Accelerating Sphere Tracing Csaba Blint , Gbor Valasek Etvs Lornd - - PowerPoint PPT Presentation

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Accelerating Sphere Tracing Csaba Blint , Gbor Valasek Etvs Lornd - - PowerPoint PPT Presentation

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Accelerating Sphere Tracing Csaba Blint , Gbor Valasek Etvs Lornd University, Hungary Table of Contents Introduction to Sphere Tracing Relaxed Sphere Tracing Current best Enhanced Sphere Tracing Our first contribution

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SLIDE 1

Accelerating Sphere Tracing

Csaba Bálint, Gábor Valasek

Eötvös Loránd University, Hungary

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➢ Introduction to Sphere Tracing ✓ Relaxed Sphere Tracing ➢ Enhanced Sphere Tracing ✓ Optimally approximates smooth surfaces ✓ ~50% faster than relaxed sphere tracing ➢ Iteratively Increasing Resolution ✓ Increases resolution while rendering ✓ ~20% speedup

Table of Contents

Our first contribution

2

Our second contribution Current best

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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SLIDE 3

Ray-Surface Intersection

  • 𝒒 is a surface point

 𝑔 𝒒 = 0  implicit representation

  • Ray:

𝒕 𝑢 = 𝒒0 + 𝑢 ⋅ 𝒘0

  • Ray-surface intersections:

𝑔 𝒕 𝑢 = 0

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

3

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SLIDE 4

Singed Distance Function

  • 𝑔(𝒒): signed distance to the surface from point 𝒒

Unbounding sphere → Center: 𝒒0 → Radius: 𝑔(𝒒0)

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Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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SLIDE 5

Sphere Tracing Step

  • Travel 𝑔(𝒒) on ray – Cannot step over an intersection

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Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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SLIDE 6

Sphere Tracing Algorithm

  • Travel 𝑔(𝒒) on ray – Cannot step over an intersection

▪ Repeat: t += 𝑔 𝒒0 + 𝑢 ⋅ 𝒘0 ; ▪ Stop:

  • After a 100 iterations
  • When 𝑔 𝒒0 + 𝑢 ⋅ 𝒘0 is small
  • When 𝑢 is large

6 [1] HART J. C.: Sphere tracing: A geometric method for the antialiased ray tracing of implicit surfaces. The Visual Computer 12 (1994), 527–545

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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SLIDE 7

SDFE – Signed Distance Function Estimate

  • 𝑔(𝒒) is a lower bound to the actual distance

7 [1] HART J. C.: Sphere tracing: A geometric method for the antialiased ray tracing of implicit surfaces. The Visual Computer 12 (1994), 527–545

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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SLIDE 8

Slowdown

  • Large overlap – could have taken a larger step

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Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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SLIDE 9
  • Take a 1.6 x larger step
  • Check if spheres overlap
  • Step back if they don’t

9 [2] KEINERT B., Schäfer H., Korndörfer J., Ganse U., Stamminger M.: Enhanced Sphere Tracing. The Eurographics Association (2014)

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

Relaxed Sphere Stracing

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  • 11 tracing steps
  • 11 function evaluations

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Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

Divergent case – Sphere Tracing

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SLIDE 11
  • 11 tracing steps
  • 13 function evaluations

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Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

Divergent case – Relaxed Sphere Tracing

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SLIDE 12
  • 11 tracing steps
  • 12 function evaluations

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Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

Divergent case – (Our) Enhanced Sphere Tracing

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SLIDE 13

Enhanced Sphere Tracing Step I

Previous

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Next ? 𝑠𝑗−1 𝑠𝑗

  • Idea: Approximate flat surfaces optimally

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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SLIDE 14

Enhanced Sphere Tracing Step II

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𝑠𝑗−1 𝑠𝑗 Tangent 𝑠𝑗 𝑒𝑗 𝑠𝑗+1 𝑠𝑗+1 = ?

  • Unbounding sphere is expected to touch

✓ The previous unbounding sphere ✓ The flat surface Next step size to try: 𝑒𝑗+1 = 𝑠

𝑗 + 𝑠 𝑗+1

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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SLIDE 15

Enhanced Sphere Tracing Step III

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Similar 𝑠𝑗−1 𝑠𝑗 Triangles 𝑠𝑗 𝑒𝑗 𝑠𝑗+1 𝑠𝑗+1

Solution: 𝑠

𝑗+1 = 𝑠 𝑗 ⋅ 𝑒𝑗 − 𝑠 𝑗−1 + 𝑠 𝑗

𝑒𝑗 + 𝑠

𝑗−1 + 𝑠 𝑗

Next step size to try: 𝑒𝑗+1 = 𝑠

𝑗 + 𝑠 𝑗+1

What if the surface is curved?

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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SLIDE 16

Enhanced Sphere Tracing Step – Relaxation

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𝑠𝑗−1 𝑠𝑗 𝑠𝑗 𝑒𝑗 𝑠𝑗+1 𝑠𝑗+1

Relax enhanced step: 𝑒𝑗+1 = 𝑠

𝑗 + 𝜕 ⋅ 𝑠 𝑗+1

Eg.: 𝜕 = 0.9 relaxation parameter. Different goal: fix slight concavity  Optimal on smooth surfaces

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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Test Scene I: CSG composition

17 Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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Test Scene II: Village

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Scene III: Mandelbulb 3D Fractal

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Non-smooth surface!

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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Enhanced Sphere Tracing: Relative runtimes

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Classic ST Faster than relaxed ST

  • But why?

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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Enahnced Sphere Tracing: Step back rato

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Enhanced sphere tracing → Only a few fallbacks → Better thread coherence → Even faster on the GPU Ratio of overshoots visualized

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

  • White pixels:

100% wrong prediction

  • Black pixels:

100% good prediction

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Enhanced Sphere Tracing: Relative Error

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Close to the surface our algorithm overtakes

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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Enhanced Sphere Tracing: Relative Performance

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1.6 times better than Relaxed ST

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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Cone Tracing

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Unbounding sphere

[1] HART J. C.: Sphere tracing: A geometric method for the antialiased ray tracing of implicit surfaces. The Visual Computer 12 (1994), 527–545

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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Iteratively Increase Resolution

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4 new cones penetrate the scene further

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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Iteratively Increasing Resolution Results

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Baseline: single res.

Using 2 different resolutions: 20%

Multiple resolution change is expencive

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

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Summary

➢ Enhanced Sphere Tracing ✓ Optimally approximates smooth surfaces ✓ 50% faster than relaxed sphere tracing ➢ Iteratively Increasing Resolution ✓ Increases resolution while rendering ✓ 20% faster & Interactive or real-time rendering

[2] KEINERT B., Schäfer H., Korndörfer J., Ganse U., Stamminger M.: Enhanced Sphere Tracing. The Eurographics Association (2014) [1] HART J. C.: Sphere tracing: A geometric method for the antialiased ray tracing of implicit surfaces. The Visual Computer 12 (1994)

Accelerating Sphere Tracing, Csaba Bálint, Gábor Valasek

Thank you for your attention!