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Realistic Image Synthesis
- Path Tracing -
Pascal Grittmann
Philipp SlusallekKarol Myszkowski
Gurprit Singh
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What are we computing? β The physical phenomenon
Β© Giuseppe Milo (CC-BY)
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What are we computing? β The mathematical formulationGiven:A point visible to the camera
Compute:Incident light from all directions, reflected to the camera
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What are we computing? β The mathematical formulationGiven:A point visible to the cameraprevious point
Compute:Incident light from all directions, reflected to the camera previous point
Recursive problem!
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Reminder: Rendering Equation
πΏπΏππ π₯π₯,ππππ = πΏπΏππ π₯π₯,ππππ + οΏ½Ξ©ππ π₯π₯,ππππ,ππππ πΏπΏππ π₯π₯,ππππ cos ππππ ππππππ
Outgoing radiance from π₯π₯ in direction ππππ
Emitted radiance from π₯π₯ in direction ππππ
Bidirectional scattering distribution function β BSDF (how much light is reflected
from ππππ towards ππππ? )
Incident radiance at π₯π₯from direction ππππ
The mysterious cosine term
Integral over sphere of all directions
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Why the cosine term?
β’ Definition of radiance: β’ βflux per unit solid angle per unit
projected areaβ
β’ We are interested in:β’ βflux per unit solid angle per unit
projected areaβ
cos ππππ
ππππππππ
ππππ
ππ
π₯π₯
βππππ
π₯π₯
βππππ
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Challenges
β’ Recursive high dimensional β’ Discontinuities (object and shadow boundaries)
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Solution: Monte Carlo integration
β’ In general, with a single sample (primary estimator)
οΏ½Ξ©ππ π₯π₯ πππ₯π₯ β
ππ π₯π₯ππ π₯π₯
β’ Applied to rendering:
πΏπΏππ π₯π₯1,ππππ β πΏπΏππ π₯π₯1,ππππ +ππ π₯π₯1,ππππ,ππππ cos ππππ
ππ πππππΏπΏππ π₯π₯2,βππππ
Estimated recursively! High (infinite) dimensionality
π₯π₯1
π₯π₯2
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How to sample ππππ?
β’ Ideally proportionally to the full integrandβ’ ππ ππππ β ππ π₯π₯,ππππ,ππππ πΏπΏππ π₯π₯,ππππ cos ππππ
β’ Usually not possible Limit to one or more factorsβ’ Simplest method: cosine
β’ ππ ππππ β cos ππππβ’ Often also possible (approximately, at least): BSDF times cosine
β’ ππ ππππ β ππ π₯π₯,ππππ,ππππ cos ππππ
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When to terminate?
β’ In reality, all surfaces reflect lightβ’ We cannot track infinitely long paths, need to terminate recursion at
some pointβ’ Options:
β’ Maximum path length (biased)β’ Russian roulette (unbiased, later)
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Russian Roulette and Splitting
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Russian roulette
β’ Randomly terminate the path with some probability ππβ’ Unbiasedβ’ Increases variance (i.e., noise in the image)
β’ Given an estimator πΉπΉ, replace by πΉπΉπ π π π :
with probability ππ
with probability (1 β ππ)πΉπΉπ π π π = οΏ½
1πππΉπΉ
0
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What criterion to use?
β’ Fixed probabilityβ’ Current path throughputβ’ Efficiency-optimized RR (Veach 97):
β’ Based on statistics of surrounding pixels
β’ βAdjoint driven Russian roulette and splittingβ Vorba et al 201β’ Based on statistics from a pre-pass, combined with splitting
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Russian roulette experimentsinput scene path depth < 11 path depth < 101
RR with p=0.3 efficiency optimized p = throughput / 0.01
32.6min
53.2min
28.8min
10.6min
Thiago Ize (University of Utah, currently Solid Angle)http://www.cs.utah.edu/~thiago/cs7650/hw12/
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Splitting
β’ Use more samples for the next recursion than the current oneβ’ Often done at the first point (the one visible to the camera)β’ Trade off: anti-aliasing vs variance
No splitting Splitting
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Example: Direct illumination
Sampling projected solid angle4 eye rays per pixel100 shadow rays
Sampling light source area4 eye rays per pixel100 shadow rays
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Example: Splitting for direct illumination
Stratified random sample locations4 eye rays per pixel
16 shadow ray
Stratified random sample locations64 eye rays per pixel
1 shadow ray
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Importance Sampling the BSDF
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Simplest method: cosine hemisphere samplingβ’ PDF proportional to the cosine:
β’ ππ ππππ = cos ππππππ
β’ Ideal for diffuse surfaces
β’ Given two uniform samples π’π’1,π’π’2:β’ ππππ = 2πππ’π’1β’ ππππ = cosβ1 π’π’2β’ (Derivation in Assignment 3)
Β© William Sherif
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Example: Phong BRDF
β’ Simple glossy BRDFβ’ Multiplies by a cosine lobe around the perfect reflection
ππ π₯π₯,ππππ,ππππ =ππ + 22ππ
cosππ πΌπΌππ
β’ πΌπΌππ is the angle formed by ππππ and ππππβ’ ππππ is the prefect reflection of ππππ πΌπΌππ
ΞΈβππππ
ππππ
ππππππ
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Sampling the cosine lobe
β’ Step 1: sample the cosine lobe about π§π§ axis β’ ππ ππβ² = cosππ ππβ² ππ+1
