Iteration Solution of the Iteration Solution of the Global Illumination ProblemGlobal Illumination Problem

László Szirmay-Kalos

Solution by iterationSolution by iteration Expansion uses independent samples

– no resuse of visibility and illumination information Iteration may use the complete previous

information

– Ln= Le+ Ln-1

– pixel = Ln

Storage of the temporary Storage of the temporary radiance: finite elementsradiance: finite elements

FEM:

Projecting to an adjoint base:

Lj(n) bj (p) = Lj

e bj (p) + Lj(n-1) bj (p)

L(p) Lj bj (p)

Li(n) = Li

e + Lj(n-1) < bj , bi’ >

p=(x,)

FEM iterationFEM iteration

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)

L(n) = Le + RL(n-1)

Problems of classical iterationProblems of classical iteration

storage complexity of the finite-element representation – 4 variate radiance: very many basis functions

error accumulation

– L/(1-q), qcontraction The error is due to the drastic simplifications

of the form-factor computation

Stochastic iterationStochastic iteration

Use instantiations of random operator *

– Ln= Le+ *n Ln-1

which behaves as the real operator in average

– *n L L

Example: x = 0.1 x + 1.8Example: x = 0.1 x + 1.8Solution by stochastic iterationSolution 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

Iteration with a single rayIteration with a single rayTransfer the whole power from x into selected with probability:L(x,) cos

x

x

x

Le(x,)

1

2

3

Making it convergentMaking it convergent

Ln= Le+ *n Ln-1

n =Ln is not convergent

pixel=( L1+ L2+...+ Lm)/m

Stochastic iteration with FEMStochastic iteration with FEMDiffuse caseDiffuse 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

Stochastic radiosityStochastic radiosity

P = Pe + HP Random transport operator: H* P|i = Hij Expected value:

E[H*P|i ]= j Hij Pi/ = H P|i

Selects a single (a few) patch with the probability of its relative power and transfers all power from here

Transillumination radiosityTransillumination radiositySelects a single (a few) directions and transfers all power into these directions

Projected rendering equation:L = Le + RL

Transport operator:Rij =< bj ,bi’ >= fi /Ai Aibj (h(x,-’) cos’ dxd’

Random transport operator:Rij

*= 4 fi /Ai Ai bj (h(x,-’) cos’dx

Ai Ai bbjj ((hh(x,-(x,-’’) cos) cos’dx’dx

Ai

’A(i,j,’)

Aj

Transilluminationplane

A(i,j,’)= projected area of path j, which is visible from path i in direction ’

Stochastic ray radiosityStochastic ray radiositySelects a single (a few) rays (points+dirs) with aprobability proportional to the power cos/areaand transfers all power by these rays

P = Pe + HP Random transport operator: if y and are selected: H* P|i = fi bi(h(y,)

Expected value of the random transport operator:E[H*P|i ]= j fi Aj bi(h(y,) cos/ dy/Aj Pj/ = j fi /Aj Aj bi(h(y,) cos dy Pj = H P|i

Stochastic iteration for the Stochastic iteration for the non-diffuse casenon-diffuse case

Ln= Le+ *n Ln-1 Reduce the storage requirements of the

finite-element representation Search * which require L not everywhere

Ln (pn+1) = Le (pn+1)+ *n (p n+1,p n) Ln-1 (p n)

Stochastic integrationStochastic 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

Ray-bundle based iterationRay-bundle based iteration

Le

pixel

Storage requirement: 1 variable per patch

Finite elements for the Finite elements for the positional variationpositional variation

FEM:

Projected rendering equation:– L(’) = Le(’) +F(’,) A(’) L(’)d’

Random transport operator:– Select a global direction ’ randomly:

– * L(’) = 4 F(’,) A(’) L(’)

L(x,) Lj () bj (x)

Ray-bundle iterationRay-bundle iterationGenerate the first random direction

FOR each patch i L[i] = Le(1)FOR m = 1 TO M Reflect incoming radiance L to the eye and add contribution/M to Image Generate random global direction m+1

L = Le(m+1)+ 4 F(m,m+1) A(m) L(m)ENDFORDisplay Image

Ray-bundle imagesRay-bundle images

10k patches500 iterations9 mins

60k patches600 iterations45 mins

60k patches300 iterations30 mins

Can we use quasi-Monte Can we use quasi-Monte Carlo samples in iteration?Carlo samples in iteration?

1/(M-1) *(pi)*(pi-1) Le2Le

1/(M-1) f(pi,pi-1) f(x,y) dxdy

pi must be infinite-distribution sequence!

Future improvements ?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