primary sample space and multiplexed mltanisb.github.io/slides.pdf · f(x,tt) tt is a parameter of...

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Kelemen style MLTMMLT

Conclusion

Primary sample space and Multiplexed MLT

Anis Benyoub

Novembre 28, 2014

Anis Benyoub Primary sample space and Multiplexed MLT

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Conclusion

Plan

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Rendering equationPath tracingImportance samplingMISBPTOriginal MLT

Kelemen style MLTMMLTConclusion


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Conclusion


Rendering equation Kajiya 1986

Lo(x, ωo, λ, t) = Le(x, ωo, λ, t) +∫

Ω fr (x, ωi, ωo, λ, t) Li(x, ωi, λ, t) (ωi · n) dωi

t: time; λ: wavelength;x : point in space;n: normal at x ;ωo : exiting light direction;ωi incident light direction;Li : incident radiant intensity;Lo : radiance at x ;Le : radiant intensity at x .


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Path tracing Kajiya 1986

Solving the rendering equation by Monte carlo sampling.I Simple implementation;I Slow convergence for indirect illumination or small light sources.


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Importance sampling

I Distributing samples according to fr or Li may be a good idea;I We don’t know the exact form of Li but it’s greatly influenced by the

light source position, size and shape.

For samples distributed along p(X):

I(x) =∫

D f (x) d x ≈ 1n∑n

i=1f (X)p(X)


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Multiple Importance sampling

I Combining several sampling methods seems like a good idea (and it is!);I The statistical model is more tricky but correct.

Let ωi be the samples’ weights:I(x) =

∫D f (x) d x ≈

∑ni=1

1ni

∑nij=1 ωi (Xij )

f (X)p(X)


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Bi-directional Path tracing Lafortune 1993

Solving the rendering equation by BPT.I Not that simple implementation.


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Efficient for indirect illumination and small lights.


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Still struggles with complex light paths (small pdf light paths, caustics, ...).


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Conclusion


Metropolis light transport Veach 1997

The rendering equation becomes a path integration problem:Φj =

∫P F (z) d z ≈ 1

M∑M

i=1F (zi )p(zi )

I Based on Metropolis sampling;I Several criteria: high acceptance probability, large changes to the path,

ergodicity, changes to the image location, stratification and low cost;I Involves combining several mutation strategies: bidirectional mutations,

perturbations, and lens subpath mutations;I Mutations are made in the path space.


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Metropolis light transport Veach 1997

Efficient where BPT fails.


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Primary sample space MLT Kelemen 2002

I Veach style MLT is considered too complex to implement and not flexibleenough;

I The idea is doing the sampling in a more convinient space.


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Let’s review the statistical model: Φj =∫

P F (z) z ≈ 1M∑M

i=1F (zi )p(zi )

I The scalar contribution function I(z) which represents the luminance ofthe carried light;

I The pdf is defined as: p(z) = I(z)/b, b =∫

P I(z) d z;I Obviously b cannot be known precisely, this introduces a start-up bias;I We generate samples using a BPT to approximate its value. The bias is

then converted to noise.


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The methodology focuses on two mutations: small and large mutations. Theissue of this choice is that each of them can lead to correlated samples whichleads to increasing the error.


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A technique that does large mutations when the contribution is low and smallmutations when the contribution is high.

→ This sounds exactly like importance sampling (Awesome!).


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I Importance sampling uses normalized values for each of its evaluations;I We keep the normalized values (u1, u2, ..., un), they define our sample in

the PSS;I We do a little parameter subsitution:

F (z) = F ∗(u) = F (S(u)) ·∣∣∣d S(u)

du

∣∣∣ = F (S(u))pS(U)

I The new scalar contribution function: I∗(u) = I(S(u))pS(U)


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Let’s focus on large steps:I Guarantee ergodicity;I Reduce the start-up bias;I The pdf is known for the sample.

After some approximations we get: a(ui → ut) ≈ T ∗(ui )T ∗(ut) ≈ 1


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Conclusion

f l o a t Pr imarySample ( i n t i )

i f ( u [ i ] . modi fy < t ime )

i f ( l a r g e s t e p )

Push ( i , u [ i ] ) ;u [ i ] . modi fy = t ime ;u [ i ] . v a l u e = random ( ) ;

e l s e

i f ( u [ i ] . modi fy < l a r g e s t e p t i m e )

u [ i ] . modi fy = l a r g e s t e p t i m e ;u [ i ] . v a l u e = random ( ) ;

w h i l e ( u [ i ] . modi fy < t ime −1)

u [ i ] . v a l u e = Mutate ( u [ i ] . v a l u e ) ;u [ i ] . modi fy++;

