Computer graphics III Light reflection, BRDF. Jaroslav Křivánek, MFF UK

Transcription

1 Cmputer graphcs III Lght reflectn, BRDF Jarslav Křvánek, MFF UK

2 Basc radmetrc quanttes Image: Wjcech Jarsz

3 Interactn f lght wth a surface Absrptn Reflectn Transmssn / refractn Reflectve prpertes f materals determne the relatn f reflected radance L r t ncmng radance L, and therefre the appearance f the bject: clr, glssness, etc.

4 Surce: MERL BRDF database Interactn f lght wth a surface Same llumnatn Dfferent materals

5 BRDF Frmal defntn Bdrectnal Reflectance Dstrbutn Functn utgng n L (w ) L r (w ) dw reflected q q ncmng f r ( w w ) L dlr ( w) ( w ) csq dw [sr 1 ] 5

6 BRDF Mathematcal mdel f the reflectn prpertes f a surface Intutn Value f a BRDF = prbablty densty, descrbng the event that a lght energy packet, r phtn, cmng frm drectn w gets reflected t the drectn w. f r Range: ( w ) 0, w

7 Westn et al. Predctng Reflectance Functns frm Cmplex Surfaces, SIGGRAPH BRDF The BRDF s a mdel f the bulk behavr f lght n the mcrstructure when vewed frm dstance

8 BRDF prpertes Helmhlz recprcty (always hlds n nature, a physcally-plausble BRDF mdel must fllw t) f r ( w w) fr ( w w )

9 BRDF prpertes Energy cnservatn A patch f surface cannt reflect mre lght energy than t receves

10 BRDF (an)strpy Istrpc BRDF = nvarant t a rtatn arund surface nrmal,,, ;,, ;, q q q q q q r r r f f f

11 Surfaces wth anstrpc BRDF

12 Anstrpc BRDF Dfferent mcrscpc rughness n dfferent drectns (brushed metals, fabrcs, )

13 Istrpc vs. anstrpc BRDF Istrpc BRDFs have nly 3 degrees f freedm Instead f and t s enugh t cnsder nly D But ths s nt enugh t descrbe an anstrpc BRDF Descrptn f an anstrpc BRDF and are expressed n a lcal crdnate frame (U, V, N) U tangent e.g. the drectn f brushng V bnrmal N surface nrmal the Z axs f the lcal crdnate frame

14 Reflectn equatn A.k.a. reflectance equatn, llumnatn ntegral, OVTIGRE ( utgng, vacuum, tme-nvarant, gray radance equatn ) Hw much ttal lght gets reflected n the drectn w? Frm the defntn f the BRDF, we have dl r ( w ) f r ( w w ) L ( w ) csq dw

15 Reflectn equatn Ttal reflected radance: ntegrate cntrbutns f ncdent radance, weghted by the BRDF, ver the hemsphere ) ( r d cs ) ( ) ( ) ( x H f r L L w q w w w w upper hemsphere ver x =

16 Reflectn equatn Evaluatng the reflectance equatn renders mages!!! Drect llumnatn Envrnment maps Area lght surces etc.

17 Energy cnservatn Mre rgrus Reflected flux per unt area (.e. radsty B) cannt be larger than the ncmng flux per unt surface area (.e. rradance E). 1 )cs ( cs )cs ( ) ( )cs ( )cs ( r r d L d d L f d L d L E B w q w w q w q w w w w q w w q w

18 Surface appearance and the BRDF Appearance BRDF lbe (fr fur dfferent vewng drectns) Suce: Ngan et al. Expermental analyss f BRDF mdels, 18

19 Surface appearance and the BRDF Appearance BRDF lbe (fr fur dfferent vewng drectns) Suce: Ngan et al. Expermental analyss f BRDF mdels, 19

20 Surface appearance and the BRDF Appearance BRDF lbe (fr fur dfferent vewng drectns) Suce: Ngan et al. Expermental analyss f BRDF mdels, 20

21 Surface appearance and the BRDF Appearance BRDF lbe (fr fur dfferent vewng drectns) Suce: Ngan et al. Expermental analyss f BRDF mdels, 21

22 Reflectance Rat f the ncmng and utgng flux A.k.a. albed (used mstly fr dffuse reflectn) Hemsphercal-hemsphercal reflectance See the Energy cnservatn slde Hemsphercal-drectnal reflectance The amunt f lght that gets reflected n drectn w when llumnated by the unt, unfrm ncmng radance. ( w ) a( w ) H ( x) f r ( w w ) csq dw

23 Hemsphercal-drectnal reflectance Nnnegatve Less than r equal t 1 (energy cnservatn) ( w ) 0,1 Equal t drectnal-hemsphercal reflectance What s the percentage f the energy cmng frm the ncmng drectn w that gets reflected (t any drectn)? Equalty fllws frm the Helmhlz recprcty

