Rendering Equation

rendering equation

• advantages – predictive simulation • can be used for architecture, engineering, … – photorealistic • if simulation if correct, images will look real • disadvantages – (really) slow • simulation of physics is computationally very expensive – need accurate geometry, materials and lights • otherwise just a correct solution to the wrong problem

• geometric optics – light particles travel in straight lines – light particles do not interact with each other – describes: emission, reflection/refraction, absorption [Stam et al., 1996] al., et [Stam

• wave optics – light particles interact with each other – describes: diffraction, interference, polarization [Gondek et al., 1997] al., et [Gondek

• quantum optics – light particles are like any other quantum particles – captures: fluorescence, phosphorescence [Glassner et al., 1997] al., et [Glassner

• describe physical behavior of light in vacuum filled with objects – based on geometric optics principles – can be extended to describe participating media – can be extended to describe wavelenght dep.

• power: energy per unit time – measured in Watts = Joules/sec dQ Φ = dt • irradiance: power per unit area – measured in Watts/meter2 dΦ E = dA

• power per unit projected area and solid angle – depends on position and direction (5D) d 2Φ d 2Φ dE L(x → Θ) = ⊥ = = dA dωΘ dAcosθΘdωΘ cosθΘdωΘ

cosθΘ = Nx ⋅Θ , Bekaert, Bala] Bala] Bekaert, , é [Dutr

• notation follows [Dutré, Bekaert, Bala]

• radiance leaving from point x in direction Θ L(x → Θ) • radiance coming to point x from direction Ψ L(x ← Ψ) • solid angle for a direction Ψ

dωΨ • in general L(x → Θ) ≠ L(x ← Θ)

• radiance is a function of wavelenght

L(x → Θ) = L(x → Θ,λ) ∫λ∈spectrum

• in practice, write equations for RGB – we will use simplified notation without carry around the wavelength explicitly

• formulation between two points 2 2 2 d Φ d Φ rxy L(x → y) = = dAx cosθ xdωx→y dAxdAy cosθ x cosθ y

dAy cosθ y dωx→y = 2 rxy , Bekaert, Bala] Bala] Bekaert, , é [Dutr

• invariance on straight paths in vacuum – from energy conservation L(x → y) = L(y ← x) • corollary: radiance does not change with distance [Shirley] [Shirley]

• materials differ in the way they scatter energy – need physical description of light scattering , Bekaert, Bala] Bala] Bekaert, , é [Dutr

• bidirectional surface distribution function dL(x → Θ) dL(x → Θ) ρ(x,Ψ → Θ) = = dE(x ← Ψ) L(x ← Ψ)dωΨ cosθΨ , Bekaert, Bala] Bala] Bekaert, , é [Dutr

• reciprocity ρ(x,Ψ → Θ) = ρ(x,Θ → Ψ)

• energy conservation

∀Ψ : dL(x → Θ)cosθΘdωΘ ≤ dE(x ← Ψ) ⇒ ∫Θ∈Ω(x)

∀Ψ : ρ(x,Ψ → Θ)cosθΘdωΘ ≤1 ∫Θ∈Ω(x)

• need outgoing radiance in a given direction – from BRDF definition dL(x → Θ) ρ(x,Ψ → Θ) = L(x ← Ψ)dωΨ cosθΨ

– determine reflected radiance Lr by integration over all incoming light L (x ) dL(x ) r → Θ = ∫ → Θ =

= L(x ← Ψ)ρ(x,Ψ → Θ)cosθΨdωΨ ∫Ψ∈Ωx

• need outgoing radiance in a given direction – also consider light spontaneously emitted by surface

Le (x → Θ)

– total radiance is the sum of emitted and reflected

L(x → Θ) = Le (x → Θ) + Lr (x → Θ)

L(x → Θ) = Le (x → Θ) +

+ L(x ← Ψ)ρ(x,Ψ → Θ)cos(Nx ,Ψ)dωΨ ∫Ψ∈Ωx

L(x → Θ) = Le (x → Θ) +

+ L(x ← Ψ)ρ(x,Ψ → Θ)cosθΨdωΨ ∫Ψ∈Ωx , Bekaert, Bala] Bala] Bekaert, , é [Dutr

L(x → Θ) = Le (x → Θ) +

+ L(x ← Ψ)ρ(x,Ψ → Θ)cosθΨdωΨ ∫Ψ∈Ωx [Bala] [Bala] x x

L(x → Θ) = Le (x → Θ) +

+ L(x ← Ψ)ρ(x,Ψ → Θ)cosθΨdωΨ ∫Ψ∈Ωx

integral equation

indicates radiance at equilibrium

• point visible from x in direction Ψ y = r(x,Ψ) • since energy is conserved in vacuum L(x ← Ψ) = L(y → −Ψ) • by substituting previous values in rendering eq.

