Fundamentals of Rendering - Reflectance Functions

Transcription

1 Fundamentals of Rendering - Reflectance Functions Image Synthesis Torsten Möller Mike Phillips

2 Reading Chapter 8 of Physically Based Rendering by Pharr&Humphreys Chapter 16 in Foley, van Dam et al. Chapter 15 in Glassner 2

3 Components of Photorealistic Rendering Transport of light Transport problem: photons are transported from light sources through the scene Global problem: each point might illuminate any other point Interaction between light and matter Computation of visibility Result: incident light at each point Scene description Geometry: surfaces, volumes Light sources (see Chapter 12) Description of surface properties, reflection properties (see Chapter 5) File formats: RenderMan, DXF, VRML, 3DS, etc. 3

4 Generic Framework Acquisition Light Transport Visual Display Observer Modeling Measure- ments Global Illum. Algorithms Tone Mapping Operators Human Perception Geometry BRDFs Lights Textures Radiometric Values Image [Greenberg et al. 97] 4

5 Surface Reflectance Measured data Gonioreflectometer (See the Cornell Lab) Phenomenological models Intuitive parameters Most of graphics Simulation Know composition of some materials simulate complicated reflection from simple basis Physical (wave) optics Using Maxwell s equations Computationally expensive Geometric optics Use of geometric surface properties 5

6 BRDF Measurement Sensor Gonioreflectometer Dense sampling of pairs of directions Large table of measurement data Two different principal configurations: Sensor Light source Light sourc e Sample Sample 6 [Images: M. Goesele, Minicourse at IMPA, 2005]

7 BRDF Measurement cont. [Image: Cornell Lab] 7

8 Surface Reflectance Diffuse Scatter light equally in all directions E.g. dull chalkboards, matte paint Glossy specular Preferred set of direction around reflected direction E.g. plastic, high-gloss paint Perfect specular E.g. mirror, glass Retro-reflective E.g. velvet or earth s moon 8

9 Types of Surface Reflectance Types represented by different classes of BRDFs Reflection at (vicinity of) surface Polished surface: specular Rough microstructure: more diffuse character Also reflection within material Diffuse Translucency (see Chapter 4) 9

10 Surface Reflectance Iso-tropic vs. anisotropic If you turn an object around a point -> does the shading change? 10

11 Surface Properties Reflected radiance is proportional to incoming flux and to irradiance (incident power per unit area). dl o ( p,ω o ) de( p,ω i ) 11

12 The BSDF Bidirectional Scattering Distribution Function: f r p,ω o,ω i ( ) Measures portion of incident irradiance (E i ) that is reflected as radiance (L o ) f r ( p,ω o,ω ) i = dl ( p,ω ) o o de( p,ω i ) Or the ratio between incident radiance (L i ) and reflected radiance (L o ) f r ( p,ω o,ω ) i = dl o( p,ω o ) de( p,ω i ) = dl o ( p,ω o ) L i ( p,ω i )cosθ i dω i 12

13 The BRDF and the BTDF Bidirectional Reflectance Distribution Function (BRDF) Describes distribution of reflected light Bidirectional Transmittance Distribution Function (BTDF) Describes distribution of transmitted light BSDF = BRDF + BTDF 13

14 Illumination via the BxDF The Reflectance Equation L o ( p,ω o ) = f ( r p,ω o,ω ) i L S 2 i ( p,ω i )cosθ i dω i The reflected radiance is the sum of the incident radiance over the entire (hemi)sphere foreshortened scaled by the BxDF 14

15 Parameterizations 6-D BRDF f r (p, ω o, ω i ) Incident direction ωi Reflected/Outgoing direction ωo Surface position p: textured BxDF 4-D BRDF f r (ω o, ω i ) Homogeneous material Anisotropic, depends on incoming azimuth e.g. hair, brushed metal, ornaments 15

16 Parameterizations 3-D BRDF f r (θ o, θ i, φ o φ i ) Isotropic, independent of incoming azimuth e.g. Phong highlight 1-D BRDF f r (θ i ) Perfectly diffuse e.g. Lambertian 16