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26 BRDF cmpnents General BRDF Ideal dffuse (Lambertan) Ideal specular Glssy, drectnal dffuse

27 Ideal dffuse reflectn

28 Ideal dffuse reflectn

29 Ideal dffuse reflectn A.k.a. Lambertan reflectn Jhann Henrch Lambert, Phtmetra, Pstulate: Lght gets reflected t all drectns wth the same prbablty, rrespectve f the drectn t came frm The crrespndng BRDF s a cnstant functn (ndependent f w, w ) f r, d ( w w) fr, d

30 Ideal dffuse reflectn Reflectn n a Lambertan surface: L ( w ) f f r, d Vew ndependent appearance E L r, d H ( x) ( w ) csq dw rradance Outgng radance L s ndependent f w Reflectance (derve) d f r, d

31 Ideal dffuse reflectn Mathematcal dealzatn that des nt exst n nature The actual behavr f natural materals devates frm the Lambertan assumptn especally fr grazng ncdence angles

32 Whte-ut cndtns Under a cvered sky we cannt tell the shape f a terran cvered by snw We d nt have ths prblem clse t a lcalzed lght surce. Why?

33 Whte-ut cndtns We assume sky radance ndependent f drectn (cvered sky) L ( x, w L ) We als assume Lambertan reflectn n snw sky Reflected radance gven by: snw snw L d L sky Whte-ut!!!

34 Ideal mrrr reflectn

35 Ideal mrrr reflectn

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37 Nshn, Nayar: Eyes fr Relghtng, SIGGRAPH 2004

38 The law f reflectn n q q q q + md 2 Drectn f the reflected ray (derve the frmula) w 2( w n) n w

39 Image: Wkpeda Dgressn: Drac delta dstrbutn Defntn (nfrmal): The fllwng hlds fr any f: Delta dstrbutn s nt a functn (therwse the ntegrals wuld = 0)

40 BRDF f the deal mrrr BRDF f the deal mrrr s a Drac delta dstrbutn ), ( ) ( ), ( r q q q L R L, cs ) ( ) cs (cs ) ( ), ;, ( q q q q q q R f m r q n q q q Fresnel reflectance (see belw) We want:

41 BRDF f the deal mrrr BRDF f the deal mrrr s a Drac delta dstrbutn Verfcatn: ), ( ) ( cs ), ( cs ) ( ) cs (cs ) ( cs (.) (.) ), ( r r, r q q w q q q q q q w q q L R d L R d L f L m r

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43 Ideal refractn

44 Ideal refractn

45 Ideal refractn Index f refractn h Water 1.33, glass 1.6, damnd 2.4 Often depends n the wavelength Snell s law h snq h snq h w h w

46 Image: wkpeda Ideal refractn Drectn f the refracted ray: w h w h csq 1 h h h h (1 cs 2 2 q ) n f < 0, ttal nternal reflectn Crtcal angle: h q, c arcsn h

47 Ideal refractn Change f radance Fllws frm the cnservatn f energy (flux) When gng frm an ptcally rarer t a mre dense medum, lght energy gets cmpressed n drectns => hgher energy densty => hgher radance h L L h 2 2

48 BRDF f deal refractn BRDF f the deal refractn s a delta dstrbutn: Change f radance Fresnel transmttance Snell s law f t ( q, ; q, ) h h 2 2 (1 R( q )) ( h snq h snq ) ( csq ) Refracted ray stays n the ncdence plane

49 Fresnel equatns

50 Fresnel equatns Read [frenel] Rat f the transmtted and reflected lght depends n the ncdent drectn Frm abve mre transmssn Frm the sde mre reflectn Extremely mprtant fr realstc renderng f glass, water and ther smth delectrcs Nt t be cnfused wth Fresnel lenses (used n lghthuses)

51 Fresnel equatns Delectrcs Image: Wkpeda

52 Fresnel equatns Delectrcs

53 Fresnel equatns Frm the sde - lttle transmssn - mre reflectn Try fr yurself!!! Frm abve - lttle reflectn - mre transmssn

54 Fresnel equatns Metals

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57 Glssy reflectn

58 Glssy reflectn Nether deal dffuse nr deal mrrr All real materals n fact fall n ths categry

59 Surface rughness and blurred reflectns The rugher the blurrer Mcrscpc surface rughness

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61 BRDF mdels

62 BRDF mdelng BRDF s a mdel f the bulk behavr f lght when vewng a surface frm dstance BRDF mdels Emprcal Physcally based Apprxmatn f measured data (a.k.a mes-scale)

