Interactive Sound Rendering

1. Interactive Sound Rendering Session5: Simulating Diffraction Paul Calamia pcalamia@cs.princeton.edu P. Calamia, M. Lin, D. Manocha, L. Savioja, N. Tsingos

2. Overview • Motivation: Why Diffraction? • Simulation Methods • Frequency Domain: Uniform Theory of Diffraction (UTD) • Time Domain: Biot-Tolstoy-Medwin Formulation (BTM) • Acceleration Techniques • UTD: Frequency Interpolation • BTM: Edge Subdivision • Both: Path Culling • Implementation Example: UTD with Frustum Tracing • Additional Resources

3. Motivation • Wavelengths of audible sounds can be comparable to (or larger than) object dimensions so diffraction is an important acoustic propagation phenomenon • Unlike wave-based simulation techniques, geometrical-acoustics (GA) techniques omit diffraction • Incorrect reflection behavior from small surfaces • No propagation around occluders / into shadow zones • Sound-field discontinuities at reflection and shadow boundaries

4. Continuity of Sound Fields with Diffraction • Example: reflection from a faceted arch with and without diffraction • Even with low-resolution geometry, GA + diffraction yields a continuous sound field Images courtesy of Peter Svensson, NTNU

5. Propagation into Shadow Zones • Example: propagation at a street crossing • Diffraction from the corner allows propagation into areas without line of sight to the source Images courtesy of Peter Svensson, NTNU

6. Propagation into Shadow Zones • Example: propagation at a street crossing • Diffraction from the corner allows propagation into areas without line of sight to the source • Note the continuous wavefronts too Images courtesy of Peter Svensson, NTNU

7. Common Diffraction Methods • Uniform Theory of Diffraction (UTD) • Keller ’62, Kouyoumjian and Pathak ‘74 • Typically used in the frequency domain although a time-domain formulation exists • Assumptions • Ideal wedge surfaces (perfectly rigid or soft) • High frequency • Infinitely long edges • Far-field source and receiver • For acoustic simulations see Tsingos et al. ’01, Antonacci et al. ’04, Taylor et al. ‘09

8. Uniform Theory of Diffraction • UTD gives the diffracted pressure as a function of incident pressure, distance attenuation, and a diffraction coefficient • Angle of diffraction = angle of incidence (θd = θi) • Ray-like paths on a cone of diffraction Images from Tsingos et al., ‘01

9. Uniform Theory of Diffraction (UTD)

10. Common Diffraction Methods • Biot-Tolstoy-Medwin (BTM) • Biot and Tolstoy ’52, Medwin ’81, Svensson et al. ‘99 • Typically used in the time domain although a frequency-domain formulation exists • Assumptions • Ideal wedge surfaces (perfectly rigid or soft) • Point-source insonification • For acoustic simulations see Torres et al. ’01, Lokki et al. ’02, Calamia et al. ’07 and ‘08

11. Biot-Tolstoy-Medwin Diffration (BTM) • Wedge • W = exterior wedge angle • ν = π/θWis the wedge index • Source and Receiver: Edge-Aligned Cylindrical Coordinates (r,, z) • r = radial distance from the edge •  = angle measured from a face • z = distance along the edge • Other • m = dist. from source to edge point • l = dist. from receiver to edge point • A = apex point, point of shortest path from S to R through the line containing the edge

12. Biot-Tolstoy-Medwin Diffration (BTM)

13. Numerical Challenge: Zone-Boundary Singularity • Four terms in UTD and BTM • When θW> π, two shadow boundaries and two reflection boundaries • When θW≤ π, only reflection boundaries but inter-reflections (order 2, 3, …) are possible • Each diffraction term is associated with a “zone boundary” • Geometrical-acoustics sound field is discontinuous • Diffracted field has a complimentary discontinuity to compensate At the boundaries: BTM: UTD:

14. Numerical Challenge: Zone-Boundary Singularity Source Position Reflection Boundary Shadow Boundary

15. Numerical Challenge: Zone-Boundary Singularity Normalized Amplitude Source Position Reflection Boundary Shadow Boundary

