Annoucements • Next Thursday: Chris software overview • Reading: Henrik Wann Jensen, "Global Illumination using Photon Maps," In "Rendering Techniques '96". Eds. X. Pueyo and P. Schrder. Springer-Verlag, pp. 21-30, 1996 http://graphics.ucsd.edu/~henrik/papers/ewr7/ [web page] http://graphics.ucsd.edu/~henrik/papers/ewr7/egwr96.pdf [paper] • Photon Mapping � SIGGRAPH 2002 Course Notes � http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15864- s04/www/slides/photonMappingCourse02.pdf
Local Illumination, Reflection, and BRDFs Adapted from… Szymon Rusinkiewicz Princeton University C0S 526, Fall 2002
Overview • Radiometry and Photometry • Definition of BRDF • BRDF properties and common BRDFs • Rendering equation
Radiometric Units • Light is a form of energy – measured in Joules (J) • Power: energy per unit time � Measured in Joules/sec = Watts (W) � Also called Radiant Flux ( Φ )
Point Light Source • Total radiant flux in Watts • How to define angular dependence? � Solid angle Area dA Solid angle d ω = dA/r 2 • Power per unit solid angle � Measured in Watts per steradian (W/sr)
Light Falling on a Surface • Power per unit area – Irradiance (E) � Measured in W/m 2 • Move surface away from light � Inverse square law: E ~ 1/r 2 • Tilt surface away from light � Cosine law: E ~ n • l
Light Emitted from a Surface • Power per unit area per unit solid angle – Radiance (L) � Measured in W/m 2 /sr � Projected area – perpendicular to given direction Φ d d ω = L ω dA d dA
Total Light Emitted from a Surface • Radiance integrated over all directions � = θ φ θ ω B L d ( , ) cos o Ω • Called Radiosity (B) � Measured in W/m 2
Radiometry vs. Photometry • These are all physical (radiometric) units • Don’t take perception into account • Eye sensitive to different colors λ (nm) 400 700 (blue) (red)
Photometric Units • Take human perception into account • Original unit: candle � Luminous intensity equal to a “standard candle” • Today: one of the basic SI units � One candela (cd) is the luminous intensity of a source producing 1/683 W at 555 nm (yellow-green).
Radiometric and Photometric Units Radiant energy Luminous energy Joule (J) Talbot Radiant flux or power ( Φ ) Luminous power Watt (W) = J / sec Lumen (lm) = talbots / sec Radiant intensity (I) Luminous intensity W / sr Candela (cd) Irradiance (E) Illuminance W / m 2 Lux = lm / m 2 Radiance (L) Luminance Nit = lm / m 2 / sr W / m 2 / sr Radiosity (B) Luminosity W / m 2 Lux = lm / m 2
Direct Illumination (i.e., Irradiance) Φ = E A A
Direct Illumination Φ Ι = E A Φ = ω I ω A
Direct Illumination Φ Ι = E A Φ = ω I n ˆ l ˆ n ⋅ l A ˆ ˆ ( ) ω = r r 2 n ⋅ l I ˆ A ˆ ( ) � = E r 2
Imaging Surface Lens Image Plane (film, CCD)
Imaging d surf Area A surf Area A aperture Radiance L A � Φ = aperture � I = L A L A surf surf d 2 surf
Imaging d surf d img Area A surf Area A aperture Area A img Radiance L A Φ � = � Φ = aperture � I = E L A L A surf surf A d 2 img surf
Imaging A Φ aperture = Φ = E I = L A L A surf surf A d 2 img surf A A aperture surf = E L d A 2 surf img 2 � � A d � � surf surf = � � A d � � img img A Depends only aperture E = L d on camera 2 img • Punch line: cameras “see” radiance
Surface Reflectance – BRDF • Bidirectional Reflectance Distribution Function ω dL ( ) ω → ω = f o o ( ) r i o ω dE ( ) i i • 4-dimensional function: also written as θ ϕ dL ( , ) θ ϕ θ ϕ = f o o o ( , , , ) r i i o o θ ϕ dE ( , ) i i i (the symbol ρ is also used sometimes)
Defining Surface Reflectance • Why is BRDF defined in this way? • Key point: BRDF is a differential quantity, so limit must exist Source Detector Φ src Φ det ω src ω det
Definition of BRDF • First attempt: Φ = f det r Φ src Source Detector Φ src Φ det ω src ω det
Definition of BRDF • Should f r vary with ω src ? No. Source Detector Φ src Φ det ω src ω det
Definition of BRDF • Should f r vary with ω det ? Yes. Source Detector Φ src Φ det ω src ω det