63 Emprcal BRDF mdels An arbtrary frmula that takes w and w as arguments w and w are smetmes dented L (Lght drectn) a V (Vewng drectn) Example: Phng mdel Arbtrary shadng calculatns (shaders)

64 Phng shadng mdel L N R V C I k ( N L) k ( V R) d s n R 2( N L) N L

65 Phng shadng mdel n the radmetrc ntatn w n r w Orgnal shadng mdel L ( w) L ( w ) k d csq k s cs csq w r r 2( nw ) n w r n q r BRDF f r L L csq f PhngOrg r k d k s n cs qr csq

66 Physcally-plausble Phng BRDF Mdfcatn t ensure recprcty (symmetry) and energy cnservatn Phngmdf r Energy cnserved when f d n 2 s cs 2 n q r d s 1 It s stll an emprcal frmula (.e. t des nt fllw frm physcal cnsderatns), but at least t fulflls the basc prpertes f a BRDF

67 Physcally-plausble BRDF mdels E.g. Trrance-Sparrw / Ck-Trrance mdel Based n the mcrfacet thery

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69 Mcrfacet BRDF Analytcally derved Used fr mdelng glssy surfaces (as the Phng mdel) Crrespnds mre clsely t realty than Phng Derved frm a physcal mdel f the surface mcrgemetry (as ppsed t because t lks gd - apprach used fr the Phng mdel)

70 Mcrfacet BRDF Assumes that the macrsurface cnssts f randmly rented mcrfacets We assume that each mcrfacet behaves as an deal mrrr. We cnsder 3 phenmena: Reflectn Maskng Shadwng

71 Mcrfacet BRDF Mcrfacet thery [Ck et Trrance 1982] A perfect mrrr Reflectn n a sngle drectn Outgng lght vsble surface nrmal algned wth the half vectr Half Vectr: H = L+V L+V Aggregatn f mcr-mrrrs (mcr-facets) Each mcr-mrrr have a mcr-nrmal Hw many mcr-mrrr have ther mcr-nrmal algned s that H = N? Statstcal dstrbutn: Nrmal Dstrbutn Functn (NDF) 71

72 Mcrfacet BRDF Fresnel term Gemetry term Mdels shadwng and maskng f F( q ) G( w, w r) D( q h) 4cs( q)cs( qr) Mcrfacet dstrbutn Part f the macrscpc surface vsble by the lght surce Part f the macrscpc surface vsble by the vewer

73 Apprxmatn f measured data We can ft any BRDF mdel t the data Sme BRDF mdels have been specfcally desgned fr the purpse f fttng measured data, e.g. Ward BRDF, Lafrtune BRDF Nnlnear ptmzatn requred t fnd the BRDF parameters

74 BRDF measurements Gn-reflectmeter UTIA Stanfrd Unversty f Bnn

75 Measured Materal Technques fr speedng measurements Mrrrs Objects cated by the materal: Sphere [Matusk et al 2003] Cylnders [Ngan et al 2005] [Matusk et al 2003] 10/24/2017 Effcent Representatn fr Measured Reflectance 75

76 Surface appearance and the BRDF Appearance BRDF lbe (fr fur dfferent vewng drectns) Suce: Ngan et al. Expermental analyss f BRDF mdels, 76

77 Surface appearance and the BRDF Appearance BRDF lbe (fr fur dfferent vewng drectns) Suce: Ngan et al. Expermental analyss f BRDF mdels, 77

78 Surface appearance and the BRDF Appearance BRDF lbe (fr fur dfferent vewng drectns) Suce: Ngan et al. Expermental analyss f BRDF mdels, 78

79 Surface appearance and the BRDF Appearance BRDF lbe (fr fur dfferent vewng drectns) Suce: Ngan et al. Expermental analyss f BRDF mdels, 79

80 BRDF, BTDF, BSDF: What s up wth all these abbrevatns? BTDF Bdrectnal transmttance dstrbutn functn Descrbed lght transmssn BSDF = BRDF+BTDF Bdrectnal scatterng dstrbutn functn

81 SBRDF, BTF SV-BRDF Spatally Varyng BRDF BRDF parameters are spatally varyng (can be gven by a surface texture) BTF Bdrectnal Texture Functn Used fr materals wth cmplex structure As ppsed t the BRDF, mdels even the mes-scale

82 BSSRDF BRDF Lght arrvng at a pnt s reflected/transmtted at the same pnt N subsurface scatterng cnsdered BSSRDF B-drectnal surface scatterng reflectance dstrbutn functn Takes nt accunt scatterng f lght under the surface

83 BSSRDF Sub-surface scatterng makes surfaces lks sfter BRDF BSSRDF

84 BSSRDF BRDF BSSRDF