Research Article Two-Dimensional Beam Tracing from Visibility Diagrams for Real-Time Acoustic Rendering

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1 Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volue 00, Article ID 3, 8 pages doi:0.55/00/3 Research Article Two-Diensional Bea Tracing fro Visibility Diagras for Real-Tie Acoustic Rendering F. Antonacci (EURASIP Meber), A. Sarti (EURASIP Meber), and S. Tubaro (EURASIP Meber) Dipartiento di Elettronica ed Inforazione, Politecnico di Milano, Piazza Leonardo da Vinci 3, 033 Milano, Italy Correspondence should be addressed to F. Antonacci, antonacc@elet.polii.it Received February 00; Revised June 00; Accepted 5 August 00 Acadeic Editor: Udo Zoelzer Copyright 00 F. Antonacci et al. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. We present an extension of the fast bea-tracing ethod presented in the work of Antonacci et al. (008) for the siulation of acoustic propagation in reverberant environents that accounts for diffraction and diffusion. More specifically, we show how visibility aps are suitable for odeling propagation phenoena ore coplex than specular reflections. We also show how the bea-tree lookup for path tracing can be entirely perfored on visibility aps as well. We then contextualize such ethod to the two different cases of channel (point-to-point) rendering using a headset, and the rendering of a wave field based on arrays of speakers. Finally, we provide soe experiental results and coparisons with real data to show the effectiveness and the accuracy of the approach in siulating the soundfield in an environent.. Introduction Rendering acoustic sources in virtual environents is a challenging proble, especially when real-tie operation is reuired without giving up a realistic ipression of the result. The literature is rich with ethods that approach this proble for a variety of purposes. Such ethods are roughly divided into two classes: the forer is based on an approxiate solution of the wave euation on a finite grid, while the latter is based on the geoetric odeling of acoustic propagation. Typical exaples of the first class of ethods are based on the solution of the Green s or Helholtz-Kirchoff s euation through finite and boundary eleent ethods [ 3]. The coputational effort reuired by the solution of the wave euation, however, akes these algoriths unsuitable for real-tie operation except for a very liited range of freuencies. Geoetric ethods, on the other hand, are the ost widespread techniues for the odeling of early acoustic reflections in coplex environents. Starting fro the spatial distribution of the reflectors, their acoustic properties, and the location and the radiation characteristics of sources and receivers (listening points), geoetric ethods cast rays in space and track their propagation and interaction with obstacles in the environent []. The seuence of reflections, diffractions and diffusions a ray undergoes constitutes the acoustic path that link source and receiver. Aong the any available geoetric ethods, a particularly efficient one is represented by bea tracing [5 9]. This ethod was originally conceived by Hanrahan and Heckbert [5] for applications of iage rendering, and was later extended by Funkhouser et al. [0] to the proble of audio rendering. A bea is intended as a bundle of acoustic rays originating fro a point in space (a real source or a wallreflected one), which fall onto the sae planar portion of an acoustic reflector. Every tie a bea encounters a reflector, in fact, it splits into a set of subbeas, each corresponding to a different planar region of that reflector or of soe other reflector. As they bounce around in the environent, beas keep branching out. The bea-tracing ethod organizes and encodes this bea splitting/branching process into a specialized data structure called bea-tree, which describes the inforation of the visibility of a region fro a point (i.e., the source location). Once the bea-tree is available, pathtracing becoes a very efficient process. In fact, given the location of the listening point (receiver), we can iediately deterine which beas illuinate it, just through a look up of the bea-tree data structure. We should notice,

2 EURASIP Journal on Advances in Signal Processing however, that with this solution the coputational effort associated to the bea tracing process and that of pathtracing are uite unbalanced. In fact if the environent is coposed by n reflectors, the exhaustive test of the utual visibility aong all the n reflectors involves O(n 3 ) tests, while the test of the presence of the receiver in the traced beas needs only O() tests. Soe solutions for a speedup of the coputation of the bea-tree have been proposed in the literature. As an exaple in [0] the authors adopt the Binary Space Partitioning Techniue to operate a selection of the visible obstacles fro a prescribed reflector. A siilar solution was recently proposed in [], where the authors show that a real-tie tracing of acoustic paths is possible even in a siple dynaic environent. In [] the authors generalized traditional bea tracing by developing a ethod for constructing the bea-tree through a lookup on a precoputed data structure called global visibility function, which describes the visibility of a region not just as a function of the viewing angle but also of the source location itself. Early reflections are known to carry soe inforation on the geoetry of the surrounding space and on the spatial positioning of acoustic sources. It is in the initial phase of reverberation, in fact, that we receive the echoes associated to the first wall reflections. Other propagation phenoena, such as diffusion, transission and diffraction tend to enrich the sense of presence in virtual walkthrough scenarios, especially in densely occluded environents. As bea tracing was originally conceived for the odeling of specular reflections only, soe extensions of this ethod were proposed to account for other propagation phenoena. Funkhouser et al. [3], for exaple, account for diffusion and diffraction through a bidirectional bea tracing process. When the two bea-trees that originate fro the receiver and the source intersect on specific geoetric priitives such as edges and reflectors, propagation phenoena such as diffusion and diffraction could take place. The need of coputing two bea-trees, however, poses probles of efficiency when using conventional bea tracing ethods, particularly when sources and/or receivers are in otion. A different approach was proposed by Tsingos et al. [], who proposed to use the unifor theory of diffraction (UTD) [5] by building secondary bea-trees originated fro the diffractive edges. This approach is uite efficient, as the tracing of the diffractive bea-trees can be based on the sole geoetric configuration