A RAY TRACING SIMULATION OF SOUND DIFFRACTION BASED ON ANALYTIC SECONDARY SOURCE MODEL

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1 19th European Signal Proessing Conferene (EUSIPCO 211) Barelona, Spain, August 29 - September 2, 211 A RAY TRACING SIMULATION OF SOUND DIFFRACTION BASED ON ANALYTIC SECONDARY SOURCE MODEL Masashi Okada, Takao Onoye and Wataru Kobayashi Graduate Shool of Info. Siene and Teh., Osaka University, 1-5 Yamadaoka, Suita, Osaka, , Japan Arnis Sound Tehnologies, Co., Ltd., Kitasenzoku, Ota-ku, Tokyo, , Japan {okada.masashi,onoye}@ist.osaka-u.a.jp, kobahan@arns.om ABSTRACT In this paper, we propose a novel method to estimate impulse response of sound diffration based on ray traing. This method is a Monte Carlo solution of multiple integration in analyti seondary soure model of edge-diffration, in whih sample values of the integrand are alulated by ray traing proess. Similarity between our method and general ray traing makes it possible to utilize various approahes developed for ray traing. In our implementation, a ray traing engine OptiX is employed, whih exhibits good aeleration on a graphis proessor. For more effiieny, we use importane sampling where insignifiant diffration paths are eliminated stohastially and thus reduing the number of samples. Based on these approahes, the proposed method ahieves a proessing performane of about 24M samples/se for the simulation of first-order diffration, whih demonstrates the appliability to interative simulation. 1. INTRODUCTION Ray traing is a popular tehnique to simulate wave propagation used in various fields suh as 3D omputer graphis (3DCG), eletromagnetis and aoustis beause of its omputational simpliity. For example, aousti analysis softwares suh as Odeon [1] and CATT [2] use ray traing. On the other hand, various aeleration and/or parallelization tehniques have been proposed aiming at real-time ray traing simulation in 3DCG field. Based on these tehniques, interative aousti simulation has been getting more pratial, whih an enhane reality and immersion in virtual reality. However, applying the general ray traing tehniques to aousti simulation results in lak of diffration whih is signifiant espeially in aoustis. For instane, aousti simulation results desribed in [3] show that the lak of first- and higher-order diffrations auses undesired shadow of sound. In addition, subjetive listening tests in [4] demonstrate that diffration is also important not only in oluded zones but also in non-oluded zones. There are several approahes to model edge-diffration suh as UTD (Uniform Theory of Diffration) [3, 5] and Analyti Seondary Soure Model (referred as the seondary soure model) [6]. Although UTD an alulate edgediffration more simply than the seondary soure model, errors in low-frequeny is signifiant [6] beause UTD is a high-frequeny approximation originally used in eletromagnetis. On the other hand, the seondary soure model an alulate edge-diffration aurately based on the exat Biot- Tolstoy solution [7]. However, this model requires us to solve multiple integrations for high-order diffration, whih results in omputational omplexity. In this paper, we fous on the seondary soure model and its auray. Based on this model, we propose a novel method to estimate edge-diffration impulse responses (IRs) using ray traing. The number of diffration paths evaluated in ray traing is suessfully redued by importane sampling, i.e. a general tehnique for Monte Carlo integration. With the use of the state-of-the-art ray traing engine running on a graphis proessor, our method is apable of exeuting a sound diffration simulation of a pratial sene in real time. 2. ANALYTIC SECONDARY SOURCE MODEL 2.1 Overview Let us onsider a system illustrated in Fig. 1(a), omprising a sound soure S, a reeiver R and an edge whose open angle is denoted as θ W. Here, (r S,θ S, z S ) and (r R, θ R, z R ) are ylindrial oordinates of S and R, respetively. In this system, wave from S is diffrated on the edge and then propagates to R. The diffrative IR observed at R an be alulated by the exat Biot-Tolstoy solution [7]. Based on this solution, the seondary soure model was derived [6], whih provides a disrete Huygens interpretation