© Machiraju/Möller Rendering Equation Unbiased MC Methods CMPT 461/770 Advanced Computer Graphics...

Rendering EquationUnbiased MC Methods

CMPT 461/770Advanced Computer Graphics

Torsten Möller

Reading

• Chapter 16 of “Physically Based Rendering” by Pharr&Humphreys

• Chapter 19, 20 in “Principles of Digital Image Synthesis,” by A. Glassner

Direct Light

• General equation of existant radiance:

• Account only for direct light sources, i.e. Replace Li(p,i) with Li(p,i):

– Most simple form of equation

– Somewhat easy to solve! (but a gross simplification)

– Kinda what we do in Whitted Ray-tracing

– Not too bad - most energy comes from direct lights

Lo p,o Le p,o f r p,o, i Li p, i cosi d iS 2

Lo p,o Le p,o f r p,o, i Ld p, i cosi d iS 2

Direct Light

• MC sampling to solve integral

• (A)

– pick one light as representative for all lights– Distribute N samples for this light

– Use multiple importance sampling for fr and Ld

– Scale the result by the total number of lights NL

f r p,o, i Ld p, i cosi d iS 2

fr p,o, j Ld p, j cosi

p j j1

Multiple Importance Sampling

Weight according to f

Weight according to Ld

fr p,o, j Ld p, j cosi

p j j1

Direct Light

• (B) uniformly sample all lights

– Do (A) for each light source

– Sum the result

– Smarter approach - non-uniform sampling of light sources: prefer lights with has higher power

f r p,o, i Ld j p, i cosi d iS 2ji

Light Transport Equation

• Incident radiance effected by all geometry and scattering within a scene (colour bleeding)

• Global illumination

• Principle– Ignore wave optics– Energy balance (radiance is in equilibrium)

o i e aPowerleaving

Powerentering

Poweremitted

Powerabsorbed

Light Transport Equation

• If no participating media - express incoming in terms of outgoing radiance:

• Need to solve for L (only one unknown)

o i e a

o e i a Lo p,o Le p,o f r p,o, i Li p, i cosi d iS 2

Li p, Lo t p, ,

L p,o Le p,o f r p,o, i L t p, i , i cosi d iS 2

Lo p ,

Example - lambertian light

• One spherical light source on the outside

• No scene geometry

• Lambertian BSDF, fr=c

• Emitted radiance Le in constant

• is the same, no matter which point on the light. Hence:

L p,o Le c L t p, i , i cosi d iH 2

L L t p, i , i cosi d iH 2

L p,o Le cL

Example cont.

• Let’s solve the lastequation a little weirdly:

• These are basically the number of bounces

• hh is the reflectance

• von Neumann series - principle solution method

L p,o Le cL

Le c Le cL Le c Le c Le c ... Le c i

L p,o Le

Example - direct lighting

• Using recursion for direct lighting:

• The last term is being ignored; represents secondary (bounced) light

L p,o Le p,o f r p,o, i Ld t p, i , i cosi d iS 2Ld t p, i , i Le t p, i , i

f r t p, i , i, Ld t t p, i , , cos d S 2

• Expressing LTE in terms of geometry within the scene

• Replacing the integrand (di)with an area integrator over the whole scene geometry and remembering:

• - visibility term (either one or zero)

LTE - Geometric Formulation

L p , L p p f p ,o, i f p p p

d i cos p p

• Geometry coupling term

• New (geometric) formulationof the Light Transport Equation (LTE)

• Randomly pick points in the scene and create a path vs. (previously)

• randomly pick directions over a sphere

G p p V p p cos cos p p

L p p Le p p f r p p p L p p G p p dA p A

L p1 p0 Le p1 p0 Le p2 p1 f p2 p1 p0 G p2 p1 dA p2

Le p3 p2 f p3 p2 p1 G p3 p2 A

f p2 p1 p0 G p2 p1 dA p2 dA p3

G p2 p1

G p3 p2

f p3 p2 p1

f p2 p1 p0

• Recursive evaluation

• compact formulation:

• For a path

• Where p0 is thecamera and pi isa light source

L p1 p0 P p i i1

p i p0 p1...pi

G p2 p1

G p3 p2

f p3 p2 p1

f p2 p1 p0

• with:

• Where

• Is called the throughput

• Special case:

P p i ... Le pi pi 1 T p i dA p2 ...dA pi A

G p2 p1

G p3 p2

f p3 p2 p1

f p2 p1 p0

T p i j1

f p j1 p j p j 1 G p j1 p j

P p 1 Le p1 p0

• Again - handle with care (e.g. point light):