of reflectors. Once source and receiver locations are given, in fact, a siple test on the diffractive bea-trees deterines the diffractive paths. Again, however, this approach inherits the advantages of bea tracing but also its liits, which are in the fact that anewbea-treeneedsbecoputedeverytieasource oves. As already entioned above, in [] we proposed a ethod for generating a bea-tree through a lookup on the global visibility function. That ethod had the rearkable advantage of coputing a large nuber of acoustic paths in real tie as both source and reflector are in otion in a coplex environent. In this paper we generalize the work proposed in [] in order to accoodate diffusion and diffraction phenoena. We do so by revisiting the concept of global visibility and by introducing novel lookup ethods and new operators. Thanks to these generalizations, we will also show how it is possible to work on the visibility diagras not just for constructing bea-trees but also to perfor the whole path-tracing process. In this paper we expand and repurpose the bea tracing ethod for applications of real-tie rendering of acoustic sources in virtual environents. Two are the envisioned scenarios: in the forer the user is wearing a headset, in the latter the whole sound field within a prescribed volue is rendered using loudspeaker arrays. We will show that the two scenarios share the sae bea tracing engine which, in the first case, is followed by a path-tracing algorith based on bea-tree lookup [], with an additional head-related transfer function. In the second case the bea tracer is used for generating the control paraeters of the bea-shaping algorith proposed in []. This bea-shaping ethod allows us to design the spatial filter to be applied to the loudspeaker arrays for the rendering of an arbitrary bea. Other solutions exist in the literature for the rendering of virtual environents, such as wave field synthesis (WFS) and abisonics. Roughly speaking, WFS coputes the spatial filter to be applied to the speakers with an approxiation of the Helholtz-Kirchoff s euation. Interestingly enough, for exaple, in [7] the task of coputing the paraeters of all the virtual sources in the environent is deanded to an iage-source algorith. Therefore, soe WFS systes already partially rely on geoetric ethods. When rendering occluded environents, however, the iage-source ethod tends to becoe coputationally deanding, while fast bea tracing techniues []can offer a significant speedup. It is iportant to notice that the ethod proposed in [] was developed for odeling coplex acoustic reflections in a specific class of 3D environents obtained as the cartesian product between a D floor plan and a D (vertical) direction. This situation, for exaple, describes a coplex distribution of vertical walls ending in horizontal floor and ceiling. When considering acoustic wall transission, a D D environent becoes useful for odeling a ulti-floored building with a repeated floor plan. Although D D environents enjoy the advantages of D odeling (siplicity, duality, etc.), the coputation of all delays and path lengths still needs to be perfored in a 3D space. While this is rather straightforward in the case of geoetric reflections, it becoes ore challenging when dealing with diffraction and diffusion phenoena. The paper is organized as follows. In Section we review and revisit the concept of global visibility and its use for efficiently tracing acoustic paths. In Section 3 we discuss the ain atheatical odels used for explaining diffusion and diffraction phenoena, and we choose the one that best suits our bea tracing approach. Sections and 5 focus on the odeling of diffusion and diffraction with visibility diagras. In Section we present two possible applications of the algorith presented in this paper. In Section 7 we prove the efficiency and the effectiveness of our odeling solution. Finally, Section 8 provides soe final coents and conclusions.

3 EURASIP Journal on Advances in Signal Processing 3 B(0, ) y (i) + x (i) A(0, ) Figure : The specialized RRP and the set of rays passing through the reference reflector in the (, ) doain (visibility region fro the reference reflector).. The Visibility Diagra Revisited In this section we review the concept of visibility diagra, as it is a key eleent for the reainder of this paper. In [] we adopted this representation for generating a specialized data structure that could swiftly provide inforation on how to trace acoustic beas and rays in real tie with the rules of specular reflection. This approach constitutes a generalization of the bea tracing algorith proposed by Hanrahan and Heckbert [5]. The visibility diagra is a reapping of the geoetric structures and functional eleents that constitute the geoetric world (rays, beas, reflectors, sources, receivers, etc.) onto a special paraeter space that is copletely dual to the geoetric one. Visibility diagras are particularly useful for iediately assessing what is in the line of sight fro a generic location and direction in space. We will first recount the basic concepts of visibility diagras and provide a general view of the path-tracing proble for the specific case of purely specular reflections. This overview will be provided in a slightly ore general fashion than in [], as all the algorithic steps will be given with reference to visibility diagras, and will constitute the starting point for the discussions in the following sections... Visibility and the Tracing Proble. ArayinaDspace is uniuely characterized by three paraeters: two for the location of its origin, and one for its direction. As we are tracing paths during their propagation, we are interested in rays eerging fro a reflector after bouncing off it. As a conseuence, the origin corresponds to the virtual source. Furtherore, because we are interested in assessing only where the ray will end up, we can afford ignoring soe inforation on where the ray is coing fro, for exaple the source distance. This eans that a ray description based on three paraeters turns out to be redundant, and can be easilyreducedtotwoparaeters.in[] we adopted the Reference Reflector Paraetrization (RRP) paraetrization based on the location of the intersection on the reference reflector and the travel direction of the ray. Although the RRP is referred to a frae attached to a specific reflector, this choice does not represent a liitation, due to the iterative nature of the visibility evaluation process. Let s i be the reference reflector. For reasons that will be clearer later on, the RRP noralizes s i through a translation, a rotation and a scaling of the axes in such a way that the reference reflector lies on the segent of the y-axis between and.the set of rays passing through s i is described by the euation y (i) = x (i) +. Figure shows the