16. Numerical Challenge: Zone-Boundary Singularity • Approximations exist to allow for numerically robust implementations • BTM (Svensson and Calamia, Acustica ’06): Serial expansion around the apex point • UTD (Kouyoumjian and Pathak ’74): Approximation valid in the “neighborhood” of the zone boundaries

17. Acceleration Techniques • Reduce computation for each diffraction component • UTD: Frequency Interpolation • BTM: Edge Subdivision • Reduce the number of diffraction components through path culling • Shadow Zone • Zone-Boundary Proximity

18. Magnitude (dB re. 1) Frequency (Hz) Frequency Interpolation • Magnitude of diffraction transfer function typically is smooth • Phase typically is ~linear • Compute UTD coefficients at a limited number of frequencies (e.g. octave-band center frequencies 63, 125, 250, …, 8k, 16k Hz) and interpolate

19. S R n2 n1 n0 n1 n2 Edge Subdivision for Discrete-Time IRs • Sample-aligned edge segments: one for each IR sample • Pros • Accurate • Good with approx for sample n0 • Cons • Slow to compute • Must be recalculated when S or R moves

20. S S R R Edge Subdivision for Discrete-Time IRs • Even edge segments • Pros • Trivial to compute • Independent of S and R positions • Cons • No explicit boundaries for n0→ harder to handle singularity • Requires a scheme for multi-sample distribution 6.1 4.9 3.3 1.5 0.8 1.5 3.3 4.9 6.1

21. Edge Subdivision for Discrete-Time IRs • Hybrid Subdivision • Use a small number of sample-aligned segments around the apex point • High accuracy for the impulsive (high energy) onset • Easy to use with approximations for h(n0) • Use even segments for the rest of the edge • Can be precomputed • Limited recalculation for moving source or receiver 6.1 4.9 3.0 3.0 4.9 6.1 n2 n1 n0 n1 n2

22. Hybrid Edge Subdivision Example • 35 1.2 m x 1.2 m rigid panels • Interpanel spacing 0.5 m • 5 m above 2 source and 2 receiver positions • Evaluate • The number of sample-aligned segments: 1 – 10 • The size of the even segments: maximum sample span of 40, 100, and 300 • The numerical integration technique • 1-Point (midpoint) • 3-Point (Simpson’s Rule) • 5-Point (Compound Simpson’s Rule with Romberg Extrapolation)

23. Hybrid Edge Subdivision

24. Path Culling Significant Growth in Paths Due to Diffraction

25. Path Culling • Option 1: For each wedge, compute diffraction only for paths in the shadow zone • Intuition: Sound field in the “illuminated” area around a wedge will be dominated by direct propagation and/or reflections, shadow zone will receive limited energy without diffraction • Pro: Allows propagation around obstacles • Con: Ignores GA discontinuity at reflection boundary • Implementations described in Tsingos et al. ’01, Antonacci et al. ’04, Taylor et al. ’09

26. Path Culling • Option 2: Compute diffraction only when amplitude is “significant” • Intuition: numerically/perceptually significant diffracted paths are those with highest amplitude and/or energy, typically those with the receiver close to a zone boundary • Pro: Eliminates large discontinuities in the simulated sound field • Con: Does not allow for propagation deep into shadow zones • Implementation described in Calamia et al. ‘08

27. Path Culling • Predict relative size based on proximity to a zone boundary and apex-point status • Significant variation in diffraction strength (~220 dB in this example) Source Position Reflection Boundary Shadow Boundary

28. Path Culling Results • Numerical and subjective evaluation in a simple concert-hall model ABX tests comparing full IRs with culled IRs, 17 subjects An angular threshold of 24° culls ~92% of the diffracted components

29. Simulation Example: Frustum Tracing • Goals • Find propagation paths around edges • Render at interactive rates • Allow dynamic sources, receivers, and geometry • Method • Frustum tracing with dynamic BVH acceleration • Diffraction only in the shadow region • Diffraction paths computed with UTD

30. Step 1: Identify Edge Types (Preprocess) • Mark possible diffracting edges • Exterior edges • Disconnected edges

31. Step 2: Propagate Frusta • Propagate frusta from source through scene

32. Step 2: Propagate Frusta • Propagate frusta from source through scene • When diffracting edges are encountered, make diffraction frustum

