A new ray tracing method based on phase space...2016/04/06 · • Convergence is 1/N instead of 1/ 𝑁 Using ray tracing on PS far fewer rays need to be traced to PS intensity with

1 CASA-day April 6, 2016

A new ray tracing method

based on phase space

Carmela Filosa

Jan ten Thije Boonkkamp

Wilbert IJzerman

Centre for Analysis, Scientific computing and Applications 2

Phase space based ray tracing

• Introduction of ray tracing

– A two-dimensional optical system: the TIR-collimator

– Physical background

– Issues with ray tracing

• The idea: phase space based ray tracing

– How to improve ray tracing

– Boundaries are mapped onto boundaries

• Implementation

– Refinement procedure

– Boundary detection

– Calculation of intensity

• Results

– Speed of convergence

• Conclusion and outlook

Outline

Centre for Analysis, Scientific computing and Applications

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Ray tracing is like playing billiards


Ray tracing for a TIR-collimator


Background

𝑠2 =𝑛1𝑛2𝑠1 + 1 −

𝑛1𝑛2

2

+𝑛1𝑛2

2

𝑛, 𝑠1 2 −𝑛1𝑛2𝑛, 𝑠1 𝑛 𝑛

𝑡2 = 𝑡1 − 2 𝑡1, 𝑛 𝑛

Law of reflection::

Law of refraction::

I = 𝑑ϕ

𝑑𝑡 𝑐𝑑 = 𝑙𝑚/𝑟𝑎𝑑

L = 𝑑2ϕ

cos(𝑡) 𝑑𝑎𝑑𝑡 𝑐𝑑/𝑚

Intensity::

Luminance::

Étendue:: 𝑑𝐸 = 𝑛 𝑑𝑎 cos(𝑡) 𝑑𝑡 𝑚 𝑟𝑎𝑑

Flux:: Measure of the perceived power of light by the human eye, (ϕ = 𝑙𝑚 ) .

n reflected ray incident ray

• Shoot rays from source to target randomly


What is ray tracing?

Our model

• Intensity calculated by counting the rays in bins

Convergence (Compared to a simulation with a large number of rays)


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• Full system is characterized by phase space

• 𝒫𝑠: 𝒮𝗑 −1,1 (position 𝑥∊𝒮 and angle τ = sin 𝑡) 𝒫𝑡: 𝒯𝗑 −1,1 (position 𝑞∊𝒯 and angle η = sin θ).

• The optical map ℳ:𝒫𝑠 𝒫𝑡 maps region to region and

boundary to boundary (Riesz)

How to improve ray tracing?

𝒫𝑠 𝒫𝑡


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Implementation: How to find the boundaries on source phase space?

• On source phase space: take a coarse grid

• If paths are different refine unless already fine enough


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Implementation: How to find the boundaries on target phase space?

Method 1: use alpha-shapes 1

For a cloud of points, like eating stracciatella ice-cream with a spoon without eating the chocolate

1 F. Bernardini and C. L. Bajaj, “Sampling and reconstructing manifolds using alpha-shapes,” (1997)

Issue: how to choose the size of your spoon? (We used an Étendue argument)


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Method 2: use that boundary is mapped on boundary for regions of phase space 1


1 H. Ries and A. Rabl, “Edge-ray principle of nonimaging optics,” JOSA A 11, 2627–2632 (1994)


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1 H. Ries and A. Rabl, “Edge-ray principle of nonimaging optics,” JOSA A 11, 2627–2632 (1994)

Method 2: use that boundary is mapped on boundary for regions of phase space 1

More points around the boundaries


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To calculate intensity: integrate along a line If we assume constant brightness (Lambertian source) We obtain:

𝐼𝑃𝑆 η = 𝐿𝑡 𝑞, η 𝑑𝑞𝑞Π𝑗,2𝑖 η

𝑞Π𝑗,2𝑖−1 η𝑖,𝑗

𝐿𝑆 𝑥, 𝜏 = 𝐿𝑡 𝑞, η = 1

𝐼𝑃𝑆 η = 𝑞Π𝑗,2𝑖 η − 𝑞Π𝑗,2𝑖−1 η

𝑖,𝑗


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• Method calculates intensity correctly

Results

• Convergence is 1/N instead of 1/ 𝑁

PS intensity with around 105 rays.

Reference intensity: MC ray tracing with 108 rays.

Using ray tracing on PS far fewer rays need to be traced to obtain the same accuracy.


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Conclusions and outlook

• This method is 2D only (meridional plane) but can be extended to 3D as well

• More rapid convergence 1/N versus 1/ 𝑁

• Future work: include Fresnel reflection and scattering phenomena

Acknowledgement:

I would like to thank Jorg Portegies for helpful discussions and providing his master thesis on which this work is based.


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How to choose the parameter α for the α-shapes method


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Stopping criterion for the triangulation refinement

Documents

A new ray tracing method based on phase space...2016/04/06 · • Convergence is 1/N instead of 1/ 𝑁 Using ray tracing on PS far fewer rays need to be traced to PS intensity with