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Advanced Techniques for Solving Rendering and Potential Equations in Computer Graphics

This document outlines various strategies for addressing rendering and potential equations, focusing on techniques like recursive ray tracing and approximations for solving the rendering equation. It explores local and global illumination models, the impact of light source visibility, and provides examples of iterative, expansion, and inversion methods for achieving approximate solutions. The text also discusses the coupling of variables in rendering equations, revealing computational alternatives for more efficient rendering while maintaining realism through light simulation.

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Advanced Techniques for Solving Rendering and Potential Equations in Computer Graphics

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  1. Strategies to solve the rendering or potential equations László Szirmay-Kalos

  2. Coupling in the 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,)

  3. Compromises in the solution • Ignore all types of coupling: • local illumination models • Follow just very specific types of coupling that have finite number: • recursive ray-tracing • Resolve the coupling: • global illumination models

  4. Example: x = 0.1 x + 1.8Solution by ignoring the coupling • Ignoring the coupling: • right side’s x: x = 0.1 x + 1.8  1.8 • Approximate solution: • x = 0.1·1.8 + 1.9 = 1.98

  5. Local illumination models • The radiance of in the right side: • is replaced by an approximation of the emission function: abstract lightsources • Single reflection of the light from the abstract lightsources are computed L(x,w)=Le(x,w)+Llightsource fr (’,x,) cos ’dw’  Le(x,w)

  6. Lightsource models • Ambient lightsource: La constant • ambient refl. model: Lref = La ka • Sky-light model: Lref = La a(x, view dir) • Directional lightsource • radiates into a single direction, the intensity is independent of the position • Positional lightsource • radiates from a single position, the intensity decreases with the square distance

  7. Local illumination algorithms pixel • Visibility computation from the eye • Lightsource visibility from the visible point • Calculation of the reflection of the light from the lightsources

  8. Result of local illumination models Without lightsource visibility computation With lightsource visibility computation

  9. Recursive ray-tracing • The radiance of in the right side is replaced by an approximation of the emission function and by the radiance from ideal refraction and refraction directions • Single incoherent reflection and multiple coherent reflections of the light from the abstract lightsources are computed L(x,w)=Le(x,w)+ + Llightsource fr (’,x,) cos ’dw’ + krLin (x, wr) + ktLin(x, wt)

  10. refracted ray reflected ray shadow rays Llightsource Recursive ray-tracing

  11. Comparison of the solution alternatives of L = Le +tL Local illumination with shadows 0.1 sec … sec Recursive ray tracing secs… mins Global illumination mins … hours

  12. Global illumination solution • Inversion • Expansion • Iteration (1- )L = Le L = (1- ) -1Le L = Le+L = Le+Le+2L= Le+Le+2 Le +3L =   nLe Ln = Le+ Ln-1

  13. Example: x = 0.1 x + 1.8Solution by inversion • Grouping the unknown terms: • 0.9 x = 1.8 • Inversion: • x = 0.9 -1 1.8 = 2 • Luck: multiplication by a scalar is invertible

  14. Inversion (1- )L = Le L = (1- ) -1Le • is infinite dimensional • it requires finite approx. of and the radiance L : • finite element techniques

  15. Example: x = 0.1 x + 1.8Solution by expansion • xi=1.8+0.1 1.8 +0.121.8 + …+ 0.1i+1x • i: 1 2 3 4 • xi sequence: 1.8 1.98 1.998 1.9998

  16. Expansion L=Le+L=Le+Le+2L=nLe ifnL 0 •  should be a contraction • Rendering equation: • radiance: • measured power: • Potential equation: • potential: • measured power: L =   nLe F= ML =  M nLe W =  ’ n We F= M’W =  M’ nWe

  17. Error of truncating the Neumann series ML =  M nLe DM nLe error= |n=D+1M nLe |=|M D+1( n=0 nLe )|  qD+1F || L||  q||L||, q is the contraction