reflector s i referred to the noralized frae in the geoetric doain (left). The set of rays passing through s i is called region of visibility fro s i and it is represented by the horizontal strip (reference visibility strip) in the (, ) doain. Due to the duality between priitives in (x, y) and(, ) doains we will soeties refer to the RRP as the dual space. We are interested in representing the utual occlusions between reflectors in the dual space. With this purpose in ind, we split the visibility strip into visibility regions, each corresponding to the set of rays that hit the sae reflector. According to the iagesource principle, all the obstacles that lie in the sae half space of the iage-source, are discarded during the visibility test. As a convention, in the future we will use the rotation of the reference reflector which brings the iage-source in the half-space x (i) < 0. The above paraeter space turns out to play a siilar role as the dual of a geoetric space. In Table we suarize the representation of soe geoetric priitives in the paraeter space. A coplete derivation of the relations of Table can be found in [, 8]. Notice that the relation between priitives in the two doains is of coplete duality. For exaple, the dual of the oriented reflector is a wedge in the (, ) doain (sort of an oriented bea in paraeter space). Conversely, the dual of an oriented bea (a single wedge in the (x, y)geoetricspace) is an oriented segent in the (, ) doain (sort of an oriented reflector in paraeter space).... Visibility Region. The paraeters describing all rays originating fro the reference reflector s i for the region of visibility fro that reflector. After noralization, this region takes on the strip-like shape described in Figure,which we refer to as reference visibility strip. Those rays that originate

4 EURASIP Journal on Advances in Signal Processing Table : Priitives in the geoetric doain and their corresponding representation in ray space. Geoetric space Onidirectional bundle of nonoriented rays y P x Onidirectional bundle of outgoing rays (source) y P Bea (double wedge) y (, ) (, ) x x Two-sided reflector y P P x Oriented bea (single wedge) y x One-sided reflector y P P x Ray space Non-oriented ray or two-sided infinite reflector l One-sided infinite reflector l Two-sided reflector (, ) (, ) Bea (double wedge) l l One-sided reflector (, ) (, ) Oriented bea (single wedge) fro the reference reflector and hit another reflector s j for a subset of this strip (see Figure ) which corresponds to the intersection between the dual of s j and the dual of s i (reference visibility strip). The intersection of the dual of s j and the visibility strip is the visibility region of s j fro s i. Once the source location is specified, the set of rays passing through s j and s i and departing fro that location will be a subset of the visibility region of s j.onekeyadvantageof the visibility approach to the bea tracing proble resides in the fact that we only need geoetric inforation about the environent to copute the visibility regions, which can therefore be coputed in advance.... Dual of Multiple Reflectors: Visibility Diagras. When there are ore than two reflectors in the environent, we need to consider the possibility of utual occlusions, which results in overlapping visibility regions. Sorting out which reflector occludes which (with respect to the reference reflector) corresponds to deterining which visibility region overrides which in their overlap. Two solutions for the occlusion proble are possible: the first, already presented in [], is based on a siple test in the geoetric doain. An arbitrary ray chosen in the overlap of visibility regions can be cast to evaluate the front-to-back ordering of visibility regions or, ore siply, to deterine which oriented reflector is first et by the test ray. An exaple is provided in Figure where, if s i is the reference reflector, we end up having an occlusion between s and s 3, which needs to be sorted out. A test ray is picked at rando within the overlapping region to deterine which reflector is hit first by the ray. This particular exaple shows that, unless we consider each reflector as the cobination of two of oppositely-facing oriented reflectors, we cannot be sure that the occlusion proble can be disabiguated. In this case, for exaple, s occludes s 3 for soe rays, and s 3 occludes s for others. As shown in Table, a two-sided reflector corresponds to a double wedge in ray space, each wedge corresponding to one of the two faces of the reflector. By considering the two sides of each reflector as individual oriented reflectors, we end up with four distinct wedgelike regions in ray space, thus reoving all abiguities. The overlap between visibility regions of two one-sided reflectors arises every tie the extree lines of the corresponding visibility regions intersect. We recall that the dual of a point P(x, y) is a line whose slope is x. The extree lines of the visibility region of reflector s j are the dual of the endpoints of s j, that are (x (i) j, y (i) j )and(x (i) j, y (i) j ) and the slopes of the extree lines of the visibility region of s j are x (i) ji and x (i) j. A siilar notation is used for the overlapping reflector s k. Under the assuption that s j and s k never intersect in the geoetric doain, we can reorder one-sided reflectors in front-to-back order by siply looking at the slopes of the extree lines of their visibility regions. If the line l (i) of euation = x (i) j + y (i) j and the line l (i) of euation = x (i) k +y(i) k intersect in the dual space, then x(i) j > x (i) k guarantees that s j occludes s k and x (i) j < x (i) k guarantees that s k occludes s j... Tracing Reflective Beas and Paths in Dual Space... Tracing Beas. In this paragraph we suarize the tracing of beas in the geoetric space using the inforation contained in the visibility diagras. Further details on this specific topic can be found in []. This can be readily done by scanning the visibility diagra along the line that represents the dual of the virtual source. In fact, that line will be partitioned into a nuber of segents, one per visibility region. Each segent will correspond to a subbea in the geoetric space. Consider the configuration of reflectors of Figure 3. The first step of the algorith consists of deterining how the coplete pencil of rays produced by the source S is partitioned into beas. This is done by evaluating the visibility fro the source using traditional bea tracing. This initial splitting

