• Radiom etry Radiometry and Transport • Local light transport • D efinition of BRD F • BRD F properties and com m on BRD Fs CO S 526, Fall 2006 • Rendering equation

Radiometric Units Isotropic Point Source

• Light is a form of energy • Radiant leaves point source in all directions – M easured in (J) • Flux distributed evenly over sphere • Pow er: energy per unit tim e Φ – M easured in Joules/sec = W atts (W ) E = – Also called (Φ) 4π

Point Light Source in a Direction Digression – Solid Angle

Length l • H ow to define radiant flux for one direction? • Angle in radians Length Angle θ = l/r – Solid angle

Solid angle dω

• Solid angle in Area A Solid angle ω = A/r2

1 Light Falling on a Surface Point Light Source in a Direction from A Direction

• H ow to define radiant flux for one direction? • Pow er per unit area – (E) – Solid angle – M easured in W /m 2 Area dA Solid angle dω = dA/r2 • M ove surface aw ay from light – Inverse square law : E ~ 1/r2

• Tilt surface aw ay from light • Irradiance (E) = radiant flux per unit solid angle – Cosine law : E ~ n · l – M easured in W atts per (W /sr)

Light Falling on a Surface Light Falling on a Surface from A Direction from A Direction

Φ Ι Φ E = Ι E = A A ω Φ = Iω


Light Falling on a Surface Light Emitted from a Surface from A Direction in A Direction

Ι Φ • Pow er per unit area per unit solid angle – Ι E = A (L) 2 Φ = Iω – M easured in W /m /sr ˆl nˆ – Projected area – perpendicular to given direction A(nˆ ⋅lˆ) r ω = r2 dΦ L = A ω ω A I (nˆ ⋅ ˆl) d dAp d Ω E = r2 dA

2 Radiance W hy the Cosine Term?

• Flux per unit projected area per unit solid angle. • Foreshortening is by cosine of angle. dΦ • Radiance gives energy by effective surface area. L = C ω dAp d • U nits – w atts per steradian m 2 • U nits – w atts per steradian m d cosθ • W e have now introduced projected area, a θ cosine term . dΦ L = C dAcosθ dω d

Irradiance from Radiance Radiance as a unit of measure

• Radiance doesn’t change w ith distance = θ ω E — Lcos d – Therefore it’s the quantity we w ant to m easure in a Ω ray tracer. • Radiance proportional to w hat a sensor • cosθ dω is projection of a differential area (cam era, eye) m easures. – Therefore it’s w hat we want to output.

Imaging Imaging


Surface Lens Im age Plane Area Asurf Area Aaperture (film , CCD ) Radiance L A Ω = Ω Φ = L A aperture I L Asurf L Asurf 2 dsurf

3 Imaging Imaging

dsurf dimg A Φ = Φ = L A aperture E = I L Asurf L Asurf 2 dsurf Aimg A A E = L aperture surf E L 2 dsurf Aimg

2 A ≈ d ’ surf = ∆ surf ÷ ∆ ÷ Aimg « dimg ◊

Aaperture D epends only Area Asurf Area Aaperture Area Aimg E = L d 2 on cam era Radiance L dimg A Φ Ω I = L A Ω Φ = L A aperture Ω E = surf surf 2 • Punch line: cam eras “see” radiance dsurf Aimg

Surface – BRDF Surface Reflectance – BRDF

• Reflected radiance is proportional to incom ing • Bidirectional Reflectance Distribution Function L (ω ) flux and to irradiance (incident pow er per unit f (ω → ω ) = o o r i o ω area). Ei ( i )

C C • 4-dim ensional function: also w ritten as ω ∝ ω dLr ( r ) dE( i ) L (θ ,ϕ ) f (θ ,ϕ ,θ ,ϕ ) = o o o r i i o o θ ϕ Ei ( i , i )

(the sym bol ρ is also used som etim es)

Surface Reflectance – BRDF Properties of the BRDF

• Energy conservation: θ ϕ θ ϕ θ ω ≤ — fr ( i , i , o, o )cos od o 1 Ω

• H elm holtz reciprocity:

ω → ω = ω → ω C fr ( i o ) fr ( o i ) C C L (ω ) f (ω → ω ) ≡ C r r r i r ω θ ω (not always obeyed by “BRD Fs” used in graphics) Li ( i )cos id i

4 Isotropy Anisotropy

• A BRD F is isotropic if it stays the sam e w hen • Anisotropic BRD Fsdo depend on surface rotation surface is rotated around norm al

• Isotropic BRD Fsare 3-dim esional functions: θ θ ϕ −ϕ fr ( i , o , i o )

