Light Field = Radiance(Ray)

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1 Page 1 The Light Field Light field = radiance function on rays Surface and field radiance Conservation of radiance Measurement Irradiance from area sources Measuring rays Form factors and throughput Conservation of throughput From London and Upton Light Field = Radiance(Ray)

2 Page Incident Surface Radiance Definition: The incoming surface radiance (luminance) is the power per unit solid angle per unit projected area arriving at a receiving surface dω da d Φi( x, ω) L (, ω) i x dω i da Exitant Surface Radiance Definition: The outgoing surface radiance (luminance) is the power per unit solid angle per unit projected area leaving at surface dω da (, Φ ω ω ) d ( o x L, ) o x dω i da Alternatively: the intensity per unit projected area leaving a surface

3 Page 3 Field Radiance Definition: 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 (, ) Environment Maps L( θ, ϕ) Miller and Hoffman, 1984

4 Page 4 Spherical Gantry Light Field Lxyθ (,,, ϕ) ( θ, ϕ) Capture all the light leaving an object - like a hologram Two-Plane Light Field D Array of Cameras L(,,,) uvst D Array of Images

5 Page 5 Multi-Camera Array Light Field Properties of Radiance

6 Page 6 Properties of Radiance 1. Fundamental field quantity that characterizes the distribution of light in an environment. Radiance is a function on rays All other field quantities are derived from it. Radiance invariant along a ray. 5D ray space reduces to 4D 3. Response of a sensor proportional to radiance. 1st Law: Conversation of Radiance The radiance in the direction of a light ray remains constant as the ray propagates d Φ 1 d Φ d Φ = Φ 1 d L 1 da 1 dω 1 d Φ 1 = L1dω1dA1 dω da L d Φ = LdωdA r da da dω da = = dω da r L = L

7 Page 7 Radiance: nd Law The response of a sensor is proportional to the radiance of the surface visible to the sensor. Ω Aperture A Sensor R= Ldω da= LT A Ω T dω da = A Ω L is what should be computed and displayed. T quantifies the gathering power of the device. Quiz Does the brightness that a wall appears to the sensor depend on the distance? No!!

8 Page 8 Irradiance from a Uniform Surface Source Irradiance from the Environment d Φ i(, x ) = Li(, x )cos dad ω ω θ ω de( x, ω) = L ( x, ω)cosθdω i θ dω L(, xω) i da E() x = Li (, xω)cosθ dω H

9 Page 9 Irradiance from an Area Source A E() x = Lcosθ dω H = L Ω = LΩ cosθ dω Ω Ω Projected Solid Angle θ dω H cosθ dω= π cosθ dω

10 Page 10 Uniform Disk Source Geometric Derivation h r sinα Ω=π sin α α Algebraic Derivation cosα π Ω= cosθ dφdcosθ 1 0 cosα cos θ = π = πsin = π r α r + h 1 Spherical Source Geometric Derivation Algebraic Derivation R α r Ω= cosθ dω = π sin r = π R α Ω=π sin α

11 Page 11 The Sun Solar constant (normal incidence at zenith) Irradiance 1353 W/m Illuminance 17,500 lm/m = 17.5 kilolux Solar angle α Ω= π sin α πα =.5 degrees =.004 radians (half angle) Solar radiance = 6 x 10-5 steradians L 3 E W / m 7 W = = = Ω 6 10 sr m sr Polygonal Source

12 Page 1 Polygonal Source Polygonal Source

13 Page 13 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 Penumbras and Umbras

14 Page 14 Measuring Rays = Throughput Throughput Counts Rays Define an infinitesimal beam as the set of rays intersecting two infinitesimal surface elements ru ( 1, v1, u, v) da1( u1, v1) da( u, v) dt da da 1 x1 x Measure the number of rays in the beam This quantity is called the throughput =

15 Page 15 Parameterizing Rays Parameterize rays wrt to receiver ru (, v, θ, φ) dω ( θ, φ ) da( u, v) da = = x x 1 dt da d da 1 ω Parameterizing Rays Parameterize rays wrt to source ru ( 1, v1, θ1, φ1) da1( u1, v1) dω ( θ, φ ) da d T = da1 = da 1dω1 x x 1

