Lecture 3: Rendering Equation

Transcription

1 Lectue 3: Rendeing Equation CS 660, Sping 009 Kavita Bala Compute Science Conell Univesity Radiomety Radiomety: measuement of light enegy Defines elation between Powe Enegy Radiance Radiosity 1

2 Hemispheical coodinates Defined a measue ove hemisphee dω diection vecto Diffeential solid angle dω da sinθdθdϕ Radiance Radiance is adiant enegy at in diection θ: 5D function ) : Powe pe unit pojected suface aea pe unit solid angle ) units: Watt / m.s d P da dω da dω

3 Why is adiance impotant? Invaiant along a staight line (in vacuum) 1 Why is adiance impotant? Response of a senso (camea, human eye) is popotional to adiance eye Piel values in image popotional to adiance eceived fom that diection 3

4 Relationships Radiance is the fundamental quantity Powe: P Radiosity: d P ) da dω B Aea Solid Angle Solid Angle ) cosθ dω da ) cosθ dω Outline Light Model Radiance Mateials: Inteaction with light Rendeing equation 4

5 Mateials - Thee Foms Ideal diffuse (Lambetian) Ideal specula Diectional diffuse Reflectance Thee Foms Ideal diffuse (Lambetian) Ideal specula Diectional diffuse 5

6 BRDF Bidiectional Reflectance Distibution Function f Light Souce (, ) N d ) de( ) Detecto W m W m s Definition of BRDF 6D function? 4D function? Why? Wavelength-dependent f (, ) d ) de( ) d ) )cos( N, dω ) 6

7 BRDF special case: ideal diffuse Pue Lambetian f (, ) ρ d π ρ d Enegy Enegy out in 0 ρ 1 d Popeties of the BRDF Recipocity: f (, ) f (, ) Theefoe, notation: f (, ) Impotant fo bidiectional tacing 7

8 Bounds: Popeties of the BRDF 0 (, ) f Enegy consevation: f (, )cos( N, ) dω 1 Outline Light Model Radiance Mateials: Inteaction with light Rendeing equation 8

9 Light Tanspot Goal Descibe steady-state adiance distibution in scene Assumptions: Geometic Optics Achieves steady state instantaneously Related: Neuton Tanspot (neutons) Gas Dynamics (molecules) Radiance epesents equilibium Radiance values at all points in the scene and in all diections epesses the equilibium 4D function: only on sufaces 9

10 Rendeing Equation (RE) RE descibes enegy tanspot in scene Input Light souces Suface geomety Reflectance chaacteistics of sufaces Output: value of adiance at all suface points in all diections Rendeing Equation L L e L ) L e ( ) L ( ) 10

11 Rendeing Equation L L L e ) L e ( ) Rendeing Equation L L e L ) L e ( ) hemisphee L ( )... 11

12 Rendeing Equation f (, ) d ) de( ) d ) f (, ) de( ) d ) f (, ) )cos( N, ) dω L ( ) f (, ) )cos( N, dω ) hemisphee Rendeing Equation L L e L ) hemisphee L ( e ) ) f, )cos( N, ) dω ( Applicable fo each wavelength 1

13 Rendeing Equation ) L ( ) e L ) f (, )cos( N ( ) hemisphee, dω incoming adiance Geometic Optics Summay Goal: to compute steady-state adiance values in scene Rendeing equation: mathematical fomulation of poblem that global illumination algoithms must solve 13

14 ) L RE: Aea Fomulation e ( ) Ω f ( ) ) cosθ dω y y ) ) Ray-casting function: what is the neaest visible suface point seen fom in diection? y vp(, ) ) vp(, ) ) ) L ( ) Rendeing Equation e Ω f ( ) ) cosθ dω y vp(, ) da y ) vp(, ) ) dω da cosθ y dω y y 14

15 Rendeing Equation: visible sufaces ) L ( ) Coodinate tansfom ) L ( ) e e Ω f ( ) ) cosθ dω f y on all sufaces ( ) y ) cosθ y vp(, ) Integation domain visible suface points y Integation domain etended to ALL suface points by including visibility function cosθ y y da y Rendeing Equation: all sufaces cosθ cosθ y L ( ) Le (...) f (...) y ) V (, y) da A y y 15

16 Two foms of the RE Hemisphee integation ) L ( ) e Ω f ( ) ) cosθ dω Aea integation (ove polygons fom set A) cosθ cosθ y ) Le ( ) f ( ) y ) V (, y) da y A y Lighting Cues colo bleeding glossy eflection efaction soft shadow Global Illumination is impotant fo ealism 16

17 Light Souces and Reflection Models Outline Light souces Light souce chaacteistics Types of souces Light eflection Physics-based models Empiical models 17

18 Souces of light adiation Themal adiation ( blackbody ) Sun, tungsten & tungsten-halogen lamps; ac lamps Electic dischage gas dischage lamps (neon, sodium, mecuy vapo) ac lamps, fluoescent lamps Othe phenomena fluoescence (fluoescent lamps, fluoescent dyes) phosphoescence (CRTs); LEDs; lases Intensity Eamples of light emission Visible Spectum 0,00 0K 10,00 0K 600 0K 300 0K 000 K 1000 K Wavelength 500 K Bightness Bightness Wavelength Mecuy vapo lamp Mecuy lines Phospho emission Wavelength 18

19 Modeling luminaies Spectal distibution Detemined by physics of souce Geneally tabulated, often RGB used Spatial distibution Modeled as point o simple aea light Also light pobes ceate high dynamic ange inputs Diectional distibution Often shaped by eflectos Tabulated when necessay, cosine lobe is common appoimation Diectional distibutions Lambetian cosine-powe abitay 19

20 Lighting w/ Envionment Maps High lighting compleity Rich: captues eal wold Image-based lighting Acquiing lighting infomation of eal scenes Image-based techniques Use light pobe Vaying eposue 0