BRDF Representations Diffuse

• Physically-based vs. phenom enological m odels • The sim plest BRD F is “ideal diffuse” or • M easured data Lambertian: just a constant ω → ω = • D esired characteristics: fr ( i o ) kd – Fast to evaluate – M aintain reciprocity, energy conservation θ • N ote: does notinclude cos( i ) – For global illum ination: easy to im portance sam ple – Rem em ber definition of irradiance

Diffuse BRDF Ideal Mirror

• Assum e BRD F reflects a fraction ρ of light • All light incident from one direction is reflected f ω → ω θ dω = ρ — fr,Lambertian ( i o )cos o d o into another Ω θ θ θ ϕ = ρ — kd cos o sin o d od o θ∈ π [0.. 2] ϕ∈[0..2π ] π θ θ θ = ρ 2 kd — sin o cos o d o θ∈ π [0.. 2] π = ρ kd • BRD F is zero everyw here except w here ρ θ =θ ∴ = o i fr,Lambertian π ϕ = ϕ +π • The quantity ρ is called the albedo o i

5 Ideal Mirror Glossy Reflection

• To conserve energy, • N on-ideal specular reflection ω → ω θ ω = ρ • M ost light reflected near ideal m irror direction — fr,Mirror ( i o )cos o d o Ω

• So, BRD F is a delta function at direction of ideal m irror reflection δ (θ −θ )δ (ϕ +π −ϕ ) f = i o i o r,Mirror θ cos( i )

Phong BRDF Torrance-Sparrow BRDF

• Phenom enological m odel for glossy reflection • Physically-based BRD F m odel O riginally used in the physics com m unity n l is a vector to the light source – O riginally used in the physics com m unity f = k (lˆ⋅ rˆ) l is a vector to the light source r,Phong s r is the direction of m irror reflection – Adapted by Cook & Torrance and Blinn for graphics DGF • Exponent n determ ines w idth of specular lobe f − = r,T −S π θ θ cos i cos o • Constant ks determ ines size of lobe • Assum e surface consists of tiny “m icrofacets” w ith m irror reflection off each

Torrance-Sparrow BRDF Torrance-Sparrow BRDF

• D term is distribution of m icrofacets • G term accounts for self-shadow ing (i.e., how m any are pointing in each direction) À 2(n ⋅ h)(n⋅v) 2(n ⋅ h)(n⋅l)¤ G = minÃ1, , ‹ • Beckm ann distribution Õ (v ⋅ h) (v ⋅ h) › − β 2 β e [(tan ) / m] is angle betw een n and h D = h is halfw ay betw een l and v 2 4 4m cos β m is “roughness” param eter

n l h v

6 Torrance-Sparrow BRDF Other BRDF Features

• F term is Fresnel term – reflection from an ideal sm ooth • BRD Fsfor dusty surfaces scatter light tow ards surface (solution of M axw ell’s equations) grazing angles • Consequence: m ost surfaces reflect (m uch) m ore strongly near grazing angles

M etal D ielectric (note behavior at Brew ster’s angle)

Other BRDF Features Beyond BRDFs

• Retroreflection: strong reflection • So far, have assum ed 4D BRD F back towards the light source • Function of : 5D • Fluorescence (absorb at one w avelength, • Can arise from bum py diffuse surfaces em it at another): 6D • Phosphorescence (absorb now , em it later): 7D • Tem poral dependence: 8D • Spatial dependence: 10D • Subsurface scattering: 12D • … or from corner reflectors • Polarization • W ave effects (diffraction, interference) • … “Cross product” of tw o plenopticfunctions

Rendering Equation Rendering Equation

G G G G G G G G L x ω = L x ω + f x ω ω L x ω ω • n dω o ( ', ') e ( ', ') — r ( ', , ') i ( ', )( ) • O riginally expressed by [Kajiya 1986] as Ω → = → + Surface I (x' x'') Ie (x' x'') Light Surface → → → Surface G(x', x'')— fr (x x' x'') I(x x')V (x, x') dA G S dω

G G G ′′ ω n ω' x x Ω x' x′

7 Rendering Equation Radiometric and Photometric Units

Radiant energy Lum inous energy • N ext 3-4 w eeks in the course: w ays to solve the (J) Talbot rendering equation Radiant flux or power (F) Lum inous power W att (W ) = J / sec Lum en (lm ) = talbot/ sec = cd· sr Radiant (I) Lum inous intensity W / sr (cd) Irradiance (E) Illum inance W / m 2 = lm / m 2 Radiance (L) Lum inance W / m 2 / sr N it = lm / m 2 / sr (B) Lum inosity W / m 2 Lux = lm / m 2