of the solution. The interpretation is reviewed as follows. First in Fig. 1(b), wave front from S reahes an edge point S. Then S is exited by the wave front and emits diffration wave. After that, the wave reahes R. The above mentioned phenomenon arises at eah edge point. Finally, the superposition of the diffration waves from those points produes the Biot-Tolstoy s IR. Aording to the seondary soure model, the disrete time IR of the diffration is alulated by the following equation, whih is derived analytially from the Biot-Tolstoy solution, h diffr [n] = ν β[α(z),γ(z),θ S,θ R ] dz (1) R n 4π m(z) l(z) m(z) + l(z) z R n, n F S < n + 1, where is sound speed, F S is sampling frequeny, ν is a parameter alled wedge-index defined as π/θ W, and α, γ, m and l are the path parameters of a diffration wave illustrated in Fig. 1(b). β[ ] is the diretivity funtion of diffration waves defined in next equation, sin[ν(π ± θ S ± θ R )] β[α,γ,θ S,θ R ] = osh[νη(α,γ)] os[ν(π ± θ S ± θ R )] ±± η(α,γ) = osh 1 ( 1 + sin(α)sin(γ) os(α) os(γ) ). EURASIP, ISSN

2 (a) wave front from wave front from Figure 1: Illustration of first-order edge diffration: (a) oordinate system (b) a disrete Huygens interpretation of diffration aording to the seondary soure model. From Eq. (1), we an find that the instantaneous value at n is expressed as the funtion of path parameters of diffration waves whih arrive at R at that time. In this model, an edge is regarded as a new soure. Based on this regard, ontributions from the new edge are integrated to alulate high-order diffration. Eq. (2) desribes the formulation to alulate seond-order diffration illustrated in Fig. 2. β 1 and β 2 are diffration fators of two edges, determined with the diretivity funtion. h diffr [n] = R 2 n ν 1 ν 2 (4π) 2 (b) β 1 (z) β 2 (z) m 1 (z) l 1 (z) l 2 (z) dz 1dz 2 (2) z = (z 1,z 2 ) R 2 n, n m 1(z) + l 1 (z) + l 2 (z) 2.2 Diffiulties for Appliations F S < n + 1 As indiated in the previous setion, alulation of Kth-order diffration needs to solve Kth-order integration. Sine the integration annot be solved analytially, general implementations of this model use the mid-point rule integration [6, 8, 9]. Based on this rule, Eq. (1) is approximated as ( h diffr [n] = ν ) β[α(z i ),γ(z i ),θ S,θ R ] z 4π m(z i ) l(z i ) i N n, n m(z i) + l(z i ) F S < n + 1, where z i is one of the mid-points in the equally subdivided regions. Let D be the number of divisions of an edge. Then this approximation needs to evaluate D K diffration paths whih grow exponentially in aordane with K. Therefore, muh more omputational ost is neessary for high-order diffration. Although an adaptive edge-subdivision strategy has been proposed for fast alulation [1], this strategy is not appliable to high-order diffration. On the other hand, path validation proess is needed for exat alulation. In this proess, diffration paths oluded by some kind of obstales (e.g. walls) must be deteted in order to eliminate the ontributions of oluded waves. In a sene with many obstales, the proess may be ompliated. Consequently, omputational osts of the integrations and/or the path validation make it diffiult to alulate IRs with this model espeially for high-order diffration. Figure 2: Interpretation of seond-order diffration. In pratie, there are multiple diffration path via all ombination of S and S. 3. PROPOSED METHOD Proposed method alulates the multiple integrations in the seondary soure model by using Monte Carlo method [11]. Fousing on the integrand whih is the funtion of diffration paths parameters (i.e. propagation lengths and input/output angles), we use ray traing to alulate an integrand s sample value (referred as V S ). In the ray traing proess, path validation is also onduted, in whih V S is set to so that the ontribution of an oluded wave is eliminated. 