17 BxDF Property 0 Ranges from 0 to (strictly positive) Infinite when incoming radiance distribution is from single incident ray f r ( p,ω o,ω ) i = dl o( p,ω o ) de( p,ω i ) = dl o ( p,ω o ) L i ( p,ω i )cosθ i dω i 17

18 BRDF Property 1 Linearity of functions Sillion, Arvo, Westin, Greenberg, A Global Illumination Solution for General Reflectance Distributions" 18

19 BRDF Property 2 Helmholtz Reciprocity f r (ω o, ω i ) = f r (ω i, ω o ) Materials are not a one-way street Incoming to outgoing pathway same as outgoing to incoming pathway 19

20 BRDF Property 3 Isotropic vs. anisotropic f r (θ i, φ i, θ o, φ o ) = f r (θ o, θ i, φ o φ i ) Reciprocity and isotropy f r (θ o, θ i, φ o φ i ) = f r (θ i, θ o, φ i φ o ) = f r (θ o, θ i, φ o φ i ) 20

21 BRDF Property 4 Conservation of Energy Materials must not add energy (except for lights) Materials must absorb some amount of energy When integrated, must add to less than one Reflected light Incoming light Surface 21

22 Blinn Phong (Blinn) Phong is not energy conserving Energy conserving Blinn Phong: Diffuse: k d L d (n l)/π Specular: ksis((n+8)/8π)(e H) n 22

23 Blinn Phong 23

24 Reflectance ratio of reflected to incident flux Is between 0 and 1 (p) = d o(p) d i (p) = Reflectance f r (p, p, i )L i (p, i ) cos i cos o d i d o L i (p, i ) cos i d i 3x3 set of possibilities for Ω: { dω i,ω i,h i } { dω o,ω o,h o } If L is isotropic and uniform there is a clear relationship between ρ and Φ. 24

25 Reflectance Hemispherical-directional reflectance Reflection in a given direction due to constant illumination over a hemisphere Total reflection over hemisphere due to light from a given direction (reciprocity) Also called albedo - incoming photon is reflected with probability less than one ρ hd ( p,ω o ) = 1 π H 2 (n ) f r ( p,ω o,ω i )cosθ i dω i 25

26 Reflectance Hemispherical-hemispherical reflectance Constant spectral value that gives the fraction of incident light reflected by a surface when the incident light is the same from all directions ρ hd ( p) = 1 π H 2 (n ) H 2 (n ) f r (p,ω o,ω i )cosθ o cosθ i dω o dω i 26

27 Representations Tabulated BRDF s Require dense sampling and interpolation scheme Factorization Into two 2D functions for data reduction (often after reparameterization) Basis Functions (Spherical Harmonics) Loss of quality for high frequencies Analytical Models Rough approximation only Very compact Most often represented as parametric equation (Phong, Cook-Torrance, etc.) 27

28 Construction of BRDF BRDF models for ideal reflection types Ideal diffuse Ideal specular Phenomenological and ad-hoc models Phong and Blinn-Phong Ward, Lafortune, Ashikhmin, Schlick, etc. Often with a few parameters that are fitted to measurements Physics-based local illumination models Simulation of surface and electrodynamics Torrance-Sparrow (Cook-Torrance) Directly from measured data Gonioreflectometer How to represent large amounts of data efficiently? Spherical harmonics, separable representations 28

29 Law of Reflection Angle of reflectance = angle of incidence I N R + I ( ) = 2cosθN = 2 I N R = I 2( I N)N ( ) ω r = R ω i,n R ϕ i ϕ r θ i θ r ( )N p θ r = θ i ϕ r = ϕ i + π ( )mod2π 29

30 Polished Metal 30

31 Ideal Reflection (Mirror) BRDF cast as a delta function ( ) ( ) L i θ i,ϕ i θ i θ r L r θ r,ϕ r L r ( r, r ) = L i ( r, r ± ) f r ( i, i, r, r ) = (cos i cos r ) cos i ( i r ± ) L r ( r, r ) = p f r ( i, i, r, r )L i ( i, i ) cos i d cos i d i (cos i cos r ) = ( i r ± )L i ( i, i ) cos i d cos i d i cos i = L i ( r, r ± ) 31

