Distribution Ray Tracing

Transcription

1 Dstbuton Ray Tacng In Whtte Ray Tacng we compute lghtng vey cuely Phong + specula global lghtng In Dstbute Ray Tacng we want to compute the lghtng as accuately as possble Use the fomalsm of Raomety Compute aance at each pxel (by ntegatng all the ncomng lght Snce ntegals ae can not be one analytcally we wll employ numec appoxmatons

2 Benefts of Dstbuton Ray Tacng Bette global ffuse lghtng Colo bleeng Bouncng hghlghts Extene lght souces Ant-alasng Moton blu Depth of fel Subsuface scatteng

3 Raance at a Pont Recall that aance (shang at a suface pont s gven by ρ ω n p L p L ( ( ( ( = If we paameteze ectons n sphecal coonates an assume small ffeental sol angle we get ρ ω n p L p L e e ( ( ( ( Ω an assume small ffeental sol angle we get ( ( ( φ φ φ φ ρ π φ π n L p p L e e = ] 02 [ ] [02 sn ( ( ( (

4 Raance at a Pont Recall that aance (shang at a suface pont s gven by ρ ω n p L p L ( ( ( ( = If we paameteze ectons n sphecal coonates an assume small ffeental sol angle we get ρ ω n p L p L e e ( ( ( ( Ω an assume small ffeental sol angle we get ( ( ( φ φ φ φ ρ π φ π n L p p L e e = ] 02 [ ] [02 sn ( ( ( ( I l ll Integal s ove all ncomng ecton (hemsphee

5 Iaance at a Pxel To compute the colo of the pxel we nee to compute total lght enegy (flux passng though the pxel (ectangle (.e. we nee to compute the total aance at a pxel Φ j = H( α β α mn α α max β mn β β max αβ Integals s ove the extent of the pxel

6 Numecal Integaton (1D Case Remembe: ntegal s an aea une the cuve We can appoxmate any ntegal numecally as follows y x f (x D x N = 1 f ( x N D 0 f ( x x

7 Numecal Integaton (1D Case Remembe: ntegal s an aea une the cuve We can appoxmate any ntegal numecally as follows y x = D N f (x D x D N f ( x x 0 = 1 D N f ( x

8 Numecal Integaton (1D Case Poblem: what f we ae eally unlucky an ou sgnal has the same stuctue as samplng? y x f (x D x D N f ( x x 0 = 1 D N f ( x

9 Monte Calo Integaton Iea: anomze ponts x to avo stuctue nose (e.g. ue to peoc textue y x f (x D x Daw N anom samples x nepenently fom unfom stbuton b t Q(x=U[0D] (.e. Q(x = 1/D s the unfom pobablty ensty functon Then appoxmaton to the ntegal becomes 1 1 w x f ( f ( x x fo w = N Q( x We can also use othe Q s fo effcency!!! (a.k.a. mpotance samplng

10 Monte Calo Integaton y x f (x D x Then appoxmaton to the ntegal becomes 1 1 w x f ( f ( x x fo w = N Q( x We can also use othe Q s fo effcency!!! (a.k.a. mpotance samplng

11 Statfe Samplng Iea: combnaton of unfom samplng plus anom jtte Beak oman nto T ntevals of wths t an N t samples n nteval t y t f (x Integal appoxmate usng the followng: T 1 N N t= 1 t j= 1 t t f ( x t j D x

12 Statfe Samplng If ntevals ae unfom t = D/T an thee ae same numbe of samples n each nteval N t = N/T then ths T Nt appoxmaton euces to: t= 1 j= 1 The nteval sze an the # of samples can vay!!! D f N ( x t j y t f (x D Integal appoxmate usng the followng: T 1 N N t= 1 t j= 1 t t f ( x t j x

13 Back to Dstbuton Ray Tacng Base on one of the appoxmate ntegaton appoaches we nee to compute Let s ty unfom samplng ( ( ( φ φ φ φ ρ π φ π n L p p L e e = ] 2 0 [ ] 2 [0 sn ( ( ( ( π φ π ] 02 [ ] [02 ( ( ( φ φ φ φ ρ Δ Δ = = sn ( ( ( 1 1 M m N n n m n m n m e n L p = = 1 1 m n π 2 / Δ Δ 1 n whee N M π φ 2 = Δ = Δ φ φ Δ = Δ = m n m n N 2 mpont of the nteval (sample pont Inteval wth

14 Impotance Samplng n Dstbuton Ray Tacng Poblem: Unfom samplng s too expensve (e.g. 100 samples/hemsphee wth epth of ay ecuson of 4 => =10 8 samples pe pxel wth 10 5 pxels =>10 15 samples Soluton: Sample moe ensely (usng mpotance samplng whee we know that effects wll be most sgnfcant Decton towa pont o extene lght souce ae sgnfcant Specula an off-axs specula ae sgnfcant Textue/lghtness gaents ae sgnfcant Sample less wth geate epth of ecuson

15 Impotance Samplng Iea: anomze ponts x to avo stuctue nose (e.g. ue to peoc textue y x f (x 1 N w f ( x f ( x x fo w = 1 Q( x