  18. Terms of the expansion of the rendering equation: 1-bounce M(Le) = P R(p,w1) Le(y,-w1) dw1dp Le y L(x,w) ’1 w1 w p x R(p,w1) = 1/Spfr (w1,x,) cos’1

  19. Terms of the expansion of the rendering equation: 2-bounce M(2Le)=P R(p,w1,w2)Le(z,-w2) dw1dw2dp Gathering walks y w2 L(x,w) ’1 z ’2 w1 w p x Le R(p,w1,w2) = 1/Spfr (w1,x,)cos’1 fr (w2,y,1)cos’2

  20. Terms of the expansion of the potential equation: 1-bounce M’(’We)= S Le(x,w)cos{ We(y,w1) fr(w,y,1)cos1dw1 }dwdx = S Le(x,w)cosfr(w,y,eye)g(y)dwdx=S Le(x,w)R(x,w)dwdx x w Le (x,w) p 1  weye y R(x,w) = cos fr (w,y,eye) g(y)

  21. Terms of the expansion of the potential equation: 2-bounce F=M’(’2We)=S R(x,w,w1) Le(x,w) dw1dwdx Shooting walks y w1 p  x 2 1 w weye z Le (x,w) R(x,w,w1) = cos fr (w,y,1)cos1 fr (w1,z,eye) c(p)

  22. Expansion is a series of high dimensional integrals R(w1,...,wn,x) = w1 w2 …wn wi= fr cos ’ pixel w 1 e w L 2 F=n...R(w1,...,wn,x)Le(w1,...,wn,x)dw1,...,dwndx 1/M iM nDR(w1(i),...,wn(i),x(i) )Le(w1(i),...,wn(i),x(i)) • It requires just discrete samples from Le and R • No finite elements are needed

  23. Example: x = 0.1 x + 1.8Solution by iteration • Iteration scheme: • xn = 0.1 xn-1 + 1.8, • n: 1 2 3 4 • xnsequence:0 1.8 1.98 1.998 1.9998

  24. Iteration Ln = Le+ Ln-1 • Requirements of convergence • Convergent: Ln = Le+ Ln-1 Ln -1 = Le+ Ln-2 |Ln-Ln-1| = |(Ln-1-Ln-2)| = |n-1(L1-L0)| lim |n-1(L1-L0)| = 0

  25. If  is a contraction • || L ||  q||L||, • q < 1contraction ratio = albedo • ||n-1(L1-L0) ||  q n-1||L1-L0|| • converges with the speed of the geometric series e.g.: • ||L||1=, q is the average albedo

  26. Error analysis of the iteration • Assume that  is only approximated by * • L1 = Le+* L= L+L • L2 = Le+* L1 = L+L+* L • … • Ln = Le+* Ln-1 = L+L+*L+*2 L+... • ||L-L||||L||+||*L||+…=||L||(1+q+q2+…) ||L-L|| ||  L- * L || / (1-q)

  27. Analitically solvable cases • Normal approach: scene  radiance solution • Inverse approach: radiance  scene • Good for testing global illumination algorithms • constant radiance • constant reflected radiance

  28. Constant radiance L (x,w) = L* L*= Le (x,w)+tL* = Le (x,w)+L*t1 = Le (x,w)+L* a(x,w) • a(x,w) = 1 - Le (x,w)/ L* • arbitrary geometry in a closed environment

  29. Constant reflected radiance L (x,w) = Le (x,w)+Lr* Le +Lr* = Le +t(Le +Lr* ) = Le +t(Le +Lr* )= Le +tLe +Lr*t1 = Le +tLe +Lr*a(x,w) • a(x,w) = 1 - tLe(x,w)/ Lr* • special case: constant albedo?

  30. R y x’ y x |x-y| Spherical geometry with diffuse emission: constant reflected radiance tLe= Le(h(x,-w’)) fr cosx’ dw’ = S fr cosx’ cosyLe(y)/|x-y|2 dy cosx’=cosy=|x-y|/2R tLe=fr S Le(y) dy 4R2  =albedo  average(Le)

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