A Mathematical Framework for Efficient Closed-Form Single Scattering. Vincent Pegoraro Mathias Schott Philipp Slusallek

Transcription

1 A Mathematical Framework for Efficient Closed-Form Single Scattering Vincent Pegoraro Mathias Schott Philipp Slusallek

2 Outline 1. Introduction 2. Related Work 3. Air-Light Integral 4. Complexity Reduction Framework 5. Computational Complexity Analysis 6. Implementation 7. Results 8. Discussion and Future Work 9. Conclusion

3 Introduction Applications Entertainment: motion-picture / video-game Industrial design: automotive / architectural Safety-oriented research: exit / traffic signs Motivation Efficient simulation of accurate light transport Closed-form solution to air-light integral High order of computational complexity

4 Related Work Numerical Quadrature Ray-marching / volume-slicing: [Max 86, ] Prone to under-sampling artifacts Analytic Approaches Directional lights: [Blinn 82, ] Point lights: Semi-analytic solutions: [Lecocq et al. 00, ] Approximate Closed-form solutions: [Pegoraro et al. 09, ] High order of computational complexity

5 Air-Light Integral Measured Radiance

6 Reduced Radiance Air-Light Integral

7 Medium Radiance Air-Light Integral

8 Air-Light Integral Angular Distributions

9 Air-Light Integral Angular Distributions Phase function Light source

10 Air-Light Integral Medium Radiance (Sequel) A few expansions and substitutions later with the change of variable of integration d(n, k) distribution w/ 0/1/3-deep nested loops

11 Air-Light Integral and the antiderivative defined as w/ complex-valued exponential integral giving Overall O(N 6 ) computational complexity

12 Complexity Reduction Framework Outline Complexity Reduction of Medium Radiance Complexity Reduction of Antiderivative Antiderivative with Coefficients Reuse

13 Complexity Reduction Framework Medium Radiance: O(N ) O(N ) Algebraic manipulations, expand, substitute, rearrange

14 Complexity Reduction Framework Antiderivative Expand factorized nested sums

15 Complexity Reduction Framework Antiderivative: O(N 4 ) O(N 3 ) A few loop permutations, rearranging of the terms, and algebraic identities later where M = 1, or M = 2m F 3 = 0 for air-light

16 tabulate Complexity Reduction Framework Coefficients Reuse: O(N 3 ) O(N 2 ) Algebraic manipulations, expand, substitute, rearrange

17 Computational Complexity Analysis Order of Complexity of Individual Schemes Antiderivative: original vs. reduced Antiderivative: reduced vs. coefficients reuse Medium radiance: original vs. reduced

18 Implementation Avoiding Redundant Computation Evaluate exponential integral before iterating Compute power/factorial terms incrementally Optimization for air-light integration

19 Results Order of Complexity of Individual Schemes Phase function Light source Performance:

20 Results Performance Characteristics GPU-based fragment shader OpenGL & Cg at on GeForce GTX 280 CPU-based implementation Offline rendering system Benchmarks evaluating chromatic scattering Table of Coefficients Up to 16-term distributions: 0.88MB in 70ms Precomputation may be carried out on the fly

21 Results Low-Degree Angular Distributions on GPU 4.51x 3.28x 2.34x 3.66x

22 Results Low-Degree Angular Distributions on GPU Closed form Sun et al. Eddington

23 Results Low-Degree Angular Distributions on GPU Closed form Lecocq et al. Rayleigh

24 Results High-Degree Angular Distributions on CPU 11.0x 14.5x Speed-up increases with problem size

25 Results High-Degree Angular Distributions on CPU Closed form

26 Results High-Degree Angular Distributions on CPU Dual formulation

27 Results High-Degree Angular Distributions on CPU Reduced closed form Original closed form

28 Discussion and Future Work Table of Coefficients Currently a straightforward regular array OK due to typically small storage requirements 4-D not so OK for higher-degree representations Relatively sparse data Indices define bounds on subsequent indices Investigate alternative storage layouts

29 Conclusion Contributions Alternative formulation of closed-form solution Algebraic patterns and mathematical identities Analytic simplifications (no ad-hoc approx.) Benefits Reduce order of computational complexity Speed-ups increase with problem size Preserve generality and accuracy of solution Efficient generation of reference images

30 Acknowledgements German Research Foundation (DFG) Anonymous reviewers Thank you!