Rendering Problem

1. RenderingProblem László Szirmay-Kalos

2. Image synthesis: illusion of watching real world objects monitor Tone mapping pixel S fr (’, x, ) We(x,)  Le(x,) Color perception  

3. Measuring the light: Flux • Power going through a boundary [Watt] • Number of photons • Spectral dependence: F [, +d] F ()

4. g() r() b() 400 500 600 700 Color perception   perception: r, g, b  r, g, b

5. Perception of non-monochromatic light F ()  r=F () r () d    F (i) r (i) i g=F () g () d  b =F () b () d 

6. Representative wavelengths Fe() F ()   Light propagation: Linear functional: F () = T(Fe()) r=F () r () d  =  F (i) r (i) i = r= T(Fe( i))r (i) i

7. Measuring the directions: 2D 2D case Direction: angle  from a reference direction Directional set: angle [rad] arc of a unit circle Size: length of the arc Total size: 2 

8. Measuring the directions: 3D Direction: angles , from two reference directions Directional set: solid angle [sr] area of a unit sphere Size: size of the area Total size: 4  

9. Size of a solid angle d d d dw = sin d d sin d

10. Solid angle in which a surface element is visible dA  r dw dA cos  dw = r2

11. Radiance: L(x,w) • Emitted power of a unit visible area in a unit solid angle [Watt/ sr/ m2] w dw dF  L(x,w) = dF / (dA cos dw) dA

12. Light propagation between two infinitesimal surfaces: Fundamental law of photometry ’  r L dw dA dA’ dF emitter receiver dA cos  dA’ cos ’ dF = LdA cos dw = L r2

13. Symmetry relation of the source and receiver dA cos  dA’ cos ’ dF = L =LdA’ cos ’dw’ r2 dw’ ’  r dA dw’ dA’ dF emitter receiver

14. Light-surface interaction w dw w’ x Probability density of the reflection w(w’,x,w) dw = Pr{photon goes to w  dw | comes from w’}

15. Reflection of the total incoming light w dw Fin (dw’) Fref (dw) dw’ w’ x Fref (dw) = Fe (dw) +  Fin (dw’) w(w’,x,w)dw

16. Rewriting for the radiance Fref (dw)= LdA cos dw Fe(dw)= LedA cos dw Fin(dw’)= LindA cos ’dw’ Visibility function h(x,-w) L(x,w) ’ Lin =L(h(x,-w’),w’) w  ’ x

17. Substituting and dividing bydA cos dw w(’,x,) cos  L(x,w)=Le(x,w)+L(h(x,-w’),w’) cos ’dw’ w ’ w(’,x,) cos   = fr (’,x,) x Bidirectional Reflectance Distribution Function BRDF: fr (’,x,) [1/sr]

18. Rendering equation L(x,w)=Le(x,w)+L(h(x,-w’),w’) fr(’,x,) cos’dw’ L(h(x,-w),w) h(x,-w) L(x,w) ’ ’ w x fr (’,x,) L = Le + tL

19. Rendering equation • Fredholm integral equation of the second kind • Unknown is a function • Function space: Hilbert space, L2 space • scalar product: L = Le + tL <u(x,w),v(x,w)> = S u(x,w) v(x,w) cos dwdx

20. Function space • Linear space (vector space) • addition, zero, multiplication by scalars • Space with norms • ||u||2 = <u,u >, ||u||1 = <|u|,1>, • ||u|| = max|u|, • Hilbert space: scalar product: • L2 space: finite square integrals

21. Measuring the light: radiance • Sensitivity of a measuring device: We(y,w’) L(y,w’)  ’ Light beam reaches the device: 0/1 „probability” We(y,w’): effect of a light beam of unit power emitted at y in direction w’ Scaling factor

22. Measured values • Single beam : F(dw’) We(y,w’) = L(y,w’)cos dAdw’ We(y,w’) • Total measured value:SWe(y,w’)dF=S L(y,w’)We(y,w’) cosdw’dy = < L, We > = ML

23. Simple eye model Pupil: e Pupil: e Wp  Wp y ’ Lp pixel ’ y F r Computer screen F Real world Lp=F / (e cosqeWp) C=1 /(e cosqeWp)if y is visible in Wp and ’ points from y to e 0 otherwise We(y,’)=

24. Simple eye model: pinhole camera Pupil: e dw’= decose /r2 dy= r2dp/ cos Wp  y ’ ’ y Pinhole camera: e, ’ 0 r Lp= ML =S L(y,w’)We(y,w’) cosdw’dy  yL(y, w’) C· cos·’ · dy = p L(h(eye, wp ),-wp) C ·cos ·e cose /r2 · r2dp/cos = p L(h(eye, wp ),-wp) ·Ce cose dp Proportional to the radiance! Camera constant: 1 /Wp

25. Why radiance Lp= pL(h(eye, wp ),-wp) /Wpdp The color of a pixel is proportional to the radiance of the visible points and is independent of the distance and the orientation of the surface!! F=LDA cosdw /r2 pixel DA  r2 / cos r

26. Integrating on the pixel Sp pixel p p f dp= dp cos p/|eye-p|2 = dp cos3p/f 2 dp/Wp dp / Sp

27. Integrating on the visible surface pixel  y r dp= dy cos /|eye-y|2 = dy g(y)

28. Measuring function F= S L(y,w’)We(y,w’) cosdw’dy = = pL(h(eye, wp ),-wp) /Wpdp= = SL(y,w’) · cos /|eye-y|2 /Wpdy (w-wyeye)/|eye-y|2 /Wpif y is visible in the pixel 0 otherwise We(y,’)= g(y)

29. Potential: W(y,w’) • The direct and indirect effects in a measuring device caused by a unit beam from y at ’ • The product of scaling factor C and the probability that the photon emitted at y in ’ reaches the device w’ y

30. Duality of radiance and potential • Light propagation = emitter-receiver interaction • radiance: intensity of emission • potential: intensity of detection

31. Potential equation w’ y C · Pr{ detection} = C · Pr{ direct detection} + C · Pr{ indirect detection} Pr{ indirect detection} =  Pr{ detection from the new point | reflection to w}· Pr{ reflection to w} dw

32. Potential equation W(y,w’)=We(y,w’)+W(h(y,w’),w) fr(’,h(y,w’),)cosdw y W(y,w’) w’ q fr (’,h(y,w’),) h(y,w’) W = We + ’W

33. Measuring the light: potential w’ y Fe(dw’) Measured values of a single beam = Fe(dw’)W(y,w’) = Le(y,w’)cos dAdw’ W(y,w’) Total measured value = M’W= S W(x,w)dFe=S Le(x,w) W(x,w) cosdwdx = < Le , W>

34. Operators of the rendering and potential equations • Measuring a single reflection of the light: • Adjoint operators: F1=< Le , W> = < Le , ’We > F1=< L, We> = < Le ,We > < Le , ’We > = < Le ,We >

35. Rendering problem: <S,Le,We ,fr> pixel S fr (’, x, ) Le(x,) We(x,) F= S We(x,w) dF=S L(x,w) We(x,w) cos dwdx F= ML