Page 1 The Light Field Rays and throughput Form factors Light field representations Hemispherical illumination Illumination from uniform area light sources Shadows: Blockers, umbras and penumbras Radiosity Rays and Throughput
Page Throughput = Measuring Rays Define an infinitesimal beam as the set of rays intersecting two differential surface elements da1 da dt = da da r 1 Measure the number of rays in the beam This quantity is called the throughput Throughput = Measuring Rays Parameterize rays wrt to source or receiver da 1 dω dt = da d = da da r ω / 1 1 1 dω 1 da dt = da d = da da r ω / 1
Page 3 Throughput = Measuring Rays Tilting the surfaces parameterizes the rays wrt to the tilted surface da1 da cosθ cosθ dt = dada = dω da r Parameterizing Rays: S R Parameterize rays by Projected area rxyθ (,,, φ) Measuring the number that hit the shape T = dωθϕ (, ) da ( x, y) = S S = 4π A R A ( θϕ, ) dωθϕ (, ) A ( ω) ( xy, ) Sphere: ω( θφ, ) T= 4π A = 4π r
Page 4 Parameterizing Rays: M S Parameterize rays by ruvθφ (,,, ) ( uv, ) ( θφ, ) N T = da(,) u v cos θ dωθϕ (, ) M H N ( ) S? Projected Solid Angle θ dω H cosθ dω= π cosθ dω
Page 5 Parameterizing Rays: M S Parameterize rays by ruvθφ (,,, ) ( uv, ) ( θφ, ) N T = da(,) u v cos θ dωθϕ (, ) M H N ( ) S π Sphere: T= πs= 4π r Crofton s Theorem: 4 πa = πs A = S 4 Types of Throughput 1. Infinitesimal beam of rays cosθ cosθ dt( da, da ) = dω da = da( x) da( x ) x x. Differential-finite beam cosθcos θ T( da, A ) da = cos θdω( x ) da( x) = da( x ) da( x) Ω A x x 3. Finite-finite beam cosθ cosθ TAA (, ) = dω da= dax ( ) dax ( ) x x AΩ A A Special case is one object with area A
Page 6 Form Factors Differential Form Factor Probability of a ray leaving da(x) hitting A TdAAdA (, ) TdAAdA (, ) TdAA (, ) Pr( A da) = = = TdA ( ) πda π cosθcosθ = da( x ) π x x A da( x ) A cosθ cosθ Gxx (, ) dax ( ) dax ( ) π x x A A da() x
Page 7 Form Factor Probability of a ray leaving A hitting A TAA (, ) Pr( A A) = TA ( ) 1 cosθcosθ = da( x ) da( x) A π x x AA da( x ) A A da() x Conservation of Radiance
Page 8 Conservation of Throughput Throughput conversed during propagation Number of rays conserved Assuming no attenuation or scattering n (index of refraction) times throughput invariant under the laws of geometric optics Reflection at an interface Refraction at an interface Causes rays to bend (kink) Continuously varying index of refraction Causes rays to curve; mirages Conservation of Radiance Radiance is the ratio of two conserved quantities: 1. Power. Throughput Radiance conserved Φ( T) d T 0 Lr ( ) = lim = T Φ dt
Page 9 Surface Radiance Definition 1: The surface radiance (luminance) is the intensity per unit projected area leaving a surface dω da Lx (, ω) d Φ ( x, ω) dω da Field Radiance Definition 1: The field radiance (luminance) at a point in space in a given direction is the power per unit solid angle per unit area perpendicular to the direction da dω Lxω (, )
Page 10 Light Field Representations Spherical Light Field Lxyθ (,,, ϕ) ( θ, ϕ) Capture all the light leaving an object - like a hologram
Page 11 Two-Plane Light Field D Array of Cameras L(,,,) uvst D Array of Images Environment Maps Miller and Hoffman, 1984 L( θ, ϕ)
Page 1 Environment Maps Interface, Chou and Williams (ca. 1985) The Sky From Greenler, Rainbows, halos and glories
Page 13 Irradiance from a Hemisphere Irradiance from a Hemisphere dφ () x = L(, xω)cosθdωda= deda de() x = L(, x ω)cosθd ω Ex () = Lx (, ω)cosθdω H θ dω Lxω (, ) Light meter da
Page 14 Irradiance Environment Maps L( θ, ϕ ) E( θ, ϕ) R N Radiance Environment Map Irradiance Environment Map Irradiance Map or Light Map Isolux contours
Page 15 Uniform Area Sources Irradiance from an Area Source Ω E() x = Lcosθ dω H = L Ω = LΩ cosθ dω Ω
Page 16 Disk Source Geometric Derivation h r Ω=π sin α α Ω= Algebraic Derivation cosα π 1 0 cos θ = π L = Lπsin α r = Lπ r + h cosθ dφdcosθ cosα 1 Spherical Source Geometric Derivation Algebraic Derivation R α r Ω= cosθ dω = π sin r = π R α Ω=π sin α
Page 17 Polygonal Source Lambert s Formula γ N i i γ i 3 Ai A1 i= 1 A A = γ N N i i i n n A = γ N N i i i i= 1 i= 1 A 3
Page 18 Penumbras and Umbras Radiosity
Page 19 Radiosity and Luminosity Definition: The radiosity (luminosity) is the energy per unit area leaving a surface. dφo Mx ( ) da W lm = lux m m This is officially referred to as the radiant (luminous) exitance. Uniform Diffuse Emitter dφ ( x) = L ( x, ω)cosθdωda= MdA o M( x) = Lo( x, ω)cosθdω H M = Lcosθ dω = πl o dm( x) = L ( x, ω)cosθdω M L = π o
Page 0 The Sun Solar constant (normal incidence at zenith) Irradiance 1353 W/m Illuminance 17,500 lm/m = 17.5 kilolux Solar angle α =.5 degrees =.004 radians (half angle) Ω = π sin α = 6 x 10-5 steradians Radiance L 3 E 1.353 10 W / m 7 W = = =.5 10 5 Ω 6 10 sr m sr