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 (θ d = θ i )  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/θ W is 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.  Four terms in UTD and BTM  When θ W > π, two shadow boundaries and two reflection boundaries  When θ W ≤ 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 Numerical Challenge: Zone-Boundary Singularity UTD: BTM: At the boundaries:

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

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

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. 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 Frequency (Hz) Magnitude (dB re. 1)

19. Edge Subdivision for Discrete-Time IRs  Sample-aligned edge segments: one for each IR sample  Pros  Accurate  Good with approx for sample n 0  Cons  Slow to compute  Must be recalculated when S or R moves n2n2 n1n1 n1n1 n0n0 n2n2 S R

20. Edge Subdivision for Discrete-Time IRs  Even edge segments  Pros  Trivial to compute  Independent of S and R positions  Cons  No explicit boundaries for n 0 → harder to handle singularity  Requires a scheme for multi-sample distribution S R S R 6.11.54.93.30.84.91.53.36.1

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(n 0 )  Use even segments for the rest of the edge  Can be precomputed  Limited recalculation for moving source or receiver 6.14.93.04.93.06.1 n2n2 n1n1 n1n1 n0n0 n2n2

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 S/RZone Segment Norm.Max. PairSizeInteg.SizeInteg.Proc.Error (samples) Time(dB) 141-point1001-point.0214.97 1all5-pointN/A 1.0000 0

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

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.