2ππ
β’ Step 2: compute ππππβ’ Reflect ππππ about ππ
β’ Step 3: transform ππβ² to coordinate system where ππππ is the π§π§ axisβ’ Matrix multiplication
πΌπΌππΞΈ
βππππππππ
ππππππ
π§π§
ππβ²
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Next Event Estimation
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Reduce variance by connecting to points on the light source
Sampling ππππ from the hemisphere
Sampling ππππ as a direction to a point on the light
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But it is a point, not a directionβ¦
β’ We sample a direction ππππ by sampling a point π¦π¦ on the lightβ’ With some pdf ππ π¦π¦ defined on the surface area of the lightβ’ But our integral is over the (hemi-) sphereβ’ We need to perform a change of variables
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From surface area to hemisphere
β’ Projected unit surface area at the light sourceβ’ cos πππ¦π¦
β’ Compute surface area on unit hemisphereβ’ Similar triangles 1
ππ
β’ 2D 1ππ2
β’ We must also account for the visibility!β’ ππ π₯π₯,π¦π¦
β’ Result: cos πππ¦π¦ππ2
ππ π₯π₯,π¦π¦
π¦π¦
π₯π₯
ππ
ππππππ
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How to select the light source
β’ Total powerβ’ Uniformly β’ Estimated unoccluded contributionβ’ Estimated occluded contribution (prepass)β’ Clustering to support many lights
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What if a BSDF sample hits the light
β’ Cannot count both: would yield twice the actual value!β’ Could average both and divide by two high varianceβ’ Or: use MIS
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Multiple Importance Sampling (MIS)Veach & Guibas 1995
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Combining multiple sampling techniques
β’ Sampling the BSDF
β’ Sampling the light surfacep(y)= cosΞΈy/rxy
2
x
y
p(Ο)= cosΞΈx/Ο
x
y
ΞΈx
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Both techniques perform well for different effects
Light surface BSDF MIS
Light surface BSDF MIS
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How to combine them?
β’ Form an affine combination of the estimators:
οΏ½π‘π‘βππ
οΏ½ππ
πππ‘π‘
π€π€π‘π‘ π₯π₯π‘π‘,ππππ π₯π₯π‘π‘,ππ
πππ‘π‘ πππ‘π‘ π₯π₯π‘π‘,ππ
β’ Theoretically, any weighting functions π€π€π‘π‘ π₯π₯ work, provided:β’ π€π€π‘π‘ π₯π₯ = 0 if πππ‘π‘ π₯π₯ = 0β’ βπ‘π‘ π€π€π‘π‘ π₯π₯ = 1 βπ₯π₯
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First, all densities need to be in the same measureβ’ ππ ππ = cos ππ
ππhas unit π π ππβ1 (density on the hemisphere)
β’ ππ π¦π¦ has unit ππβ2 (density on surface area)
β’ Applying the same logic as before, we can transform them:
β’ ππ π¦π¦|ππ = cos ππππ
cos πππ¦π¦ππ2
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Balance heuristic
β’ Provably good choice: minimizes upper bound of the variance
π€π€π‘π‘ π₯π₯ =πππ‘π‘πππ‘π‘ π₯π₯
βππ ππππππππ π₯π₯
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Power, maximum, and cuttoff heuristics
β’ Can sometimes perform better for low-variance techniquesβ’ Amplify the weights to remove residual noise from βbadβ techniques
β’ Maximum: only consider technique with largest πππ‘π‘πππ‘π‘ π₯π₯β’ Cutoff: Balance, but set πππ‘π‘πππ‘π‘ π₯π₯ to zero if below some threshold
β’ Power heuristic: π€π€π‘π‘ π₯π₯ = πππ‘π‘πππ‘π‘ π₯π₯2
βππ ππππππππ π₯π₯ 2
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Comparison β Image of a light source on surfaces with different roughness
more diffusemore specular
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Optimal weights
β’ It is possible to derive the optimal set of functions π€π€π‘π‘ π₯π₯β’ Result: linear system involving many integralsβ’ Kondapaneni et al 2019β’ (Our recent SIGGRAPH paper)
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Putting it all together
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A less basic path tracer
β’ Given a point visible to the cameraβ’ (1) Compute direct illumination
β’ Select light source and point on the lightβ’ Trace a shadow ray for ππ π₯π₯,π¦π¦β’ Sample BSDF and check if a light was intersectedβ’ Compute MIS weights and add estimate
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A less basic path tracer
β’ Given a point visible to the cameraβ’ (1) Compute direct illumination
β’ Select light source and point on the lightβ’ Trace a shadow ray for ππ π₯π₯,π¦π¦β’ Sample BSDF and check if a light was intersectedβ’ Compute MIS weights and add estimate
β’ (2) Terminate with Russian roulette
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A less basic path tracer
β’ Given a point visible to the cameraβ’ (1) Compute direct illumination
β’ Select light source and point on the lightβ’ Trace a shadow ray for ππ π₯π₯,π¦π¦β’ Sample BSDF and check if a light was intersectedβ’ Compute MIS weights and add estimate
β’ (2) Terminate with Russian rouletteβ’ (3) Continue the path
β’ Could sample a new direction from the BSDF, or re-use the one from (1)
β’ Go back to (1), repeat until RR terminates
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Many improvements possible
β’ Splitting (e.g., at first intersection)β’ Combine multiple light sampling strategies (via MIS)β’ Store statistics to approximate πΏπΏππ for importance sampling (βguidingβ,
later)β’ And so onβ¦