32 η i, η t indices of refraction (ratio of speed of light in vacuum to the speed of light in the medium) I Snell s Law η i sinθ i = η t sinθ t η i N I = η t N T N ω t = T( ω i,n) θ i ϕ i ϕ r p θ t T ϕ r = ϕ i + π ( )mod2π 32

33 Law of Refraction Starting at Snell s law: We conclude that Assuming a normalized T: Solving this quadratic equation: Leads to the total reflection condition: η i η t N I = N T ( ) = 0 N T µi T = µi + γn ( ) T 2 =1= µ 2 + γ 2 + 2µγ I N γ = µ I N ( ) ± 1 µ 2 1 ( I N) 2 = µcosθ i ± 1 µ 2 sin 2 θ i ( ) 1 µ 2 ( 1 ( I N) 2 ) 0 = µcosθ i ± cosθ o 33

34 Optical Manhole Total Internal Reflection For water η w = 4/3 Livingston and Lynch 34

35 Fresnel Reflection At top layer interface Some light is reflected, Remainder is transmitted through Simple ray-tracers: just given as a constant Physically based - relative contrib. depends on incident angle Polarization of light (dispersion effects) wavelength Solution of Maxwell s equations to smooth surfaces Dielectrics vs. metals (conductors) 35

36 Fresnel Reflection - Dielectrics Objects that don t conduct electricity (e.g. glass) Fresnel term F for a dielectric is proportion of reflection (e.g. glass, plastic) grazing angles: 100% reflected (almost mirror-like) normal angles: 5% reflected (see the material well!) 36

37 Fresnel Reflection - Dielectrics Polarized light: r = η t cosθ i η i cosθ t η t cosθ i + η i cosθ t r = η i cosθ i η t cosθ t η i cosθ i + η t cosθ t Where ω t is computed according to Snell s law F Unpolarized light: ( r ω ) i = 1 2 r r F t ω i ( ) ( ) ( ) = 1 F ( r ω ) i 37

38 Fresnel Reflection - Dielectrics 38

39 Fresnel Reflection - Conductor Typically metals No transmission Absorption coefficient k 39

40 Fresnel Reflection - Conductor Polarized light: ( r 2 = η2 + k 2 )cos 2 θ i 2ηcosθ i +1 ( η 2 + k 2 )cos 2 θ i + 2ηcosθ i +1 ( ) 2ηcosθ i + cos 2 θ i ( η 2 + k 2 ) + 2ηcosθ i + cos 2 θ i r 2 = η2 + k 2 Unpolarized light: F r ω i ( ) = 1 2 r ( r ) 40

41 Fresnel Reflection - Conductor How to determine k or η? Measure F r for θ i =0 (normal angle) 1. Assume k = 0 r 2 = r 2 = η 1 η Assume η = 1 ( ) 2 ( ) 2 r 2 = r 2 = k 2 k η = 1+ F ( r 0) 1 F ( r 0) k = 2 F r 0 ( ) 1 F r 0 ( ) 41

42 Fresnel Normal (Dielectric) 10% reflected Air 90% transmitted Material 42

43 Fresnel Grazing (Dielectric) 90% reflected Air 10% transmitted Material 43

44 Fresnel Mid (Dielectric) 60% reflected Air 40% transmitted Material 44

45 Fresnel Reflection Conductor (Aluminum) Dielectric (N=1.5) Schlick Approximation: F( θ) = F( 0) + 1 F( 0) ( ) 1 cosθ ( ) 5 45

46 Fresnel Reflection 46

47 Fresnel Reflection Example - Copper color shift as θ goes from 0 to π/2 at grazing, specular highlight is color of light π /2 θ λ Measured Reflectance Approximated Reflectance 47

48 Fresnel Reflection Cook, Robert L. & Torrance, Kenneth E. (1982): A Reflectance Model for Computer Graphics 48

49 Ideal Specular - Summary Reflection: f r Transmission: ( ) ( p,ω i,ω o ) = F r ( ω i ) δ ω i R( ω o,n) cosθ i f t ( ) = η 2 o p,ω i,ω o η i 2 ( ) ( 1 F r( ω i )) δ ω o T( ω i,n) cosθ i 49

