ; x o, ω o ). x i xo An element of reflected radiance dl r is proportional to an element of incident flux dφ i. S (the BSSRDF) is the factor of proportionality. d 2 Φ Using the definition of radiance L =, we have cos θ da dω L r (x o, ω o ) = S(x i, ω i ; x o, ω o )L i (x i, ω i ) cos θ dω i da. References A 2π - Nicodemus, F. E., Richmond, J. C., Hsia, J. J., Ginsberg, I. W., and Limperis, T. Geometrical considerations and nomenclature for reflectance. Tech. rep., National Bureau of Standards (US), n 2

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5 BRDF [Jensen et al. 2001] BSSRDF

6 [Donner and Jensen 2006]

7 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 where S = T 12 (S (0) + S (1) + S d )T 21. 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). Sd 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 δe + S d )T 21, and generalize the model such that S d = S d (x i, ω i ; x o ).

8 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 volume of the scattering medium. Total flux, or fluence, is defined by y dz φ(x) = L(x, ω) dω. dy 4π dx x z We find an expression for φ by solving the diffusion equation (D 2 σ a )φ(x) = q(x) + 3D Q(x), where σ a and D are absorption and diffusion coefficients, while q and Q are zeroth and first order source terms. x

9 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, where Φ 4πC φ (1/η) Φ is the flux entering the medium at xi. no is the surface normal at the point of emergence x o. Cφ 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).

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

11 Our BSSRDF when disregarding the boundary d Using x = x o x i, r = x, cos θ = x ω 12 /r, we take the gradient of φ(r, θ) (the expression for ray source diffusion) and insert to find S d (x, ω 12, r) = 1 C E (η) ( 1 e σtrr 4C φ (1/η) 4π 2 r 3 [ 3D(1 + σ tr r) ω 12 n o C φ (η) ( r 2 D + 3(1 + σ trr) x ω 12 ) ((1 + σ tr r) + 3D 3(1+σtrr)+(σtrr)2 r 2 x ω 12 ) x n o )], which would be the BSSRDF if we neglect the boundary.

12 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 = D/ 1 3Dσ a. In case of a refractive boundary (η 1 η 2 ), the distance is Ad e with A = 1 C E(η) 2C φ (η).

13 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 n i = x o x i x o x i n i (x o x i ) n i (x o x i ), or n i = n i if x o = x i. and the virtual source is given by x v = x i +2Ad e n i, d v = x v x i, ω v = ω 12 2( ω 12 n i ) n i.

14 Distance to the real source (handling the singularity) z-axis zr real source n dr d n 2 n 1 r z = 0 incident light ω i -z v dv virtual source ω o observer standard dipole d r = r 2 + z 2 r. our model 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.

15 r2 + d 2 e Distance to the real source (handling the singularity) d e β n o x i θ r x o ω 12 n o θ 0 Since we neither have normal incidence nor x o in the tangent plane, we modify the distance correction: R 2 = r 2 + z 2 + d 2 e 2z d e cos β. It is possible to reformulate the integral over z to an integral along the refracted ray. We can approximate this integral by choosing an offset D along the refracted ray. Then z = D cos θ 0.

16 Our BSSRDF when considering boundary conditions Our final distance to the real source becomes { dr 2 r = 2 + Dµ 0 (Dµ 0 2d e cos β) for µ 0 > 0 (frontlit) r 2 + 1/(3σ t ) 2 otherwise (backlit), with µ 0 = cos θ 0 = n o ω 12 and r r cos β = sin θ = 2 (x ω 12 ) 2 r 2 + de 2 r 2 + de 2. The diffusive part of our BSSRDF is then S d (x i, ω i ; x o ) = S d (x o x i, ω 12, d r ) S d (x o x v, ω v, d v ), while the full BSSRDF is as before: S = T 12 (S δe + S d )T 21.

17 Previous Models Previous models are based on the point source solution of the diffusion equation and have the problems listed below. 1. Ignore incoming light direction: Standard dipole [Jensen et al. 2001]. Multipole [Donner and Jensen 2005]. Quantized diffusion [d Eon and Irving 2011]. 2. Require precomputation: Precomputed BSSRDF [Donner et al. 2009, Yan et al. 2012]. 3. Rely on numerical integration: Photon diffusion [Donner and Jensen 2007, Habel et al. 2013]. Using ray source diffusion, we can get rid of those problems.

18 Results (Grapefruit Bunnies) ours reference dipole quantized

19 Results (marble Bunnies) ours reference dipole quantized

20 Results (Simple Scene) Path traced single scattering was added to the existing models but not to ours. Faded bars show quality measurements when single scattering is not added. The four leftmost materials scatter light isotropically RMSE dipole btpole qntzd ours apple marble potato skin1 choc milk soy milk w. grape milk (r.) SSIM skin1 choc apple marble potato milk soy milk w. grape milk (r.)

21 Results (2D plots, 30 Oblique Incidence) quantized ours Our model is significantly different when the angle of incidence changes when the direction toward the point of emergence changes.

22 Results (2D plots, 45 Oblique Incidence) quantized ours Our model is significantly different when the angle of incidence changes when the direction toward the point of emergence changes.

23 Results (2D plots, 60 Oblique Incidence) quantized ours Our model is significantly different when the angle of incidence changes when the direction toward the point of emergence changes.

24 Results (Diffuse Reflectance Curves) dipole btpole qntzd ours ptrace dipole btpole qntzd ours ptrace R d (x) Our model comes closer than the existing analytical models to measured and simulated diffuse reflectance curves.

25 Results (Image Based Lighting) quantized ours

26 The 3Shape Buddha! (scanned with a TRIOS Scanner) matte milk-coloured mini milk

27 Conclusion First BSSRDF which... Considers the direction of the incident light. Requires no precomputation. Provides a fully analytical solution. Much more accurate than previous models. Incorporates single scattering in the analytical model. Future work: Consider the direction of the emergent light. Real-time approximations. Directional multipole and quadpole extensions. Directional photon diffusion. Anisotropic media (skewed dipole).