• E.g. Whitted raytracing only usesspecular BSDF’s

P p 2 Le p2 p1 f p2 p1 p0 G p2 p1 dA p2 A

plight p2 Le p2 p1

p plight f p2 p1 p0 G p2 p1

Le plight p1 f plight p1 p0 G plight p1

Specular distributions

G plight p1

plight

f plight p1 p0

Solving the LTE

• Many different algorithms proposed to deal with

• Most energy in the first few bounces:

• emitted radiance at p1

• one bounce to light (direct lighting)

P p i i0

L p1 p0 P p 1 P p 2 P p i i3

Solving the LTE

• Simplify according to small and large light sources:

• Can be handled separatly (different number of samples)

Le Le,s Le,l

P p i ... Le pi pi 1 T p i dA p2 ...dA pi A

... Le,s pi pi 1 T p i dA p2 ...dA pi A

... Le,l pi pi 1 T p i dA p2 ...dA pi A

Solving the LTE

• Similarly, we can split BxDF into delta and non-delta distributions:

• This creates many factors that cannot be ignores

T p i j1

f f G p j1 p j

Measuring over a pixel

• Integrating over pixel j:

• Where We is the filter function around a pixel:

I j We p film, Li p film, cos ddA p film S 2

A film

We p0 p1 Li p1 p0 G p1 p0 dA p0 dA p1 S 2

A film

We p0 p1 f j p0 t p0,camera p0 p1

Integrating over a pixel

• And hence the integral:

• A - light bounces around until it hits the camera• B - a quantity from the sensor bounces around and

only makes a contribution if it hits a light

I j We p0 p1 Li p1 p0 G p1 p0 dA p0 dA p1 S 2

A film

We p0 p1 P p i G p1 p0 dA p0 dA p1 S 2

A film

... We p0 p1 T p i Le pi1 pi dA p0 ...dA pi A

Path Tracing

• Kajiya 1986; solution to

A. How to deal with infinite sum?

B. For a particular i - how to generate one or more paths to estimate

L p1 p0 P p i i1

A) Infinite Sum

• In general => the longer the path the less the impact

• Use Russian Roulette after a finite number of bounces– always compute first few terms– Stop after that with probability q

L p1 p0 P p 1 P p 2 P p 3 1

1 qP p i

A) Infinite Sum

• Use probability qi after each

L p1 p0 1

P p 1 1

P p 2 1

P p 3 ...

B) Path Generation

• Pick a surface in the scene at random (I.e. uniform probability distribution):

• Pick a point on this surface at random (uniform) with probability 1/Ai

• Overall probability of picking a random surface point in the scene:

pA pi Ai

B) Path Generation

• This is repeated for each point on the path• Last point should be sampled on a light source• If we know things about the scene (which objects

are contributing most indirect lighting to the scene) we can sample more smartly

• Problems:– High variance - only few points are mutually visible

– Incorrect integral - for delta distributions

• For path– At each pj find pj+1 according to BSDF

– At pi-1 find pi by multiple importance sampling of BSDF and Ld

• This algorithm distributes samples according to solid angle () instead of area

• Hence prob. Distribution pA needs to be adjusted:

p i p0 p1...pi

B) Incremental Path Generation

pA pi ppi pi1

• MC estimator:

• Implementation:– Re-use path for new path

• Introduces correlation, but speed makes up for it

B) Incremental Path Generation

Le pi pi 1 pA pi j1

f p j1 p j p j 1 cosi

p p j1 p j

Path Tracing

Direct lighting only

1024 samples per pixel 8 samples per pixel

Pure Path Tracing

• Best for bigluminaires.

• If lights small,few hits andlarge variance.

Results

• Objects are gray,except forspheres and base.– Color bleeding– Caustics

Results

401 minutes 533 minutes

256 x 256 image

Ray Traced(no ambient)

Path Traced

Light scatteredby sphere

Bi-directional path tracing

• Compose one path from two paths– started at the camera p0 and

– started at the light source q0

• Modifications for efficiency:a) Use all i+j paths:

b) For all pathsreplace q1 with directlighting computationlike normal path tracing

q jq j 1...q1

p p1p2...pi,q jq j 1...q1

p1p2...pi

p1...pi,q j ...q1

p1...pi 1,q j ...q1

p1...pi 2,q j ...q1

p1,q j ...q1

p1...pi,q j ...q1

p1...pi,q j 1...q1

p1...pi,q1

p1p2...pi,q1

Bi-directional path tracing

• When useful?– Light sources difficult to reach– Specific BSDF evaluations (caustics)

Bi-direct RT : Pseudo-Code

Pure Bi-Directional: Analysis

• Advantages:– Each ray cast contributes to many paths– Building from both ends can catch difficult cases

• All specular paths• Caustics

– Extends to participating media (anisotropic, heterogeneous)

• Disadvantages:– Still using lots of effort to catch slow varying diffuse

components– May not sample difficult to find paths– Does allow for much noise to be present

Still Tough Cases

• Caustics– How do you know which

direction to cast eye rays toreach the interesting light?

