NULL-COLLISION ALGORITHMSPART 2 TRANSMITTANCE ESTIMATION DELTA - - PowerPoint PPT Presentation

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Sep 10, 2023 276 likes •476 views

NULL-COLLISION ALGORITHMSPART 2 TRANSMITTANCE ESTIMATION DELTA TRACKING Extinction A B A B ransmittance T Distance A B MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING DISTANCE SAMPLING 2 DELTA TRACKING Extinction A

SLIDE 1

NULL-COLLISION ALGORITHMS—PART 2 TRANSMITTANCE ESTIMATION

SLIDE 2

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

DELTA TRACKING

2 — DISTANCE SAMPLING

Distance Extinction T ransmittance A B A B A B

SLIDE 3

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

DELTA TRACKING

3 — DISTANCE SAMPLING

Distance Extinction T ransmittance A B A B A B

2 samples

SLIDE 4

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

DELTA TRACKING

4 — DISTANCE SAMPLING

Distance Extinction T ransmittance A B A B A B

3 samples

SLIDE 5

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

DELTA TRACKING

5 — DISTANCE SAMPLING

Distance Extinction T ransmittance A B A B A B

4 samples

SLIDE 6

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

DELTA TRACKING

6 — DISTANCE SAMPLING

Distance Extinction T ransmittance A B A B A B

Each collision provides only binary inf

rmation

SLIDE 7

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

RATIO TRACKING

7 — DISTANCE SAMPLING

Distance Extinction T ransmittance A B A B A B

1) Remove termination 2) Compute weight

Y

µn(xi) ¯ µ

[Cramer 1978, Novák et al. 2014]

SLIDE 8

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

RATIO TRACKING

8 — DISTANCE SAMPLING

Distance Extinction T ransmittance A B A B

1) Remove termination 2) Compute weight

Y

µn(xi) ¯ µ

SLIDE 9

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

RATIO TRACKING

9 — DISTANCE SAMPLING

Distance Extinction T ransmittance A B A B

1) Remove termination 2) Compute weight

Y

µn(xi) ¯ µ

SLIDE 10

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

RATIO TRACKING

10 — DISTANCE SAMPLING

Distance Extinction T ransmittance A B A B

1) Remove termination 2) Compute weight

Y

µn(xi) ¯ µ

Extra steps => higher cost than delta tracking

SLIDE 11

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

RATIO TRACKING

11 — DISTANCE SAMPLING

Probabilistic TERMINATION replaced by WEIGHTING

Rational score instead of binary
Requires more steps than a delta-tracking estimator (must reach B)
Reduces the need for tight majorants
Loose majorants produce (more null collisions and therefore) finer estimates

SLIDE 12

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

RESIDUAL RATIO TRACKING

12 — DISTANCE SAMPLING

Compute part of the transmittance analytically

[Novák et al. 2014]

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MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

RESIDUAL RATIO TRACKING

13 — DISTANCE SAMPLING

CONTROL   component RESIDUAL   component Distance

Control transmittance  computed analytically Residual transmittance  estimated via  ratio tracking

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MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

RESIDUAL RATIO TRACKING

14 — DISTANCE SAMPLING

Distance

Piecewise exponential transmittance T rue transmittance

hT(t)i = Tcontrol(t) hTresidual(t)i

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MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

RESIDUAL RATIO TRACKING

15 — DISTANCE SAMPLING

HOMOGENEOUS and RESIDUAL HETEROGENEOUS components

Reduces noise by handling part of the transmittance analytically
Requires a space-partitioning data structure (e.g. octree) to be practical
Can handle negative residual extinctions

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MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

NEXT-FLIGHT ESTIMATORS

16 — DISTANCE SAMPLING

Score a weight at every tentative collision

Cramer [1978] combines next-flight estimation with delta and ratio tracking

NEXT-FLIGHT DELTA TRACKING

hT(t)i = T¯

µ(0, t) + n

X

j=1

µn(tj) ¯ µ(tj) T¯

µ(tj, t)

T ransmittance along the remaining segment through real + fictitious matter A B Fraction of fictitious matter

SLIDE 17

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

SUMMARY

17 — DISTANCE SAMPLING

DELTA TRACKING estimator

Relatively cheap but binary, inefficient w/ loose majorants

RATIO TRACKING estimator

More expensive, but also more accurate especially w/ loose majorants

RESIDUAL TRACKING estimators

Reduces variance by employing analytic computation for part of the transmittance function

NEXT-FLIGHT estimators

Further improve performance by scoring a weight at each step
Not fully explored yet in the context of rendering…

SLIDE 18

MONTE CARLO METHODS FOR PHYSICALLY BASED VOLUME RENDERING

ACKNOWLEDGEMENTS

18 — DISTANCE SAMPLING

Peter Kutz f

r tracing down many of the early delta tracking papers

Maurizio Nitti f

r help w/ illustrations