50 Ideal Diffuse Reflection Uniform Sends equal amounts of light in all directions Amount depends on angle of incidence Perfect all incoming light reflected no absorption f r (ω i,ω o ) k d 50

51 L o,d (ω o ) = M = Ideal Diffuse Reflection Ω f r,d (ω i,ω r )L i (ω i )cosθ i dω i = f r,d L i (ω i )cosθ i dω i Ω = f r,d E Ω L o,d (ω o )cosθ o dω o = L o,d cosθ o dω o Ω = L o,d π Lamberts Cosine Law: M = ρ d E = ρ d E s cosθ s ρ d = M E = L o,dπ E f r,d = ρ d π = f r,d Eπ E = f r,d π 51

52 Diffuse Helmholtz reciprocity? Energy preserving? ρ d 1 f r,d = ρ d π 1 π 52

53 Reflectance Models Ideal Diffuse Specular Ad-hoc: Phong Classical / Blinn Modified Ward Lafortune Microfacets (Physically-based) Torrance-Sparrow (Cook-Torrance) Ashikhmin 53

54 Classical Phong Model L o ( p,ω o ) = ( k ( d N ω ) i + k ( s R( ω o,n) ω ) e ) L ( p,ω ) i i i Where 0<k d, k s <1 and e>0 Cast as a BRDF: ( ) e ( ) ω i f r (p,ω i,ω o ) = k d + k s R ω o,n ( N ω ) i Not reciprocal Not energy-preserving Specifically, too reflective at glancing angles, but not specular enough But cosine lobe itself symmetrical in ω i and ω o 54

55 Blinn-Phong Like classical Phong, but based on half-way vector H( ω o,ω i ) N f r (p,ω i,ω o ) = k d + k s ( N ω i ) ω h = H( ω o,ω i ) = norm( ω o + ω i ) ( ) e Implemented in OpenGL Not reciprocal Not energy-preserving Specifically, too reflective at glancing angles, but not specular enough But cosine lobe itself symmetrical in ω i and ω o 55

56 Phong Illumination Model cont. BRDF for Phong illumination 56

57 Phong Illumination Model cont. Illustration of ambient, diffuse, and specular components Different angles of incident light 57

58 Modified Phong ( ) f r (p,ω i,ω o ) = k d π + k s e + 2 2π ( R( ω o,n) ω ) e i For energy conservation: k d + k s < 1 (sufficient, not necessary) Peak gets higher as it gets sharper, but same total reflectivity 58

59 Ward-Phong Based on Gaussians f r (p,ω i,ω o ) = k d π + k s cosθ i cosθ o exp tan2 ω h α 2 4πα 2 α: surface roughness, or blur in specular component. 59

60 Ward Model Properties Can be extended to an anisotropic model Phenomenological model, but pretty close to experimental measurements Conservation of energy Can be inverted analytically 60

61 Lafortune Model Phong cosine lobes symmetrical (reciprocal), easy to compute Add more lobes in order to match with measured BRDF How to generalize to anisotropic BRDFs? weight dot product: f r (p,ω i,ω o ) = k d π + nlobes i=1 ω o R iω i ( ) e i 61

62 Physically-based Models Some basic principles common to many models: Fresnel effect Surface self-shadowing Microfacets Different physical models required for different kinds of materials Some kinds of materials don t have good models Remember that BRDF makes approximation of completely local surface reflectance! To really model well how surfaces reflect light, need to eventually move beyond BRDF 62

63 Cook-Torrance Model Based in part on the earlier Torrance-Sparrow model, but adds wavelength dependence of Fresnel term Neglects multiple scattering f r (p,ω i,ω o ) = F r( ω h )D( ω h )G( ω o,ω i ) 4 cosθ i cosθ o D - Microfacet Distribution Function how many cracks do we have that point in our (viewing) direction? G - Geometrical Attenuation Factor light gets obscured by other bumps F - Fresnel Term 63

