Lecture 5: Rendering Equation Chapter 2 in Advanced GI

Transcription

1 Lectue 5: Rendeing Equation Chapte in Advanced GI Fall 004 Kavita Bala Compute Science Conell Univesity Radiomety Radiomety: measuement of light enegy Defines elation between Powe Enegy Radiance Radiosity

2 Powe Enegy: Symbol: Q; unit: Joules Powe: Enegy pe unit time (dq/dt) Aka. adiant flu in this contet Symbol: P o Φ; unit: Watts (Joules / sec) Photons pe second All futhe quantities ae deivatives of P (flu densities) Iadiance Powe pe unit aea (dp/da) That is, aea density of powe It is defined with espect to a suface Symbol: E; unit: W / m Measuable as powe on a small-aea detecto Aea powe density eiting a suface is called adiant eitance (M) o adiosity (B) but has the same units

3 Radiance Radiance is adiant enegy at in diection θ: 5D function L( Θ) : Powe pe unit pojected suface aea pe unit solid angle L( Θ) units: Watt / m.s d P da dω Θ da Θ 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

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

5 Mateials - Thee Foms Ideal diffuse (Lambetian) Ideal specula Diectional diffuse BRDF Bidiectional Reflectance Distibution Function Light Souce Ψ N Θ Detecto f (, Ψ Θ) dl( Θ) de( Ψ) dl( Θ) L( Ψ)cos( N, Ψ dω ) Ψ

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

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

8 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 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 Θ + L( Θ) L e ( Θ) + L ( Θ)

10 Rendeing Equation L Θ L L e + L( Θ) L e ( Θ) + Rendeing Equation Θ L L e L + L( Θ) L e ( Θ) + hemisphee L ( Ψ)...

11 Rendeing Equation f (, Ψ Θ) dl( Θ) de( Ψ) dl( Θ) f (, Ψ Θ) de( Ψ) dl( Θ) f (, Ψ Θ) L( Ψ)cos( N, Ψ) dωψ L ( Θ) f (, Ψ Θ) L( Ψ)cos( N, Ψ) dω hemisphee Ψ Rendeing Equation Θ L L e L + L( Θ) hemisphee L ( e Θ) L( Ψ) + f, Ψ Θ)cos( N, Ψ) dω ( Applicable fo each wavelength Ψ

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

13 Shading Models Reflectance Thee Foms Ideal diffuse (Lambetian) Ideal specula Diectional diffuse

14 Ideal Specula Reflection Calculated fom Fesnel s equations Eact fo polished sufaces Basis of ealy ay-tacing methods Fesnel Equations η 1 sinθ1 η sin θ F F p s η cosθ1 η1 cosθ η cosθ + η cosθ η1 cosθ1 η cosθ η cosθ + η cosθ 1

15 Fesnel Reflectance ( F s + F p ) F fo unpolaized light Equations apply fo metals and nonmetals fo metals, use comple inde η n+ik fo nonmetals, k0 Metal vs. Nonmetal 1 Fesnel eflectance Metals 0 0 Nonmetals (k0) θ 90

16 Mies van de Rohe s unbuilt Coutyad House Ideal Diffuse Reflection Chaacteistic of multiple scatteing mateials An idealization but easonable fo matte sufaces Basis of most adiosity methods BRDF is a constant function

17 Diectional Diffuse Reflection Chaacteistic of most ough sufaces Descibed by the BRDF Classes of Models fo the BRDF Plausible simple functions Phong 1975; Physics-based models Cook/Toance, 1981; He et al. 199; Empiically-based models Wad 199, Lafotune model

18 Phong Reflection Model L Diffuse Specula Mio Reflection R Vecto V Diffuse k d ( N L) Specula k ( R V ) s n n ( R. Θ) f ( Θ Ψ) ks + k ( N. Ψ) d The Blinn-Phong Model L H Half-Vecto Specula V n ( N. H ) f ( Θ Ψ) ks + k ( N. Ψ) d

19 Phong: Reality Check Phong model Physics-based model Computationally simple, visually pleasing Doesn t epesent physical eality Enegy not conseved Not ecipocal Maimum always in specula diection The Modified Blinn-Phong Model n f ( Θ Ψ) ks ( N. H ) + k d

20 Real photogaphs Phong: Reality Check Phong model Theefoe, physically-based models Cook-Toance BRDF Model A micofacet model Suface modeled as andom collection of plana facets Incoming ay hits eactly one facet, at andom Input: pobability distibution of facet angle

21 Facet Reflection H vecto used to define facets that contibute N α H L θ θ V Cook-Toance BRDF Model R s Fesnel Reflectance F DG π ( N L)( N V ) Specula tem (eally diectional diffuse) Fesnel eflectance fo smooth facet

22 Facet Distibution D function descibes distibution of H Fomula due to Beckmann deivation based on Gaussian height distibution D tanα 1 m e m cos 4 α Masking and Shadowing G ( N H )( N min 1, ( V H ) V ) ( N H )( N, ( V H ) L )

23 Rob Cook s vases Souce: Cook, Toance 1981 Empiical BRDF Repesentation Genealized Phong model (Lafotune 1997) Used to epesent: Measued data Wave optics eflectance model Featues: Efficient and compact Easily added to endeing algoithms

24 Wad Model Physically valid Enegy conseving Satisfies ecipocity: f Θi Θ ) f ( Θ Based on empiical data Isotopic and anisotopic mateials ( Θi ) Wad Model: Isotopic f s tan θ h ep( ) 1 ρ α s 4πα ( N V )( N L) whee, α is suface oughness

25 Wad Model: Anisotopic whee, α, α y ae suface oughness in ae mutually pependicula to the nomal ) 1 ˆ ˆ ep( 4 1 N H y H H V N L N f y y s s + + α α α πα ρ y ˆ ˆ, y ˆ ˆ,