33. Step 3: Auralization • If receiver is inside frustum • Calculate path back to source • Attenuate path with UTD coefficient and add to IR • Convolve audio with IR • Output final audio sample

34. System Demo

35. Future Work • Direct comparison of UTD and BTM • Numerical accuracy • Computation time • Subjective Tests • Limited subjective tests of auralization with diffraction • Static scenes • Torres et al. JASA ‘01 • Calamia et al. Acustica ‘08 • Dynamic scenes • None

36. Additional Resources • F. Antonacci, M. Foco, A. Sarti, and S. Tubaro, “Fast modeling of acoustic reflections and diffraction in complex environments using visibility diagrams. In Proc. 12th European Signal Processing Conference (EUSIPCO ‘04), pp. 1773 - 1776, 2004. • P. Calamia, B. Markham, and U. P. Svensson, “Diffraction culling for virtual-acoustic simulations,” Acta Acustica united with Acustica, Special Issue on Virtual Acoustics, 94(6), pp. 907 - 920, 2008. • P. Calamia and U. P. Svensson, “Fast time-domain edge-diffraction calculations for interactive acoustic simulations,” EURASIP Journal on Advances in Signal Processing, Special Issue on Spatial Sound and Virtual Acoustics, Article ID 63560, 2007. • A. Chandak, C. Lauterbach, M. Taylor, Z. Ren, and D. Manocha, “ADFrustum: Adaptive frustum tracing for interactive sound propagation,” IEEE Trans. on Visualization and Computer Graphics, 14, pp. 1707 - 1722, 2008. • R. Kouyoumjian and P. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface. In Proc. IEEE, vol. 62, pp. 1448 - 1461, 1974.

37. Additional Resources • T. Lokki, U. P. Svensson, and L. Savioja, “An efficient auralization of edge diffraction,” In Proc. Aud. Engr. Soc. 21st Intl. Conf. on Architectural Acoustics and Sound Reinforcement, pp. 166 - 172, 2002. • D. Schröder and A. Pohl, “Real-time hybrid simulation method including edge diffraction,” In Proc. EAA Symposium on Auralization, Otaniemi, 2009. • U. P. Svensson, R. I. Fred, and J. Vanderkooy, “An analytic secondary-source model of edge diffraction impulse responses,” J. Acoust. Soc. Am., 106(5), pp. 2331 - 2344, 1999. • U. P. Svensson and P. Calamia, “Edge-diffraction impulse responses near specular-zone and shadow-zone boundaries,” Acta Acustica united with Acustica, 92(4), pp. 501 - 512, 2006. • M. Taylor, A. Chandak, Z. Ren, C. Lauterbach, and D. Manocha, “Fast edge-diffraction for sound propagation in complex virtual environments,” In Proc. EAA Symposium on Auralization, Otaniemi, 2009.

38. Additional Resources • R. Torres, U. P. Svensson, and M. Kleiner, “Computation of edge diffraction for more accurate room acoustics auralization,” J. Acoust. Soc. Am., 109(2), pp. 600 - 610, 2001. • N. Tsingos, T. Funkhouser, A. Ngan, and I. Carlbom, “Modeling acoustics in virtual environments using the Uniform Theory of Diffraction,” In Proc. ACM Computer Graphics (SIGGRAPH ’01), pp. 545 - 552, 2001. • N. Tsingos, I. Carlbom, G. Elko, T. Funkhouser, and R. Kubli, “Validation of acoustical simulations in the Bell Labs box,” IEEE Computer Graphics and Applications, 22(4), pp. 28 - 37, 2002. • N. Tsingos and J.-D. Gascuel, “Soundtracks for computer animation: Sound rendering in dynamic environments with occlusions,” In Proc. Graphics Interface97, Kelowna, BC, 1997. • N. Tsingos and J.-D. Gascuel, “Fast rendering of sound occlusion and diffraction effects for virtual acoustic environments,” In Proc. 104th Aud. Engr. Soc. Conv., 1998. Preprint no. 4699.