64 Microfacet Models Microscopically rough surface Specular facets oriented randomly measure of scattering due to variation in angle of microfacets a statistic approximation, I.e. need a statistic distribution function 64

65 Rough Surface Diffuse Scattering Reduced Specular 65

66 Microfacet Distribution Function D Blinn (Phong) D( h ) = cos c h, where c = ln 2 ln cos D( h ) = e (c h) 2, where c = 2 Torrance-Sparrow (Gaussian) where m is the root mean square (standard deviation) slope of the facets (as an angle) beta is angle where distribution is one half of the total (0.5) 66

67 Microfacet Distribution Function D Beckmann (most effective) 1 D( h ) = m 2 cos 2 Represents a distribution of slopes But θ h = tanθ h for small θ h h e ( tan h m )2 67

68 Microfacet Distribution Function D 68

69 Multiscale Distribution Function May want to model multiple scales of roughness: D( ω ) h = w j D ( j ω ) h w j =1 j j Bumps on bumps 69

70 Self-Shadowing (V-Groove Model) Geometrical Attenuation Factor G how much are the cracks obstructing themselves? No interference G = min 1, 2 N ω h ω o ω h ( )( N ω ) o ( ) ( )( N ω ) i ( ), 2 N ω h ω o ω h 70

71 Cook-Torrance - Summary 71

72 Cook-Torrance: Examples Reflections on metal surfaces 72

73 Cook-Torrance - Summary 73

74 Ashikhmin Model Modern Phong Ad-hoc, but: Physically plausible Anisotropic Good for both Monte-Carlo and HW implementation 74

75 Ashikhmin Model Weighted sum of diffuse and specular part: f r (p,ω i,ω o ) = k d 1 k s ( ) f d (p,ω i,ω o ) + k s f s (p,ω i,ω o ) Dependence of diffuse weight on k s decreases diffuse reflectance when specular reflectance is large Specular part f s not an impulse, really just glossy Diffuse part f d not constant: energy specularly reflected cannot be diffusely reflected For metals, f d = 0 75

76 Ashikhmin Model k s : Spectrum or colour of specular reflectance at normal incidence. k d : Spectrum or colour of diffuse reflectance (away from the specular peak). q u, q v : Exponents to control shape of specular peak. Similar effects to Blinn-Phong model If an isotropic model is desired, use single value q A larger value gives a sharper peak Anisotropic model requires two tangent vectors u and v The value q u controls sharpness in the direction of u The value q v controls sharpness in the direction of v 76

77 Ashikhmin Model φ is the angle between u and ω h D( ω ) = ( q +1) ( q +1) ( ω N) ( q u cos2 φ +q v sin 2 φ) h u v h 77

78 Ashikhmin Model Diffuse term given by: f d (p,ω i,ω o ) = 28 23π 1 1 ω o N ( ( ( )) 5 ) 1 1 ( ω i N) ( ( ) 5 ) Leading constant chosen to ensure energy conservation Form comes from Schlick approximation to Fresnel factor Diffuse reflection due to subsurface scattering: once in, once out 78

79 Complex BRDF Combination of the three

80 BRDF Illustrations Phong Illumination Oren-Nayar 80

81 BRDF Illustrations Cook-Torrance-Sparrow BRDF Hapke BRDF 81

82 BRDF illustrations lumber cement 82

83 BRDF illustrations Surface microstructure 83

84 bv = Brdf Viewer Diffuse Torrance-Sparrow Anisotropic Szymon Rusinkiewicz, Princeton U. 84

85 Other Local Illumination Models HTSG He, Torrance, Sillion, Greenberg Physically most accurate model Wave-optics and diffraction Difficult to implement Difficult choice of parameters General anisotropic illumination Brushed metal Velvet General description of wave effects and diffraction CDs Thin-film effects (e.g. oil on water surface) 85

86 BRDF cannot Spatial variation of reflectance 86

87 BRDF cannot Transparency and Translucency (depth) Glass: transparent Wax: translucent BTDF Opaque milk (rendered) BSSRDF Translucent milk (rendered) 87