L(x → Θ) = Le (x → Θ) +

+ L(r(x,Ψ) → −Ψ)ρ(x,Ψ → Θ)cosθΨdωΨ ∫Ψ∈Ωx

L(x → Θ) = Le (x → Θ) +

• compute solid angle visible from x to y

• by changing domain from hemisphere to scene – and introducing explicit visibility evaluation V

L(x → Θ) = Le (x → Θ) +

cosθΨ cosθΘ + L(y → yx)ρ(x,Ψ → Θ) V (x,y)dAy ∫y∈S 2 rxy

xy N yx N cosθΨ cosθΘ ( ⋅ x )( ⋅ y ) G(x,y) = 2 = 2 rxy | x − y |

L(x → Θ) = Le (x → Θ) +

2 3 Le TLe T Le T Le [CornellPCG]

• direct illumination: radiance reaching a surface directly from the light – often efficient to sample using area formulation • indirect illumination: radiance reaching a surface after bouncing at least once on another surface – often efficient to sample using hemisphere formulation

L(x → Θ) = Le (x → Θ) + Lr (x → Θ)

L (x → Θ) = L(x ← Ψ)ρ(...)cosθ dω = r ∫ Ψ Ψ = L (x ← Ψ) + L (x ← Ψ) ρ(...)cosθ dω = ∫( e r ) Ψ Ψ

= Ld (x → Θ) + Li (x → Θ)

Ld (x → Θ) = Le (x ← Ψ)ρ(x,Ψ → Θ)cosθΨdωΨ ∫Ψ∈Ωx rewrite in area formulation

Ld (x → Θ) =

Le (x ← yx)ρ(x,xy → Θ)G(x,y)V (x,y)dAy ∫y∈lights surface

Ld (x → Θ) =

Ld (x → Θ) = Lr (x ← Ψ)ρ(x,Ψ → Θ)cosθΨdωΨ ∫Ψ∈Ωx

since Lr (x ← Ψ) = Lr (r(x,Ψ) → −Ψ)

Ld (x → Θ) =

Lr (r(x,Ψ) → −Ψ)ρ(x,Ψ → Θ)cosθΨdωΨ ∫Ψ∈Ωx

• 2D square

x1 y1 I = f (x)dAX = f (x, y)dxdy ∫x∈S ∫x0 ∫y0 • 2D hemisphere

2π π I = f (Θ)dωΘ = f (ϕ,θ )sinθdϕdθ ∫Θ∈Ω ∫0 ∫0

• capture realistic appearance is necessary [CornellPCG]

• light is reflected equally in all directions ρ ρ(x,Ψ → Θ) = d π , Bekaert, Bala] Bala] Bekaert, , é [Dutr

• Lambertian shading model motivation dL(x → Θ) = ρ(x,Ψ → Θ)dE(x ← Ψ) = ρ = d cosθ L(x ← Ψ)dω = C k cosθ π Ψ Ψ l d

• light is reflected only in one direction

ρ(x,Ψ → Θ) ∝δ (Ψ,Θ) , Bekaert, Bala] Bala] Bekaert, , é [Dutr

• light is reflected in many directions unequally – many models exist , Bekaert, Bala] Bala] Bekaert, , é [Dutr

• Phong model

n n ρ(x,Ψ → Θ) = kd + ks cos θr = kd + ks (R⋅Θ)

• Blinn-Phong model

n ρ(x,Ψ → Θ) = kd + ks (N⋅H)

• issues: – non reciprocal – non energy conserving

ρ n + 2 n ρ(x,Ψ → Θ) = d + ρ (H ⋅Θ) π 2π s

• energy conservation

ρd + ρs ≤1

• is modified Phong physically accurate?

photograph Phong accurate BRDF [LaFortune et al., 1997] [LaFortuneal., et

• is modified Phong physically accurate?

Phong

accurate BRDF [LaFortune et al., 1997] [LaFortuneal., et

• analytic model – physically motivated – hard to capture every material

• data-driven – measure light reflectance – encode in lookup table or fit – resample when rendering