Push ( i , u [ i ] ) ;u [ i ] . v a l u e = Mutate ( u [ i ] . v a l u e ) ;u [ i ] . modi fy++;

r e t u r n u [ i ] . v a l u e ;

f l o a t Mutate ( f l o a t v a l u e )

f l o a t s1 = 1 . / 1 0 2 4 , s2 = 1 . / 6 4 ;f l o a t dv = s2∗exp(− l o g ( s2 / s1 )∗ random ( ) ) ;i f ( random ( ) < 0 . 5 )

v a l u e += dv ; i f ( v a l u e > 1) v a l u e −= 1 ; e l s e

v a l u e −= dv ;i f ( v a l u e < 0) v a l u e += 1 ;

r e t u r n v a l u e ;Sample Next ( f l o a t I , C o n t r i b c o n t r i b )

f l o a t a = min ( 1 , I / o l d I ) ;newsample . c o n t r i b = c o n t r i b ;newsample . w = ( a+l a r g e s t e p ) / ( I /b+p l a r g e )/M;o l d s a m p l e . w += (1−a ) / ( o l d I /b+p l a r g e )/M;i f ( random ( ) < a )

o l d I = I ;c o n t r i b s a m p l e = o l d s a m p l e ;o l d s a m p l e = newsample ;i f ( l a r g e s t e p ) l a r g e s t e p t i m e = t ime ;t ime++;C l e a r S t a c k ( ) ;

e l s e c o n t r i b s a m p l e = newsample ;w h i l e ( ! I sStackEmpty ( ) ) Pop ( i , u [ i ] ) ;

l a r g e s t e p = ( random ( ) < p l a r g e ) ? 1 : 0 ;r e t u r n c o n t r i b s a m p l e ;


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Conclusion

Multiplexed MLT Hachisuka 2014

I BPT is inefficient for complex light paths;I Kelemen-style MLT struggles with very complex light paths (noisy

results);I Original MLT has some difficulties with glossy surface.

The idea is doing the sampling in an other space that would be more efficientfor those paths.


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Conclusion

Multiplexed MLT Hachisuka 2014

I The relation between Markov Chain Monte Carlo and MIS is said to beunclear for MLT;

I PSSMLT is a special case of MCMC where MIS is simply importancesampling.

I Even if BPT is used in PSSMLT, the different available importancesampling techniques are not used.

I A point in the sampling space u matches only point x in the path space.I It should match a point per combination of IS techniques!


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Serial Tempering

I MCMC sampling from a sum of parametrized distributions [Brooks 2011](which matches pretty well the case of MCMC and MIS);

I∑M

t=1 f (x ,Tt) Tt is a parameter of the distribution f (x ,T );I The acceptance probability from a to b in this case would be

min(1, r(Tta → Ttb )) where:

r(Tta → Ttb ) = f ∗(x ,Ttb )q(Ttb → Tta )f ∗(x ,Tta )q(Tta → Ttb ) ;

I The model can not be used for global illumination in the path space forstatistical reasons.


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Primary Sample Space Tempering

I A modified version of the PSS is introduced: the PSST!;I Each ditribution is warped according to it’s CDF in the primary sample

space. f ∗(u,Tt) = ωt(x)C∗t (x)pt(x)dν(x) = ωt(u)C∗

t (u)du;I The acceptance probability from a to b in this case would be

min(1, r(Tta → Ttb )) where:

r(Tta → Ttb ) =ωtb (u)C∗

tb (u)q(Ttb → Tta )ωta (u)C∗

ta (u)q(Tta → Ttb ) .


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MMLT

Two markov chains in practive:I One for sampling the distribution;I One for sampling the samples.

Specified path length (then multiple rendering for each then combining).Simple implementation over a PSSMLT.


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Algorithm


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Implementation details

I The average image contribution is evaluated using BPTbk ≈

1Nk

∑Nki=1 w∗(u)C∗(u);

I One MCMC per length ends up to be more efficient;I Light and camera subpaths are only connected at ending.


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Comparison


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Limitations and improvements

I Still inefficient for specular-diffuse-specular paths (photon techniques aremore efficient);

I Does not guarantee better convergence than Veach MLT;I The weights for MCMC MIS should be adapted to the scene for better

convergence;I Subpath mutations are possible;I Parallelisation of one metropolis estimator is tough, but combining several

output images is a form of parallelisation.


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Demo


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Conlusion

I The theory behind PSSMLT are MMLT are interesting;I MMLT: Small improvement but very simple and costless implementation→ worth it!

At the Boundary Between Light Transport Simulation and ComputationalStatistics - Hachisuka 2014


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Conclusion

Questions?


primary sample space and multiplexed mltanisb.github.io/slides.pdf · f(x,tt) tt is a parameter of...

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