5 EURASIP Journal on Advances in Signal Processing 5 y (i) r a S S i r b S 3 x (i) Figure : Abiguity in the occlusion between two nonoriented reflectors s and s 3. For soe rays (e.g., r a ) s occludes s 3. For other rays (e.g., r b ) s 3 occludes s. b b 5 b S b b S b 3 b b b 3 b b 5 b Figure 3: Beas traced fro the source location and the corresponding bea-tree. process produces two classes of beas: those that fall on a reflector and those that do not. The beas and the corresponding bea-tree are shown in Figures 3 and 3, respectively. We consider the splitting of bea b 3, shown in Figure. The iage-source is represented in the dual space by the line l P.Thebeab 3 will therefore be a segent on that line, which will be partitioned in a nuber of segents, one for each region on the visibility diagra. In Figure the bea splitting is accoplished in the (, ) doain, while in Figure we can see the corresponding subbeas in the geoetric doain. This process is iterated for all the beas that fall onto a reflector. Further details can be found in []. At the end of the bea tracing process we end up with a tree-like data structure, each node b k of which contains inforation that identifies the corresponding bea: (i) the one-sided reference reflector s i, (ii) the one-sided illuinated reflector s j (if any), (iii) the position of the virtual source S(x s (i), y s (i) ) in the noralized reference frae, (iv) the segent [, ] that identifies the illuinating region on the y (i) -axis, (v) the parent node (if any), (vi) a list of the children nodes (if at least one exists). The last two ites are useful when reclaiing the reflection history of a bea. Given the above inforation we are iediately able to represent the beas (i.e., segents) in the (, ) doain.... Tracing Paths. In [] the construction of the beatree was accoplished in the dual space but path-tracing was entirely done in the geoetric doain. We will now derive an alternate and ore efficient procedure for tracing the acoustic paths directly in the dual space. The goal is to test the presence of the receiver R in the bea b k, originating fro the reflector s i. The coordinates of the receiver in the noralized reference frae of s i are (x r (i), y r (i) ). In order for R to be in b k, there ust exist a ray in b k that passes through R, that is, (, ) b k : y (i) r = x (i) r +. () This eans that the ray (, ) fros to R, isrepresented in the dual space by a point resulting fro the intersection of the dual of b k (a segent) and the dual of R (a line). The presence test is thus perfored by coputing the intersection of two lines in the paraeter space. If b k does not fall onto a reflector, then the condition () is sufficient. If b k falls on reflector s j, then we ust also ake sure that s j does not occlude R. Assuing that s j lies on

6 EURASIP Journal on Advances in Signal Processing b 9 l P b 7 b8 b 0 b b Figure : Bea subdivision perfored in the (, ) doain for the bundle of rays corresponding to the reflection of the bea b 3 of Figure 3. Corresponding subbeas in the geoetric doain. P S S S 5 S S S b b b 3 b b 5 b R S 3 b 7 b 8 b 9 b 0 b b Figure 5: Path-tracing fro source S to the receiver R. The bea-tree on the right-hand side is the sae as in Figure 3. the line l (i) j : y = j x (i) + (i) j, we can easily conclude that R is not occluded by s j if the distance between S and R is saller than the distance between S and the intersection of the (, )raywiths j, which eans that: b8 b 7 (x s x r ) + ( y s y r ) ( x s ) j j + ( ) j y s j j j. The conditions in () and (possibly) () are tested for all the beas. However, if R falls onto b k, then we know that it cannot fall in other beas that share the virtual source of b k. This speeds up the path-tracing process a great deal. As an exaple of the tracing process, consider the situation shown in Figure 5. HereS and R are not in the line of sight, but a reflective path exists through the reflection fro s 3. Firstorder beas are traced directly in the geoetric doain, as done in [], therefore the presence of R in beas originating directly fro S is tested directly in the geoetric doain. Let us now test the presence of R in the reflected beas eerging fro b 3 (see Figure 3). The intersection between () b b 0 b Figure : Tracing paths in the visibility diagra: the paraeters of the outgoing ray are found by eans of the intersection of the dual of the bea (a segent) and of the receiver (a line). Paths falling in beas liited by a reflector, also () ust hold. the line l r,dualofr, and the dual of b (a segent) is easily found (see Figure on the right). Once we have checked the presence of the receiver in b, the position of the receiver and the inforation encoded in b are sufficient to deterine the delay and the aplitude of the echo associated to that acoustic path. More details on this aspect will be provided in Section.. b 9 l r

7 EURASIP Journal on Advances in Signal Processing 7 Incoing wavefront Outcoing wavefront Wall Figure 7: Diffusion phenoenon: a planar wavefront falls onto a rough surface fro a perpendicular direction. The wavefront that bounces off the surface, resulting fro a cobination of secondary wavefronts, will not propagate just in the specular direction with respect to the exciting wave. 3. Matheatical Models of Diffraction and Diffusion In this section we investigate soe atheatical odels used in the literature to uantitatively describe the causes of diffraction and diffusion. Later we will choose the odel which best works for our bea tracing ethod. 3.. Models of Diffusion. When a wavefront encounters a rough or nonhoogeneous surface, its energy is diffused in nonspecular directions (see Figure 7). Let us consider a flat surface with a single localized unevenness whose size is bigger than the wavelength λ of the incident wavefront. The Huygens principle interprets the diffused wavefront as the superposition of the local wavefronts associated to reflections on each point of the surface. As we can see in Figure 7, the direction of propagation of the outgoing wavefront differs fro the direction of the incident one. Conseuently, a sensor facing the wall will pick up energy not just fro the incident wavefront but also fro a direction that is not specular. A rough surface can be characterized through a statistical description of the speckles (in ters of size and density). In fact, the acoustic properties of the scattering aterial can be predicted or easured using various techniues [9 ]. Diffusion can also be associated to local variations of ipedance (e.g., a flat reflective surface that exhibits areas of acoustically absorbing aterial) []. Fro the listener s standpoint, diffusion tends to greatly increase the nuber of paths between source and receiver and, conseuently, the sense of presence [3]. Different odels have been proposed in the literature to account for diffusion. A reflection is said to be totally diffusive (Labertian) if the probability density function of the direction of the outgoing rays does not depend on the direction of the incoing ray. Totally diffused reflections are described by Labert s cosine law. A