Definition of BRDF • Thus, Φ ω = f det det r Φ src Source Detector Φ src Φ det ω src ω det
Definition of BRDF • What about surface area? f r must be independent of surface area Source Detector Φ src Φ det ω src ω det dA
Definition of BRDF ( ) Φ ω ⋅ dA L = = f det det r Φ dA E src Source Detector Φ src Φ det ω src ω det dA
Properties of the BRDF • Energy conservation: � θ ϕ θ ϕ θ ω ≤ f d ( , , , ) cos 1 r i i o o o o Ω • Helmholtz reciprocity: ω → ω = ω → ω f f ( ) ( ) r i o r o i (not always obeyed by “BRDFs” used in graphics)
Isotropy • A BRDF is isotropic if it stays the same when surface is rotated around normal • Isotropic BRDFs are 3-dimensional functions: θ θ ϕ − ϕ f ( , , ) r i o i o
Anisotropy • Anisotropic BRDFs do depend on surface rotation
Diffuse • The simplest BRDF is “ideal diffuse” or Lambertian : just a constant ω → ω = f k ( ) r i o d • Note: does not include cos( θ i ) � Remember definition of irradiance
Diffuse BRDF • Assume BRDF reflects a fraction ρ of light � → = ω ω θ ω ρ f d ( ) cos r Lambertian i o o o , Ω � θ θ θ ϕ = ρ k d d cos sin d o o o o θ ∈ π [ 0 .. ] 2 ϕ ∈ π [ 0 .. 2 ] � π θ θ θ = ρ k d 2 sin cos d o o o θ ∈ π [ 0 .. ] 2 π = ρ k d ρ ∴ = f r Lambertian π , • The quantity ρ is called the albedo
Ideal Mirror • All light incident from one direction is reflected into another • BRDF is zero everywhere except where θ = θ o i ϕ = ϕ + π o i
Ideal Mirror • To conserve energy, � ω → ω θ ω = f d ( ) cos 1 r Mirror i o o o , Ω • So, BRDF is a delta function at direction of ideal mirror reflection δ θ − θ δ ϕ − ϕ ( ) ( ) = f i o i o r Mirror θ , cos( ) i
Glossy Reflection • Non-ideal specular reflection • Most light reflected near ideal mirror direction
Phong BRDF • Phenomenological model for glossy reflection l is a vector to the light source = ⋅ ˆ n f k l r ˆ ( ) r Phong s , r is the direction of mirror reflection • Exponent n determines width of specular lobe • Constant k s determines size of lobe
Torrance-Sparrow BRDF • Physically-based BRDF model � Originally used in the physics community � Adapted by Cook & Torrance and Blinn for graphics DGF = f − r T S π θ θ , cos cos i o • Assume surface consists of tiny “microfacets” with mirror reflection off each
Torrance-Sparrow BRDF • D term is distribution of microfacets (i.e., how many are pointing in each direction) • Beckmann distribution β is angle between n and h − β m 2 e [(tan ) / ] = D h is halfway between l and v β m 2 4 4 cos m is “roughness” parameter n h l v
Torrance-Sparrow BRDF • G term accounts for self-shadowing � � ⋅ ⋅ ⋅ ⋅ n h n v n h n l 2 ( )( ) 2 ( )( ) = G � � min 1 , , ⋅ ⋅ v h v h � � ( ) ( )
Torrance-Sparrow BRDF • F term is Fresnel term – reflection from an ideal smooth surface (solution of Maxwell’s equations) • Consequence: most surfaces reflect (much) more strongly near grazing angles Metal Dielectric (note behavior at Brewster’s angle)
Aside: Brewster’s Angle
Other BRDF Features • BRDFs for dusty surfaces scatter light towards grazing angles
Other BRDF Features • Retroreflection: strong reflection back towards the light source • Can arise from bumpy diffuse surfaces • … or from corner reflectors
BRDF Representations • Physically-based vs. phenomenological models • Measured data • Desired characteristics: � Fast to evaluate � Maintain reciprocity, energy conservation � For global illumination: easy to importance sample
Beyond BRDFs • So far, have assumed 4D BRDF • Function of wavelength: 5D • Fluorescence (absorb at one wavelength, emit at another): 6D • Phosphorescence (absorb now, emit later): 7D • Temporal dependence: 8D • Spatial dependence: 10D • Subsurface scattering: 12D • Polarization • Wave optics effects (diffraction, interference) • …
Rendering Equation � ω = ω + ω → ω ω θ ω L x L x f x L x d ( , ) ( , ) ( , ) ( , ) cos o o e o r i o i i i Ω Outgoing Emitted BRDF Irradiance radiance radiance
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