1. Rendering Problem László Szirmay-Kalos

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

3. Measuring the light: Flux l Power going through a boundary [Watt] l Number of photons Spectral dependence:  d 

4. Color perception perception: r, g, b 400700500600 r(r( g(g( b(b( r, g, b

5. Perception of non-monochromatic light r =   r  d  i  r  i  i g =   g  d b =   b  d 

6. Representative wavelengths r =   r  d  i  r  i  i  r =  T  e  i  r  i  i  ee Light propagation: Linear functional:  = T  e 

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d dd sin  d  d  sin  d  d 

10. Solid angle in which a surface element is visible dA  dd r d  dA cos  r2r2

11. Radiance: L(x,  ) l Emitted power of a unit visible area in a unit solid angle [Watt/ sr/ m 2 ] dd dAdA   dd L(x,  ) = d  dA cos  d 

12. Light propagation between two infinitesimal surfaces: Fundamental law of photometry dd dAdA  dd dA’ ’’ r d  L dA cos  d  L dA cos  dA’ cos  ’ r2r2 L emitterreceiver

13. Symmetry relation of the source and receiver dd dAdA  d’d’ dA’ ’’ r d  L dA cos  dA’ cos  ’ r2r2 =L dA’ cos  ’ d  ’ d’d’ emitter receiver

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

15. Reflection of the total incoming light x  dd ’’ d’d’  ref (d  ) =  e (d  ) +    in (d  ’) w(  ’,x,  ) d   in (d  ’ )  ref (d  )

16. Rewriting for the radiance  ref (d  ) = L dA cos  d   e (d  ) = L e dA cos  d   in (d  ’ ) = L in dA cos  ’ d  ’ ’’ Visibility function h(x,-  L(x,  ) x ’’ L in =L(h(x,-  ’ ,  ’)  

17. Substituting and dividing by dA cos  d  L(x,  )=L e (x,  )+   L(h(x,-  ’ ,  ’) cos  ’d  ’ = f r (  ’,x,  ) x  ’’ w(  ’,x,  ) cos   w(  ’,x,  ) cos  Bidirectional Reflectance Distribution Function BRDF: f r (  ’,x,  ) [1/sr]

18. Rendering equation L(x,  )=L e (x,  )+   L(h(x,-  ’ ,  ’) f r (  ’,x,  ) cos  ’d  ’ L = L e +  L ’’ f r (  ’,x,  ) h(x,-  L(x,  ) x  ’’ L(h(x,- ,  )

19. Rendering equation l Fredholm integral equation of the second kind l Unknown is a function l Function space: Hilbert space, L 2 space –scalar product: L = L e +  L =  S   u(x,  ) v(x,  ) cos  d  dx

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

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

22. Measured values Single beam :  (d  ’) W e (y,  ’) = L(y,  ’)cos  dA d  ’ W e (y,  ’) Total measured value:  S   W e (y,  ’)d    S   L(y,  ’)W e (y,  ’) cos  d  ’dy = = M L

23. Simple eye model r  pp y ’’ ’’ yy Pupil:  e pp Real world Computer screen pixel LpLp L p =   e cos  e   p    W e (y,  ’)= C=  e cos  e  p  if y is visible in  p and  ’ points from y to  e 0 otherwise

24. Simple eye model: pinhole camera L p  M L =  S   L(y,  ’)W e (y,  ’) cos  d  ’dy    y L(y,  ’) C · cos  ·  ’ · dy =   p L(h(eye,  p ),-  p ) C · cos  ·  e cos  e /r 2 · r 2 d  p /cos  =   p L(h(eye,  p ),-  p ) · C  e cos  e d  p r  pp y ’’ ’’ yy Pupil:  e d  ’= de cos  e /r 2 d y= r 2 d  p / cos  Pinhole camera:  e,  ’  0 Camera constant:  p Proportional to the radiance!

25. Why radiance 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!! L p =   p L(h(eye,  p ),-  p ) /  p d  p r pixel  =L  A cos  d  /r 2  A  r 2 / cos 

26. Integrating on the pixel f pixel pp d  p = dp cos  p /|eye-p| 2 = dp cos 3  p /f 2 d  p /  p  dp / S p SpSp p

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

28. Measuring function   S   L(y,  ’)W e (y,  ’) cos  d  ’dy = =   p L(h(eye,  p ),-  p ) /  p d  p = =  S L(y,  ’) · cos  /|eye-y| 2 /  p dy W e (y,  ’)=  (  -  y  eye )/|eye-y| 2 /  p if y is visible in the pixel 0 otherwise g(y)

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

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

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

32. Potential equation W(y,  ’)=W e (y,  ’)+  W(h(y,  ’ ,  ) f r (  ’,h(y,  ’ ,  )cos  d  W = W e +  ’ W  y h(y,  ’  ’’ f r (  ’,h(y,  ’ ,  ) W(y,  ’)

33. Measuring the light: potential Measured values of a single beam =  e  (d  ’ ) W(y,  ’) = L e (y,  ’)cos  dA d  ’ W (y,  ’) Total measured value = M’W=  S   W (x,  )d  e   S   L e (x,  ) W(x,  ) cos  d  dx = y ’’  e  (d  ’ )

34. Operators of the rendering and potential equations l Measuring a single reflection of the light: l Adjoint operators:  1  = =

35. Rendering problem:  =  S   W e (x,  ) d    S   L(x,  ) W e (x,  ) cos  d  dx Le(x,)Le(x,) pixel f r (  ’, x,  ) S We(x,)We(x,)  =   L