survey on the typical acoustic characteristics of aterials, however, reveals that Labertian reflections turn out to be uite unrealistic. For this reason, in the literature we find two odeling descriptions: the scattering coefficient and the diffusion coefficient [, 5]. The diffusion coefficient easures the siilarity between the polar response of a Labertian reflection and the actual one. This coefficient is expressed as the correlation index between the actual and the diffusive polar responses corresponding to a wavefront coing fro a perpendicular direction with respect to the surface. The scattering coefficient easures the ratio between the energy diffused in nonspecular directions and the total (specular and diffused) reflected energy. This paraeter is useful when we are interested in odeling diffusion in reverberant enclosures but it does not account for the directions of the diffused wavefronts. This approxiation is reasonable in the presence of a large nuber of diffusive reflections, but tends to becoe a bit restrictive when considering first-order diffusion only (i.e., ignoring diffusion of diffused paths). This is why in this paper we consider the additional assuption that diffusive surfaces be wide. This way the range of directions of diffused propagation turns out to be wide enough to iniize the ipact of the above approxiation. We will use the scattering coefficient to weight the contribution coing fro totally diffuse reflections (odeled by Labert s cosine law) and specular reflections. 3.. Models of Diffraction. Diffraction is a very iportant propagation ode, particularly in densely occluded environents. Failing to properly account for this phenoenon in such situations could result in a poorly realistic rendering or even in annoying auditory artifacts. In this section we provide a brief description of three techniues for rendering diffraction phenoena: the Fresnel Ellipsoid, the line of sources, and the Unifor Theory of Diffraction (UTD). We will then explain why the UTD turns out to be the ost suitable approach to the odeling of diffraction in conjunction with bea tracing Fresnel Ellipsoids. Let us consider a source S and a receiver R with an occluding obstacle in between. According to thefresnel-kirchhoff theory, the portion of the wavefront that is occluded by the obstacle does not contribute to the signal easured in R, which therefore differs fro what we would have with unoccluded spherical propagation. In order to avoid using the Fresnel-Kirchhoff integral, we can adopt a sipler approach based on Fresnel ellipsoids. If d is the distance between S and R, only objects lying on paths whose length is between d and d + λ/ are considered as obstacles, where λ is the wavelength. If x s is the generic location of the secondary source, the locus of points that satisfy the euation Sx s R SR λ/ is an ellipsoid with foci in S and R. The portion of the ellipsoid that is occluded by obstacles provides an estiate of the absolute value of the diffraction filter s response. It is iportant to notice that the size of the Fresnel ellipsoid depends on the signal wavelength. As a conseuence, in order to study diffraction in a given configuration, we need to estiate the occluded portion of the Fresnel ellipsoids at the freuencies of interest. In [] the author proposes to use the graphics hardware to estiate the hidden portions of the ellipsoids. The ain liit related to the Fresnel ellipsoid is the absence of inforation related to the phase of the signal: fro the hidden portions of the ellipsoid, in fact, we can only infer the absolute value of the diffraction filter. If we need a ore accurate rendering of diffraction, we ust resort to other techniues.

8 8 EURASIP Journal on Advances in Signal Processing z () x () P S 00 P 50 0 Figure 8: Geoetric Theory of Diffraction: an acoustic source is in S. The acoustic source interacts with the obstacle, producing diffracted rays. Given the source position S, the points on the edge behave as secondary sources (e.g., P and P in the figure). According to the geoetrical theory of diffraction, the angle between the outgoing rays and the edge euals the angle between the incoing ray and the edge. The envelope of the outgoing rays fors a cone, known in the literature as Keller cone Line of Sources. In [7] the authors propose a fraework for accurately uantifying diffraction phenoena. Their approach is based on the fact that each point on a diffractive edge receives the incident ray and then re-eits auffled version of it. The edge can therefore be seen as a line of secondary sources. The acoustic wave that reaches the receiver will then be a weighed superposition of all wavefronts produced by such edge sources. In order to uantitatively deterine the ipact of diffraction in closed for, we need to be able to evaluate the visibility of a region (environent) fro a line (edge of secondary sources). As far as we know, there are no results in the literature concerning the evaluation of regional visibility fro a line. There are, however, several works that siplify the proble by sapling the line of sources. This way, visibility is evaluated fro a finite nuber of points [8 30]. This last approach can be readily accoodated into our fraework. However, as we are interested in a fast rendering of diffraction, we prefer to look into alternate forulations Unifor Theory of Diffraction. The Unifor Theory of Diffraction (UTD) was derived by Kouyoujian and Pathak [5] fro the Geoetric Theory of Diffraction (GTD), proposed by Keller in 9 [3]. As shown in Figure 8, according to the GTD, an acoustic ray that falls onto an edge with an angle θ i produces a distribution of rays that lies on the surface of a cone. The axis of this cone is the edge itself, and its angle of aperture is θ i = θ d. The GTD assues that the edge be of infinite extension, therefore, given a source and a receiver we can always find a point on the edge such that the diffracted path that passes through it will satisfy the constraint θ i = θ d. The Keller cones for the source S and two points P and P on the edge are shown in Figure 8. Inaway,theGTDallowsustocopactlyaccountfor all contributions of a line distribution of sources. In fact, if we were to integrate all the infinitesial contributions 0 0 y () over an infinite edge, we would end up with only one significant path, which is the one that coplies with the Keller condition, as all the other contributions would end up canceling each other out. The ipact of diffraction on the source signal is rendered by a diffraction coefficient that depends on the freuency and on the angle between the incident ray and the angular aperture of the diffracting wedge (see Section. and [3] for further details). This geoetric interpretation of diffraction is alsoadopted by theutd. The difference