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5D COVARIA NCE TRACING FOR EFFICIENT DEFOCUS AND MOTION BLUR. Laurent Belcour 1 Cyril Soler 2 Kartic Subr 3 Nicolas Holzschuch 2 Frédo Durand 4. 1 Grenoble Université, 2 Inria , 3 UC London, 4 MIT CSAIL. Blur is costly to simulate !. t ime integration. space reconstruction.

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5d covaria nce tracing for efficient defocus and motion blur

5D COVARIANCE TRACINGFOR EFFICIENT DEFOCUS AND MOTION BLUR

Laurent Belcour1 Cyril Soler2Kartic Subr3 Nicolas Holzschuch2Frédo Durand4

1 Grenoble Université, 2 Inria, 3 UC London, 4 MIT CSAIL



5d covaria nce tracing for efficient defocus and motion blur

time

integration

space

reconstruction


Previous works a posteriori
Previousworks: a posteriori

  • Image spacemethods

    • [Mitchell 1987], [Overbeck et al. 2009],

    • [Sen et al. 2011], [Rousselle et al. 2011]

  • Integrationspace

    • [Hachisukaet al. 2008]

  • Reconstruction

    • [Lehtinenet al. 2011], [Lehtinenet al. 2012]

  • Easy to plug

  • Requirealready dense sampling

  • Rely on point samples


Previous work a priori
Previouswork: a priori

  • First orderanalysis[Ramamoorthiet al. 2007]

  • Frequencyanalysis[Durand et al. 2005]


Previous work a priori1
Previouswork: a priori

  • First orderanalysis[Ramamoorthiet al. 2007]

  • Frequencyanalysis[Durand et al. 2005]

Fourier transform

zoom


Previous work a priori2
Previouswork: a priori

Predict full spectrum

Predictbounds

Compact & efficient

Special cases formula

  • Anisotropic information

  • Unwieldy

  • [Soler et al. 2009]

  • [Egan et al. 2009], [Bagheret al. 2013], [Mehaet al. 2012]

None canworkwith full global illumination!


Our idea 5d covariance representation
Our idea: 5D Covariance representation


5d covariance representation
5D Covariance representation

  • Use second moments

    • 5x5 matrix

    • Equivalent to Gaussianapprox.

  • Formulate all interactions

    • Analytical matrix operators

    • Gaussianapprox. for reflection

  • Nice properties

    • Symmetry

    • Additivity

angle (2D)

space (2D)

time


Contributions
Contributions

  • Unified temporal frequencyanalysis

  • Covariance tracing

  • Adaptive sampling & reconstruction algorithm


Our algorithm
Our algorithm

Accumulate 5D Covariance

in screenspace


Our algorithm1
Our algorithm

Accumulate 5D Covariance

in screenspace

angle

Estimate 5D

samplingdensity

time

angle

time

angle

time


Our algorithm2
Our algorithm

Accumulate 5D Covariance

in screenspace

Estimate 5D

samplingdensity

Estimate 2D reconstruction filters


Our algorithm3
Our algorithm

Accumulate 5D Covariance

in screenspace

Estimate 5D

samplingdensity

Estimate 2D reconstruction filters

Acquire5D samples

Reconstruct image


5d covaria nce tracing for efficient defocus and motion blur

Accumulate 5D Covariance

in screenspace

Estimate 5D

samplingdensity

Estimate 2D reconstruction filters

Acquire5D samples

Reconstruct image


C ovariance tracing
Covariance tracing

  • Add information to light paths

  • Update the covariance along light path

  • Atomicdecomposition for genericity


C ovariance tracing1
Covariance tracing

Free transport

Free transport


C ovariance tracing2
Covariance tracing

Free transport

Reflection


C ovariance tracing3
Covariance tracing

Free transport

Reflection

Free transport


C ovariance tracing4
Covariance tracing

Free transport

Reflection

Occlusion

Free transport

spatial visibility


C ovariance tracing5
Covariance tracing

Free transport

Occlusion

Free transport


C ovariance tracing6
Covariance tracing

Free transport

Free transport

Reflection


C ovariance tracing7
Covariance tracing

Free transport

Reflection

Free transport


Just a chain of operators
Just a chain of operators

Free transport

Occlusion

Curvature

Symmetry

BRDF

Lens


What about motion
What about motion?


We could rewrite all operators
Wecould rewrite all operators…

Occlusion

withmovingoccluder

Curvaturewithmovinggeometry

BRDF withmovingreflector

Lens withmoving camera


We will not rewrite all operators
Wewill not rewrite all operators!

Occlusion

Curvature

BRDF

Lens

Motion

Inverse Motion


Motion operator

angle

angle

Motion operator

space

space

time

time

Reflectionwithmovingreflector


Motion operator1

angle

Motion operator

space

time

Reflection

Motion


Motion operator2

angle

angle

Motion operator

space

space

time

time

Inverse Motion

Reflection

Motion


Accumulate covariance
Accumulate covariance

first light path

second light path

final covariance


5d covaria nce tracing for efficient defocus and motion blur

Accumulate 5D Covariance

in screenspace

Estimate 5D

samplingdensity

Estimate 2D reconstruction filters

Acquire5D samples

Reconstruct image


Using covariance information
Using covariance information

  • How canweextractbandwidth ?

    • Using the volume

    • Determinant of the covariance

  • How canweestimate the filter ?

    • Frequencyanalysis of integration [Durand 2011]

    • Slicing the equivalentGaussian

space

space

time


5d covaria nce tracing for efficient defocus and motion blur

Accumulate 5D Covariance

in screenspace

Estimate 5D

samplingdensity

Estimate 2D reconstruction filters

Acquire 5D samples

Reconstruct image


Implementation details occlusion
Implementationdetails: occlusion

  • Occlusion using a voxelizedscene

  • Use the 3x3 covariance of normals distribution

  • Evaluateusing ray marching


Results the helicopter
Results: the helicopter

Our algorithm

Equal time Monte-Carlo


Results the snooker
Results: the snooker

defocusblur

motion blur

BRDF blur

Equal-time Monte Carlo

Our method


Results the snooker1
Results: the snooker

  • Our method: 25min

  • Eq. quality Monte Carlo: 2h25min

    • 200 light fieldsamples per pixel

  • Covariance tracing: 2min 36s

    • 10 covariance per pixel

  • Reconstruction: 16s


Conclusion
Conclusion

  • Covariance tracing

    • Generatebetterlight paths

    • Simple formulation

  • Unifiedfrequencyanalysis

    • Temporal light fields

    • No specialcase


Future work
Future work

  • Tracing covariance has a cost

    • Mostly due to the local occlusion query

  • New operators

    • Participatingmedia