3.1 Proedure As input, our method requires information of a simulation sene: sound soure S, reeiver R and an edge E. Figure 3 illustrates ray traing proedure of the proposed method. First, (i) a ray is shot to E from S whereas V S is initialized to 1. The intersetion S of the ray and E (or the oordinate z) is determined randomly. If the ray intersets with E, (ii) a ray is shot to R from S. At the same time, oeffiients suh as diffration fator is multiplied to V S. If the ray reahes R, (iii) V S is determined. In ase that a ray does not reah E or R, (ii) /(iii) V S is set to. For Kth-order diffration (S E 1 E K R), R is replaed by the next edge E j ( j = 2,,K) in (ii). The above mentioned proedure is repeated M times, where M is the number of evaluation paths (or the number of samples). After the ray traing proess, IR estimation is onduted. Based on the average of the obtained sample values, the Monte Carlo estimates h diffr [n] by the next equation, h diffr [n] L 1 L K M V S,i (3) i N n, n d i F S < n + 1, where i is the index of an evaluated path. V S,i and d i are the sample value and the total length of the path i, respetively. L j ( j = 1,2,,K) is the length of eah edge. 3.2 Features Considering that general ray traing tehniques are the Monte Carlo solutions of integral equation alled rendering equation [12, 13], we an find that the objetive and the proedure of the proposed method are similar to those of the general tehniques. Beause of this similarity, our method an be integrated into existing aousti analysis software. Furthermore, various approahes dediated to ray traing an be appliable to our method. Let P be the number of primitives in a simulation sene. Then the omputational ost of the proposed method is proportional to K M P. Differing from the mid-point rule, 1654

3 (ii) Shoot a ray to (i) Shoot a ray to (ii)' Terminate the ray (iii). Terminate the ray (iii)'. Terminate the ray Figure 3: Proedure of ray traing. our method does not require the ost whih inreases exponentially with K. On the other hand, the term M an be redued with importane sampling introdued in Set Furthermore, the term P, whih arises from brute-fore intersetion detetion for all primitives, an be redued to logp by using so-alled aeleration strutures suh as BVH (Bounding Volume Hierarhy) [14] and kd-tree [15]. 3.3 Importane Sampling In this setion, importane sampling, whih is a general tehnique for Monte Carlo integration [11], is introdued to redue M required for high-order simulation. In ase that integration variables z = (z 1,,z K ) are sampled based on probability density funtion (p.d.f.) p(z), Eq. (3) is modified to h diffr [n] 1 M V S,i p(z i ), where z i is a set of the oordinate points on edges the path i traed. The definition of a p.d.f. used in this paper is desribed below. Let us onsider the IR of first-order edge diffration illustrated in Fig. 4(a). From this figure, we an notie that the IR attenuates rapidly after the onset. In other words, the ontribution of the diffration wave whih propagates along a long path is insignifiant. In order to eliminate suh paths illustrated in Fig. 4(b), we introdue a p.d.f. p (z) whih is defined in the next equation, [ { ( ) }] z 2 1 z p (z) = Aw 1 +. w This is a modified Cauhy distribution whose range is limited to edge length L. A is a normalization onstant whih is determined so that L p (z)dz = 1. Figure 5 illustrates the distribution and the geometri interpretation. In Fig. 5, a ray is shot from S to z-axis with uniformly sampled launhing angle θ then z is Cauhy distributed. Based on this distribution, insignifiant diffration paths an be eliminated stohastially sine the shorter a path is, the higher the sampling probability is, as shown in Fig. 5. For multiple integration, a binding distribution of p (z) is used (i.e. p(z) = p (z 1 )p (z 2 ); K = 2). 3.4 Disussion of Errors Let us onsider estimation errors in Monte Carlo integration and the mid-point rule whose orders are respetively expressed as O(N 1/2 ) and O(N 2/K ), where N( M = D K ) (a) Figure 4: Examination of insignifiant path: (a) IR of firstorder diffration whih attenuates rapidly after the onset (b) an example of long path (or insignifiant path). Cauhy distributed intersetion uniformly sampled launhing angle (b) Figure 5: Cauhy distribution and the geometri interpretation. w is length of the perpendiular line between S and an edge. z is the foot of the line. is the number of evaluation paths. Therefore, the proposed method takes advantage for high-order diffration (i.e. K > 4) in terms of the error. Furthermore, the importane sampling an also