between GTD and UTD is in how such diffraction coefficients are coputed (see Section.). The use of the UTD in bea tracing is uite convenient as it only involves one incident ray per diffractive path. The UTD, however, assues that the wedge be of infinite extension and perfectly reflective, which in soe cases is too strong an assuption. Nonetheless, the advantages associated to considering only the shortest path ake the UTD an ideal fraework for accounting for diffraction in bea tracing applications. Notice that when the incident ray is orthogonal to the edge (θ i = 90 ), the conic surface flattens onto a disc. This particular situation would be of special interest to us if we were considering an inherently D geoetry. This, however, is NOT our case. We are, in fact, considering the situation of separable 3D environents [], which result fro the cartesian product between a D environent (floor ap) and a D (vertical) direction. This special geoetry (sortofanextrudedfloorap)reuirestheodelingof diffraction and diffusion phenoena in a 3D space. The Unifor Theory of Diffraction is, in fact, inherently threediensional, but our approach to the tracing of diffractive rays akes use of fast bea tracing, whose core is two diensional. In order to be able to odel UTD in fast bea tracing, we need therefore to first flatten the 3D geoetry onto a D environent and later to adapt the D diffractive rays to the 3D nature of UTD. In order to clap down the 3D geoetry to the floor ap, we need to establish a correspondence between the 3D geoetric priitives that contribute to the Unifor Theory of Diffraction and soe D geoetric priitives. For exaple, when projected on a floor ap, an infinitely long diffracting edge becoes a diffractive point, and a 3D diffracted ray becoes a D diffracted ray. When tracing diffractive beas, each wedge illuinated (directly or indirectly) by the source will originate a disk of diffracted rays, as shown in Figure 8. At this point we need to consider the 3D nature of the environent. We do so by lifting the diffracted rays in the vertical direction. We will end up with sort of an extruded cylinder containing all the rays that are diffracted by the edge. However, when we specify the locations of the source and the receiver, we find that this set includes also paths that do not honor the Keller cone condition θ i = θ d, and are therefore to be considered as unfeasible. The reoval of all unfeasible diffracted rays can be done during the auralization phase. During the auralization, in fact, we select the paths coing fro the closer diffractive wedges, as they are considered to be ore perceptually relevant. The validation is a costly iterative process, therefore we only apply it to paths that are likely to be kept during the auralization.

9 EURASIP Journal on Advances in Signal Processing New Needs and Reuireents. As already said above, we are interested in extending the use of visibility aps for an accurate odeling and a fast rendering of diffusion and diffraction phenoena. As visibility diagras were conceived for odeling specular reflections, it is iportant to discuss what needs and reuireents need to be considered. S S R Diffraction. Visibility regions can be used for accoodating and odeling diffraction phenoena. In fact, according to the UTD, when illuinated by a bea, a diffractive edge becoesavirtualsourcewithspecificcharacteristics.our goal is to odel the indirect illuination of the receiver by eans of secondary paths: wavefronts are eitted fro the source, after an arbitrary nuber of reflections they fall onto the diffractive edge, which in turn illuinates the receiver after an arbitrary nuber of reflections. A coon siplification that is adopted in works that deal with this phenoenon [] consists of assuing that second and higher-order diffractions are of negligible ipact onto the auralization result. This, in fact, is a perceptually reasonable choice that considerably reduces the coplexity of the proble. In fact, a siple solution for ipleenting the phenoenon using the tracing tools at hand, consists of deriving a specialized bea-tree for each diffractive source. We will see later how. Another iportant aspect to consider in the odeling of diffraction is the Keller-cone condition [3], as briefly otivated above: with reference to Figure 8 we have to retain paths for which θ i = θ d. Tsingos et al. in [] proposed to account for it by generating a reduced beatree, as constrained by a generalized cone that conservatively includes the Keller-cone. The excess rays that do not belong to the Keller cone, are reoved afterwards through an appropriate check. We will see later that this approach can be ipleented using the visibility diagras. Diffusion. Let us consider a source and a receiver, both facing a diffusive surface. In this case, each point of the surface generates an acoustic path between source and receiver. This eans that the set of rays that eerge fro the diffusing surface no longer for a bea (i.e., no virtual source can be defined as they do not eet in a specific point in space). In fact, according to Huygens principle, all points of the diffusive surface can be seen as secondary sources on a generally irregular surface, therefore we no longer have a single virtual source. Unlike diffraction, diffusion indeed poses new probles and challenges, as it prevents us fro directly extending the bea tracing ethod in a straightforward fashion. One ajor difference fro the specular case is the fact that the interaction between ultiple diffusive surfaces cannot be described through an approach based on tracing, as we would have to face the presence of closed-loop diffusive paths. On the other hand, the ipact of a diffusive surface on the acoustic field intensity is rather strong, therefore we cannot expect an acoustic path to still be of soe significance after undergoing two or ore diffusive reflections. It is thus uite reasonable to assue that any relevant acoustic paths would not include ore than one relevant diffusive reflection along its way. We will see later on S Figure 9: The real source S and the receiver R face the diffusive reflector s.reflectors partially occludes s with respect to the receiver. S is the iage-source of S irrored over the prolongation of s.thesegentss and R for a diffusive path. that this assuption, reasonably adopted by other authors as well (see [3]) opens the way to a viable solution to the real-tie rendering of such acoustic phenoena. In fact, even if a diffusive surface does not preserve bea-like geoetries, it is still possible to work on the visibility regions to speed up the tracing process between a source and a receiver through a diffusive reflection.this can be readily generalized to the case in which a chain of rays go fro a source through a series of specular reflections and