16 Shaows n Ray Tacng Recall we shoot a ay towas a lght souce an see f t s ntecepte c k = j n k p k l no shaow ays Images fom the sles by Duan an Cutle one shaow ay

17 Ant-alasng n Dstbuton Ray Tace Lets shoot multple ays fom the same pont an attenuate the colo base on how many ays ae ntecepte c k = j n k p k Same woks fo ant-alasng of Textues!!! l one shaow ay Images fom the sles by Duan an Cutle w/ ant-alasng

18 Ant-alasng by Detemnstc Integaton Iea: Use multple ays fo evey pxel Algothm Subve pxel (j nto squaes Cast ay though squae centes Aveage the obtane lght Susceptble to stuctue nose epeatng textues

19 Ant-alasng by Monte Calo Integaton Iea: Use multple ays fo evey pxel Algothm Ranomly sample pont nse the pxel (j Cast ay though squae centes Aveage the obtane lght Does not suffe fom stuctue nose epeatng p g textues

20 How many ays o you nee? 1 ay/lght 10 ay/lght 20 ay/lght 50 ay/lght Images taken fom

21 Soft Shaows wth Dstbuton Ray Tacng Lets shoot multple ays fom the same pont an attenuate the colo base on how many ays ae ntecepte c k = j p k n n k one shaow ay Images fom the sles by Duan an Cutle lots of shaow ays

22 Antalasng Supesamplng jagges w/ antalasng pont lght aea lght Images fom the sles by Duan an Cutle

23 Specula Reflectons Recall we ha to shoot a ay n a pefect specula eflecton ecton (wth espect to the camea an get the aance at the esultng ht pont k c k = j = 2( s n n k k k p k n k s k s k m s = 2( c n n k k k c k

24 Specula Reflectons wth DRT Same but shoot multple ays k c k = j = 2( s n n k k k p k n k s k s k Spea s ctate by BRDF Pefect Reflectons (Metal Pefect Reflectons (glossy polshe suface Justn Legaks

25 Depth of Fel So fa wth ou Ray Taces we only consee pnhole camea moel (no lens o altenatvely t l lens but tny apetue Image Plane Lens optcal axs

26 Depth of Fel So fa wth ou Ray Taces we only consee pnhole camea moel (no lens o altenatvely t l lens but tny apetue What happens f we put a lens nto ou camea o ncease the apetue e Remembe the thn lens equaton? Image Plane Lens f = 1 z + 1 z optcal axs z z 0 1

27 Depth of Fel So fa wth ou Ray Taces we only consee pnhole camea moel (no lens o altenatvely t l lens but tny apetue What happens f we put a lens nto ou camea o ncease the apetue e Remembe the thn lens equaton? Image Plane Lens f = 1 z + 1 z optcal axs z z 0 1

28 Changng the focal-length n DRT nceasng focal length optcal axs 220x400 pxels 144 samples pe pxel ~4.5 mnutes to ene z z 0 1

29 Changng the apetue n DRT eceasng apetue optcal axs 220x400 pxels 144 samples pe pxel ~4.5 mnutes to ene z z 0 1

30 Depth of Fel

31 Depth of Fel

32 Depth of Fel

33 Depth of Fel

34 Depth of Fel

35 Camea Shutte We gnoe the fact that t takes tme to fom the mage We gnoe ths fo aomety Dung that tme the shutte s open an lght s collecte We nee to ntegate tempoally not only spatally t αβ H( α β t αβt

36 Moton Blu

37 Moton Blu

38 Moton Blu (long exposues

39 Moton Blu (shot exposues

40 Sub-suface Scatteng

41 Sub-suface Scatteng Bectonal Suface Scatteng Reflectance Dstbuton Functon

42 Bectonal Suface Scatteng Reflectance Dstbuton Functon [Images taken fom Wkpea]

43 Sem-Tanspaences Image fom

44 Caustcs Ha to o n Dstbuton Ray Tacng Why?

45 Caustcs Ha to o n Dstbuton Ray Tacng Why? Ha to come up wth a goo mpotance functon fo samplng Hence VERY VERY slow

46 Caustcs Often one usng b-ectonal ay tacng (a.k.a. photon mappng Shoot lght ays fom lght souces Accumulate the amount of lght (aance at each suface Shoot ays though mage plane pxels to look-up the aance (an ntegate t aance ove the aea of the pxel

47 Photon Mappng Smulates nvual photons In DTR we wee smulatng aance (flux Photons ae emtte fom lght souces Photons bounce off of specula sufaces Photons ae eposte on ffuse sufaces Hel n a 3-D spatal ata stuctue Sufaces nee not be paameteze Photons collecte by ay tacng fom eye

48 Photons A photon s a patcle of lght that caes flux whch s encoe as follows magntue (n Watts an colo of the flux t caes stoe as an RGB tple locaton of the photon (on a ffuse suface the ncent ecton (use to compute aance Example (pont lght souce photons emtte unfomly Powe of souce (n Watts stbute evenly among photons Flux of each photon equal to souce powe ve by total # of photons 60W lght bulb woul senng 100 photons wll esult n 0.6 W pe photon

49 How oes ths actually wok? Specal ata stuctues ae eque to o fast look-up (KD-tees

50 Photon Mappng Results Raance estmate usng 50 photons Raance estmate usng 500 photons