ompensate the disadvantage in ase of K 4, whih is demonstrated later in Set Implementation We implemented the proposed method using OptiX [16] whih is a general purpose ray traing engine distributed by NVIDIA. Owing to OptiX s parallelism utilizing a graphi proessor, we have ahieved fast implementation of our method. The graphis proessor used in this researh is GTX SIMULATION AND EVALUATION 4.1 Validation and Proessing Time Evaluation First, we ondut a test to evaluate simulation auray of our method. Figure 6 illustrates a onert hall sene used in this test. The sene onsists of P = 4 primitives (or triangle meshes) and 39 diffration edges whose open angles are θ W < π. For eah edge, first-order diffration is simulated. In this test, other wave phenomena (i.e. speular refletion and diffusion from walls) are not inluded. The number of samples assigned to eah edge is 2, and thus totally M = 78,(= 39 2,) samples are evaluated. In order to validate the results, the estimated IR is ompared with referene alulated by Edge diffration toolbox for Matlab [9] whih uses Simpson s rule (for K = 1) and the mid-point rule (for K > 1). Figure 7 shows errors between the referene and the estimated IR. The IR is partially shown in Fig. 8, in whih the errors are relatively ompared with the IR. From Fig. 7, we an find that the estimated IR ontains fleetly flutuating noise. 1655

4 2 z x y 1 Figure 6: Simulation sene in Set. 4.1 where S : ( 7.5,5,3), R : (5,35,5). The details of this model an be found in [17] Error proessing time [ms] # total samples: M (# edges # samples per edge) Figure 9: Number of samples vs. proessing times required to simulate first-order diffration in Fig m 15.25m 6 35mm m (a) perspetive view (b) side view () top view Figure 7: Errors between IRs estimated by Edge diffration toolbox and the proposed method Error IR Figure 8: Comparison of errors and estimated IR fousing on the beginning of the IR. Root mean squared error normalized by the range of the referene value is 62.6 [db]. Howerver the noise is suffiiently smaller than IR values as indiated in Fig. 8. From these results, we onfirmed that the proposed method adequately estimates diffrative IRs. We also evaluate the proessing time of our method with varying Ms. Figure 9 summarizes the evaluation result, from whih we an onfirm that the proessing time of the above simulation (i.e. M = 78,) is approximately 4ms. Even in ase with muh larger samples M 5,, the proessing time is about 21ms and thus our method ahieves a proessing performane of about 24M samples/se. In terms of update rate in dynami senes, the rate approximates 5Hz in the ase of M 5,. Based on these performane results, we an say that the method is suffiiently appliable to interative simulation. 4.2 Evaluation of Importane Sampling In this setion, we ondut a test to evaluate the effetiveness of the importane sampling (IS). Figure 1 shows the simulation sene, where two diffration edges are adjaently Figure 1: Simulation sene of seond-order diffration in Set The similar sene is originally used in [6]. arranged. In this test, we simulate seond-order diffration with the proposed method where IS is enabled or disabled. The number of samples M is set to 9, 1, and 1,,. For omparison, estimation using the mid-point rule is also onduted. The number of edge division D is set to 3, 1 and 1, so that the number of evaluation paths D 2 is equal to M. Figures 11 through 13 respetively show the estimation results of the proposed method (IS is disabled/enabled) and the mid-point rule. In these figures, arrival time of the diffration wave from the shortest path is regarded as. Although the IRs of M = 1, are noisy in Figs. 11 and 13, the IR in Fig. 12 is suffiiently onverged even though the same number of paths are evaluated. Furthermore in Fig. 12, the proposed method obtains IR onverged muh better than the others with only M = 9 samples. From these results, we an onfirm the following effetiveness desribed in Sets. 3.3 and 3.4; (1) the number of samples required for well onverged results an be redued based on IS, (2) in terms of the errors, the proposed method with IS an obtain better results than the mid-point rule if finite samples are available. 5. CONCLUSION In this paper, we proposed an estimation method for edgediffration IRs based on ray traing. This method is a Monte Carlo solution of multiple integration in the seondary soure model, where sample values alulation of integrand and path validation are onduted in ray traing proess. Fousing on the proedures similarity with general ray traing, we implemented the proposed method by using a ray traing engine OptiX. Consequently fast implementation has been ahieved. In our test, we demonstrated that the proessing time required to evaluate M 5, samples is about 1656