finally undergoes a diffusive reflection before reaching the receiver (diffusive path between a virtual source and a real receiver). A further generalization will be given for the case in which the rays undergo all specular reflections but one, which could be a diffusive reflection soewhere in between the chain. This last case corresponds to one diffusive path between a virtual source and a virtual receiver, which can be coputed by eans of two intersecting bea-trees (a forward one fro the source to the diffusive reflector and a backward one fro the receiver to the diffusive reflector).. Tracing Diffusive Paths Using Visibility Diagras As already said before, the rendering of diffusion phenoena is coonly based on Bidirectional Bea Tracing, fro both the source and the receiver. The need of tracing beas not just fro the source but also fro the receiver reuires a certain degree of syetrization in the definitions. For exaple, we need to introduce the concept of virtual receiver, which is the location of the receiver as it gets iteratively irrored against reflectors. Let us consider the situation shown in Figure 9: a (virtual) source S and a (virtual) receiver R face the diffusive reflector s.reflectors partially occludes s with respect S

10 0 EURASIP Journal on Advances in Signal Processing y S R + I S x o i l R l S S Figure 0: Noralized geoetric doain and corresponding dual space. The lines l R and l S are the dual of the points S and R. to R. In order to siplify the proble we singled out the diffusive path S R. Figure 9 also shows the iage-source S, obtained by irroring S over (the prolongation of) s. Notice that the geoetry (lengths and angles) of the path S R is preserved if we consider the path S R. Wecan therefore consider the virtual source S instead of S, and tracethe diffuse paths onto the visibility diagra fro the reference diffusing reflector. If the coordinates of S and R in the noralized geoetric doain are (x S, y S )and(x R, y R ), respectively, then the set of diffuse rays can be represented by {( i, o, ) : y S = i x S +, y R = o x R + }. (3) In other words, we are searching for the directions i and o of the rays that originate fro the point on the reference reflector and pass through S and R. The diffuse paths can be uite easily represented in the RRP fro the reference reflector. The path fro a point P on the reflector is, in fact, the intersection of the dual of P,which is the line = ; with the dual of S, which is the line l S : = x S + y S. Siilarly, the ray fro P to R is the intersection between the line = and the line l R : = x R + y R.As we can see, we do not just have the ray that corresponds to the intersection of the two lines l R and l S (sae point, sae direction), but a whole collection of rays corresponding to the horizontal segent that connects the source line l S and the receiver line l R (sae point but different directions). Notice, however, that we have not yet considered potential occlusions of the diffuse paths fro other reflectors in the environent. In Figure 9 we can see that only a portion of s contributes to diffusion. In fact there is a portion of s that is not visible fro R,asitisoccludedbys. This occlusion can be easily identified in the dual space by following a siilar reasoning to that of () for the tracing of specular reflective paths. The set of rays that are potentially occluded by s is represented in the dual space by the bea obtained by intersecting l R with the visibility region of s.inorderto test whether the bea is actually occluded by s or not, we can siply pick any ray within that area and check whether it reaches s before R. The dual space representation of the proble of Figure 9 is described in the right-hand side of Figure 0 (the geoetric description of the sae proble is shown on the left-hand side of the sae Figure, for reasons of convenience). In the exaple of Figure 0 the line of sight between R and s is partially occluded by s, therefore only the segent [, I ] contributes to diffusive paths. Let us consider, for exaple, the path S R, which is the line = P. The directions of the rays fro to R and S are given by i and o, respectively. Notice that until now, for reasons of siplicity, we have considered R to be the receiver, which forces the diffusion to occur last along the acoustic path. In order to ake the proposed approach euivalent to bidirectional bea tracing, we will consider that R is the receiver or a virtual receiver obtained by building a bea-tree fro the receiver location. In order to contain the coputational cost, we only consider low-order virtual sources and virtual receivers. In Section 7, for exaple, we liit the order of virtual sources and virtual receivers to three. 5. Tracing Diffractive Beas and Paths Using Visibility Diagras In this section we extend the use of visibility diagras to odel diffractive paths and, using the UTD, we generalize the fast bea tracing ethod of [] to account for this propagation phenoenon. 5.. Selection of the Diffractive Wedges. As already discussed in [], the diffractivefieldturnsouttobelessrelevant when source and receiver are in direct visibility. The very first step of the algorith consists, therefore, of selecting which edges are likely to generate a perceptually relevant diffraction. In what follows, we will refer to a wedge as a geoetric configuration of two or ore walls eeting into

11 EURASIP Journal on Advances in Signal Processing I S 7 S Wedge S S 5 S Figure : An exaple of diffractive wedge and beas departing fro it. Notice that if either the source or the receiver fall outside the regions arked as I and II, then there is direct visibility, therefore diffraction can be neglected. a single edge. If the angular opening of the wedge is saller than π and both the receiver and the source fall inside the wedge, then source and receiver are in direct visibility. Not all wedges are, therefore, worth retaining. Even if a wedge is diffractive, we can still find configurations where source and receiver are in direct visibility. When this happens, diffraction is less relevant than the direct path and we discard these diffractive paths. With reference to Figure, we are interested in auralizing diffraction in thetworegions arked as I and II, where source and receiver are not necessarily in conditions of utual visibility. For each of the two regions we will build a bea-tree. This selection process returns a list i =,..., M of diffractive wedges and their coordinates. 