• Bleeding– How do you know which

rays to reflect to reach theinteresting parts of diffusereflections?

MC Algorithms Revisited

• Provide ability to sample light transport paths.

• MC algorithms studied so far sample a function to compute the value of an integral.

• Problems:– Many of the paths are unimportant.– Pure Bi-Directional Path algorithm does not

reject.– It may weigh down some paths.

Metropolis’s Idea

• Circa 1953• Generate a distribution of samples proportional to

the unknown function• For rendering –

– sample image with ray density proportional to radiance

– random walk through path space

• Eric Veach’s PhD thesis and Veach&Guibas paper in Siggraph introduced work of Metropolis to Graphics

Why Does It Make Sense ?

• In the final image: – More detail in brighter areas– Value of a pixel will be proportional to the

number of times it was sampled in a path

Metropolis sampling

Metropolis Light Transport

• Generate a random walk through path space

• For each path deposit a constant amount of energy at the corresponding pixel

• Obtain desired image by distributing paths according to image contribution

x p0 p1...pi

• Propose a mutation of current path

• Compute acceptance probability

• Choose as new sample if

• Samples are correlated

• we can exploit coherence

)()(,1min),(

xyTyfxy

Metropolis Sampling

• Where is the image contribution function (contribution made to the image by light flowing through )

• tentative transition function - probability of given

T(x y )y

Metropolis Sampling

f (y )

Metropolis light transport

Mutations

MLT: Pseudo-Code

Scattering Perturbations Propagation Perturbations

Sensor Perturbations Caustic Perturbations

Mutation Strategies

• Bidirectional Mutations– large changes to the current path– ensures ergodicity

• Perturbations– high acceptance probability– changes to image location– low cost

Bidirectional Mutations

• Given a path

1. Choose a random subpath to delete.

2. Add random numbers of new vertices to the new interior endpoints of X.

3. Try to connect up the two innermost vertices.

4. Test for acceptance of the new path.

x p0 p1...pi

Mutations – Step 1

• Random subpath to delete.

• Weigh probability so that smaller subpaths are chosen more frequently.

• Add random numbers of new vertices.• Choose number to add from a distribution

centered around old number.• Choose where to put the break (i.e. how

many to add at end of first segment vs. beginning of last segment).

• Add each vertex by sampling a direction according to a BRDF, and then casting a ray.

• Chose to add 2 vertices, with the break between them.

• Connect two innermost vertices

• Test if two new endpoints are visible

• If the path is obstructed, reject the mutation

• Test for acceptance of the new path Y over old path X.

• Use

• Note that much of this can be pre-computed.

)()(,1min),(

xyTyfxy

Good Mutations

• Don’t make changesthat are too small

• Don’t get stuck

Perturbations

• Are needed when bidirectional mutations will nearly always be rejected.

• When there are small regions of the path space in which paths contribute much more than average.– Lenses– Caustics

• Smaller mutations keep them within high contribution region

Propagation Perturbations

image plane

medium

light sourceeye

Perturbations

• Move the pixel location (i.e. the path’s endpoint) by a random distance in a random direction.

• Recast rays through all the specular bounces so that it retains the same length.

• Mutation strategy depends on what is encountered• For example in the caustic LS+DE case make

small change in ray connecting the diffuse and specular surface

Caveat

• Run several copies of the algorithm in parallel.– Helps to remove startup bias.– Helps to test for convergence.

• Ignore the first several path samples.– Helps to remove startup bias.

• Don’t bother with MLT for the direct lighting in an image.– Standard techniques for direct lighting usually provide

better quality at lower cost.– This way MLT can devote more effort to the indirect

lighting.

Results

• Light for thisexample comesonly throughcrack in doorway

Results

• There are specificmutations tocapture caustics.

Good For

• Portions of space where light comes through

• hole in the wall kinds – mutations will explore new paths once it finds a

path through the hole– Bi-directional

and Plain PathTracing userandom directions

Not Good For

• Cornell Box scenes will not benefit

• Caustics will certainly help.

• Too many holes does not help

• Also caustics from mirror reflections

Results

© Machiraju/Möller Rendering Equation Unbiased MC Methods CMPT 461/770 Advanced Computer Graphics...

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