16 Page 16 Parameterizing Rays Tilting the surfaces reparameterizes the rays da ( u, v ) ru (, v, u, v) 1 1 cosθ1cosθ d T = da da x1 x = dω1 i da1 = dω i da All these throughputs must be equal. 1 da ( u, v ) Parameterizing Rays: S R Parameterize rays by Projected area rxyθ (,,, φ) A ( ω) ω Measuring the number or rays that hit a shape T = dωθϕ (, ) da( x, y) S = A ( θϕ, ) dωθϕ (, ) S = 4π A R Sphere: T= 4πA= 4π R

17 Page 17 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 Form Factors

18 Page 18 Types of Throughput 1. Infinitesimal beam of rays cosθ cosθ d T( da, da ) da( x) da( x ) x x. Infinitesimal-finite beam cosθcosθ dt( da, A ) da da( x ) da( x) A x x 3. Finite-finite beam cosθ cosθ TAA (, ) dax ( ) dax ( ) x x AA Probability of Ray Intersection Probability of a ray hitting A given it hits A TAA (, ) Pr( A A) = TA ( ) TAA (, ) = πa da( x ) A cosθ cosθ TAA (, ) = dax ( ) dax ( ) x x AA A da() x TA ( ) = πa

19 Page 19 Probability of Ray Intersection cosθ cosθ TAA (, ) = dax ( ) dax ( ) x x AA = π AA Gxx (, ) dax ( ) dax ( ) da( x ) A cosθcosθ Gxx (, ) Vxx (, ) π x x 0 visible Vxx (, ) = 1 visible A da() x Form Factor Probability of a ray hitting A given it hits A Pr( A ATA ) ( ) = Pr( AA ) TA ( ) = TAA (, ) Form factor definition FAA (, ) = Pr( A A) FAA (, ) = Pr( A A ) da( x ) A Form factor reciprocity FAAA (, ) = FAAA (, ) A da() x TA ( ) =πa

20 Page 0 Form Factor Power transfer from a constant source Φ ( AA, ) = LT( AA, ) TAA (, ) = LT( A ) TA ( ) =Φ( A ) FAA (, ) da( x ) A A da() x TA ( ) =πa Differential Form Factor Probability of a ray leaving da(x) hitting A TdAA (, ) da TdAA (, ) da TdAA (, ) Pr( A da) = = = TdA ( ) πda π Gxx (, ) dax ( ) = A A da( x ) da() x

21 Page 1 Form Factor Power transfer from a constant source dφ ( da, A ) = LT( da, A ) TdAA (, ) = LT( A ) TA ( ) =Φ( A ) FdAA (, ) da( x ) A A da() x Radiant Exitance Definition: The radiant (luminous) exitance is the energy per unit area leaving a surface. dφo Mx ( ) da W lm = lux m m In computer graphics, this quantity is often referred to as the radiosity (B)

22 Page Uniform Diffuse Emitter M = L cosθ dω = H L o = πl H o o cosθ dω θ dω L (, o x ω ) L o = M π da Form Factor Irradiance from a constant source dφ ( da, A ) =Φ( A ) F( da, A ) Φ( A ) = A FdAA (, ) A = M( A ) F( A, da) da da( x ) A EdA ( ) = MA ( ) FA (, da) A da() x

23 Page 3 Form Factors and Throughput Throughput measures the number of rays in a set of rays Form factors represent the probability of ray leaving a surface intersecting another surface Only a function of surface geometry Differential form factor Irradiance calculations Form factors Radiosity calculations (energy balance) Conservation of Throughput Throughput conserved 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

24 Page 4 Conservation of Radiance Radiance is the ratio of two quantities: 1. Power. Throughput Φ( T) Lr () = lim = T 0 T dφ dt Since power and throughput are conserved, Radiance conserved Quiz Does radiance increase under a magnifying glass? No!!

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