5 M=9 M=1, M=1,, Figure 11: IR estimated by the proposed method (IS: disabled) M=9 M=1, M=1,, Figure 12: IR estimated by the proposed method (IS: enabled) D 2 =3x3 D 2 =1x1 D 2 =1x1 Figure 13: IR estimated by the mid-point rule. 21ms, whih indiates the appliability to interative simulation. For more effiieny, importane sampling was introdued where insignifiant diffration paths are eliminated stohastially. As for our future work, we will try to apply adaptive importane sampling strategies used in 3DCG field suh as Metropolis Light Transport [18] and Photon Mapping [19] in order to make our method more flexible for any omplex senes. We also need to ondut detailed evaluation of estimated IRs qualities, inluding subjetive listening test. Aknowledgment This researh has been supported by the Global COE program of the Ministry of Eduation, Culture, Sports, Siene and Tehnology, Japan, under the title Founding Ambient Information Soiety Infrastruture. REFERENCES [1] Odeon A/S In. Odeon Room Aoustis Software for predition and auralisation. [2] CATT In. CATT-Aousti v8.. [3] N. Tsingos, T. Funkhouser, A. Ngan, and I. Carlbom. Modeling aoustis in virtual environments using the uniform theory of diffration. In Proeedings of the 28th annual onferene on Computer graphis and interative tehniques, page 552, 21. [4] R. Torres, U. Svensson, and M. Kleiner. Computation of edge diffration for more aurate room aoustis auralization. The Journal of the Aoustial Soiety of Ameria, 19:6, 21. [5] R. Kouyoumjian and P. Pathak. A uniform geometrial theory of diffration for an edge in a perfetly onduting surfae. Proeedings of the IEEE, 62(11): , [6] U. Svensson, R. Fred, and J. Vanderkooy. An analyti seondary soure model of edge diffration impulse responses. The Journal of the Aoustial Soiety of Ameria, 16:2331, [7] M. Biot and I. Tolstoy. Formulation of wave propagation in infinite media by normal oordinates with an appliation to diffration. J. Aoust. So. Am, 29(3): , [8] T. Lokki, U. Svensson, and L. Savioja. An effiient auralization of edge diffration. In AES 21st Int. Conf. on Arhitetural Aoustis and Sound Reinforement, pages , 22. [9] U. Svensson. Edge diffration toolbox for Matlab. svensson/software/index.html. [1] P. Calamia and U. Svensson. Fast time-domain edgediffration alulations for interative aousti simulations. EURASIP Journal on Applied Signal Proessing, 27(1):186, 27. [11] W. Press, B. Flannery, S. Teukolsky, W. Vetterling, et al. Numerial reipes, volume 3. Cambridge university press Cambridge, 27. [12] J. T. Kajiya. The rendering equation. SIGGRAPH Computer Graphis., 2(4):143 15, [13] S. Siltanen, T. Lokki, S. Kiminki, and L. Savioja. The room aousti rendering equation. The Journal of the Aoustial Soiety of Ameria, 122: , 27. [14] T. Kay and J. Kajiya. Ray traing omplex senes. ACM SIGGRAPH Computer Graphis, 2(4): , [15] T. Foley and J. Sugerman. KD-tree aeleration strutures for a GPU raytraer. In Proeedings of the ACM SIGGRAPH EUROGRAPHICS onferene on Graphis hardware, pages 15 22, 25. [16] S. Parker, J. Bigler, A. Dietrih, H. Friedrih, J. Hoberok, D. Luebke, D. MAllister, M. MGuire, K. Morley, A. Robison, et al. OptiX: a general purpose ray traing engine. ACM Transations on Graphis (TOG), 29(4):1 13, 21. [17] F. Mehel. Improved mirror soure method in room aoustis. Journal of Sound and Vibration, 256(5):873 94, 22. [18] E. Veah and L. Guibas. Metropolis light transport. In Proeedings of the 24th annual onferene on Computer graphis and interative tehniques, pages ACM Press/Addison-Wesley Publishing Co., [19] H. Jensen. Realisti image synthesis using photon mapping. AK Peters, Ltd. Natik, MA, USA,