Iteration Solution of the Global Illumination Problem

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Iteration Solution of the Global Illumination
Problem
László Szirmay-Kalos
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Solution by iteration

• Expansion uses independent samples
• no resuse of visibility and illumination
  information
• Iteration may use the complete previous
  information
• Ln Le t Ln-1
• ?pixel M Ln

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Storage of the temporary radiance finite elements

• FEM
• Projecting to an adjoint base

L(p) ? ??Lj bj (p)
p(x,w)
??Lj(n) bj (p) ??Lje bj (p) t ??Lj(n-1) bj
(p)
Li(n) Lie ?? Lj(n-1) ltt bj , bi gt
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FEM iteration
L(n) Le R L(n-1)

• Matrix form
• Jacobi iteration
• Complexity O(steps 1 step) O(c N 2)
• Other iteration methods
• Gauss-Seidel iteration O(c N 2)
• Southwell iteration O(N N)
• Successive overrelaxation O(c N 2)

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Problems of classical iteration

• storage complexity of the finite-element
  representation
• 4 variate radiance very many basis functions
• error accumulation
• ???L/(1-q), q???contraction
• The error is due to the drastic simplifications
  of the form-factor computation

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Stochastic iteration

• Use instantiations of random operator t
• Ln Le t n Ln-1
• which behaves as the real operator in average
• Et n L t L

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Example x 0.1 x 1.8Solution by stochastic
iteration

• Random transport operator
• xn T n xn 1.8, T is r.v. in 0, 0.2
• n 1 2 3
  4 5
• Tn sequence 0 0.1 0.2 0.15
  0.05
• xn sequence 0 ?1.8 1.91 2.18 2.13 1.9
• Not convergent!
• Averaging 1.8 1.85 1.9 2.04 1.96
  

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Iteration with a single ray
Transfer the whole power from x into w selected
with probabilityL(x,w) cos ?
1
F1
w
3
x
?
Le(x,w)
x
w
F3
x
F2
2
w
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Making it convergent

• Ln Le t n Ln-1
• ?n M Ln is not convergent
• ?pixel(M L1 M L2... M Lm)/m

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Stochastic iteration with FEMDiffuse case

• Projected transport operator
• directional integral of the transport operator
• surface integral of the projection
• Alternatives
• both explicitely classical iteration, stochastic
  radiosity
• surface integral explicitely transillumination
  radiosity
• both implicitely stochastic ray-radiosity

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Stochastic radiosity
Selects a single (a few) patch with the
probability of its relative power and transfers
all power from here
P Pe HP Random transport operator H
Pi Hij F Expected value EHPi ?j Hij F
? Pi/F H Pi
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Transillumination radiosity
Selects a single (a few) directions and transfers
all power into these directions
Projected rendering equation L Le R
L Transport operator Rij ltt bj ,bi gt fi /Ai
?? ?Ai bj (h(x,-w)) cos? dxdw Random
transport operator Rij 4? fi /Ai ?Ai bj
(h(x,-w)) cos?dx
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?Ai bj (h(x,-w)) cos?dx
?
?
A(i,j,?)
Ai
Aj
Transillumination plane
A(i,j,?) projected area of path j, which is
visible from path i in direction ?
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Stochastic ray radiosity
Selects a single (a few) rays (pointsdirs) with
a probability proportional to the power ?
cos?/area and transfers all power by these rays
P Pe HP Random transport operator
if y and w are selected H Pi fi?
bi(h(y,w)) F Expected value of the random
transport operator EHPi ?j fi ? ?Aj
bi(h(y,w)) F cos?/? dy/Aj Pj/F ?j fi /Aj ?Aj
bi(h(y,w)) cos? dy Pj H Pi
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Stochastic iteration for the non-diffuse case

• Ln Le t n Ln-1
• Reduce the storage requirements of the
  finite-element representation
• Search t which require L not everywhere
• Ln (pn1) Le (pn1)t n (p n1,p n) Ln-1 (p n)

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Stochastic integration

• Projected transport operator
• directional integral of the transport operator
• directional-surface integrals of the projection
• Alternatives
• all integrals explicitely classical iteration
• all integrals implicitely iteration with a
  single ray
• directional integral of the transport operator
  implicitely, integral of the projection
  explicitely

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Ray-bundle based iteration
pixel
e
L
Storage requirement 1 variable per patch
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Finite elements for the positional variation

• FEM
• Projected rendering equation
• L(w) Le(w) ?? F(w,w) A(w) L(w)dw
• Random transport operator
• Select a global direction w randomly
• t L(w) 4? F(w,w) A(w) L(w)

L(x,w) ? ??Lj (w) bj (x)
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Ray-bundle iteration
Generate the first random direction w1 FOR each
patch i Li Le(w1) FOR m 1 TO M Reflect
incoming radiance L to the eye and add
contribution/M to Image Generate random
global direction wm1 L Le(wm1) 4?
F(wm,wm1) A(wm) L(wm) ENDFOR Display Image
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Ray-bundle images
10k patches 500 iterations 9 mins
60k patches 600 iterations 45 mins
60k patches 300 iterations 30 mins
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Can we use quasi-Monte Carlo samples in iteration?
1/(M-1) ?t(pi)t(pi-1) Le?t 2Le 1/(M-1) ?
f(pi,pi-1) ? ?? f(x,y) dxdy pi must be
infinite-distribution sequence!
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Future improvements ?

• Problem formulation
• Monte-Carlo integral
• Expansion versus iteration
• Same accuracy with fewer samples
• importance sampling
• very uniform sequences, stratification
• Making the samples cheaper
• fast visibility computations
• global methods coherence principle