5.. Tracing Diffractive Bea-Trees. With reference to Figure, our goal is to split the bundle of rays departing fro the virtual source and directed towards the regions I and II into subbeas. In order to do this, we take advantage of the visibility diagras. As seen in Section 3, the virtual sourceisplacedonthetip ofthe diffracting wedge. The dual of this virtual source is the sei-infinite line of euation = + or =, the sign depending on the noralization of the RRP. We are interested in auralizing the diffracted field only in regions I and II, therefore we evaluate the regional visibility along the lines (virtual sources in the geoetric space) =±onlyfor> 0 or < 0. By scanning these sei-infinite lines along the visibility diagras we can iediately deterine all the visible reflectors of the above beas, and therefore deterine their branching. Figure shows the first-level beas and their dual representations for the configuration of Figure (which refers to regions I and II). The propagation and the branching of these diffractive beas will follow the usual rules, according to the iterative tracing echanis defined in Section forspecular reflections. The nuber of levels (branching order) of the diffractive bea-trees is indeed to be specified in a different S S 3 II fashion copared with reflective bea-trees, for reasons of relevance and coputational load. We notice that, at this stage, the location of source or receiver does not need to be specified. As a conseuence, we can build diffractive beatrees in a precoputation phase Diffractive Paths Coputation. Once source and receiver are specified, we can finally build the diffractive paths. Let P (i,) S,I (P (i,) S,II ) denote the ith reflective path between the source and the bea-tree to the region I (to the region II) departing fro the diffractive wedge arked with in the environent. As far as the auralization of diffraction is concerned, the path P (i,) S,I is copletely defined if we specify the source location, the position of the point of incidence of the ray on the walls (possibly in noralized coordinates) and the location of the diffractive edge. The set of paths between the source and the diffractive region inside the bea-tree of the region I (II) is P () S,I (P () S,I ). Siilarly, P (j,n) R,I (P (j,n) R,I ) is the jth path between the receiver and the nth diffractive wedge in the bea-tree I (II), and P (n) R,I (P R,II) (n) is the set of paths between receiver and the diffractive edge inside the region I (II). We are interested in diffractive paths where source and receiver fall in opposite regions: if the source falls within the bea-tree I, then we will check whether the receiver is in the bea-tree II, as diffraction is negligible when both source and receiver are in the sae region. Let P () I II be the set of paths associated to the diffractive edge (when the source is in bea-tree I, and the receiver in II). Siilarly, P () II I is the set of paths where the source is in the bea-tree II and the receiver is in I. These sets are obtained as P () I II = P () S,I P () R,II, P () II I = P () S,II P () R,I, where denotes the cartesian product. Finally, the set of paths for the diffractive edge is the union of P () I II and P () II I: () P () = P () I II P () II I. (5) In order to preserve the Keller cone condition, we have to deterine the point P d on the diffractive edge that akes the angle θ i between the incoing ray and the edge eual to the angular aperture of the Keller cone θ d. When dealing with diffractivebea-treeswhichincludealsooneorore reflections, the condition θ i = θ d is no longer sufficient for deterining the location of the virtual source, as we ust also preserve the Snell s law for the reflections inside the path. In [] the authors propose to copute the diffraction and reflection points along the diffractive path through a syste of nonlinear euations. This solution is obtained through an iterative Newton-Raphson algorith. In this paper we adopt the sae approach.

12 EURASIP Journal on Advances in Signal Processing Wedge S S S 3 S II S 7 I S S 5 S S 3 b b b 3 b I S 3 S S 5 b b 7 b b5 S Wedge S S S 7 S S 5 S 7 II (c) (d) Figure : When the source and the receiver fall outside the regions I and II, diffraction becoes negligible. The diffractive bea-trees will therefore cover regions I and II only.. Applications to Rendering In this section we discuss the above bea tracing approach in the context of acoustic rendering. As anticipated, we first discuss the ore traditional case of channel-based (pointto-point) rendering. This is the case of auralization when the user is wearing a headset. We will then discuss the case of geoetric wavefield rendering... Path-Tracing for Channel Rendering. In this section we propose and describe a siple auralization syste based on the solutions proposed above. Due to their different nature, we will distinguish between reflective, diffusive and diffractive echoes. In particular, diffraction involves a perceptually relevant low-pass filtering on the signal, hence we will stress diffraction instead of diffusion and reflection. The diffraction is rendered by a coefficient, whose value depends on the freuency. Each diffractive wedge, therefore, acts like a filter on the incoing signal, with an apparent ipact on the overall coputational cost. We therefore ade our auralization algoriths select the ost significant paths only, based on a set of heuristic rules that take the power of each diffractive filter into account.... Auralization of Reflective Paths. As far as reflections are concerned, we follow the approach proposed in []: the echo i is characterized by a length l i. The agnitude of the ith echo is A i = r k /l i, r being the reflection coefficient and k being the nuber of reflections (easily deterined by inspecting the bea-tree). The delay associated to the echo i is d i = l i /c,wherec is the speed of sound (c = 30 /s in our experients).... Auralization of Diffraction. As otivated in Section 5, we have resorted to UTD to render diffraction. In order to auralize diffraction paths, we have to copute a diffraction coefficient, which exhibits a freuency-dependent behavior. A coprehensive tutorial on the coputation of the diffraction coefficient in UTD ay be found in []. We reark here that in order to copute the diffractive filter, for each path we need soe geoetrical inforation about the wedge (available at a precoputation stage) and about the path, which is available only after the Newton Raphson algorith described in the previous Section...3. Auralization of Diffusive Paths. Accurate odeling of diffusion is typically based on statistical ethods [33]. Our

13 EURASIP Journal on Advances in Signal Processing 3 odeling solution is, in fact, aied at auralization and rendering applications, therefore we resort to an approxiated but still reliable approach that does not significantly ipact on the coputational cost: we ake use of the Labert cosine law, which works for energies, to copute the energy response. Then we convert the energy response onto a pressure response by taking its suare root. A siilar idea was developed in [3], where the author cobines bea tracing and radiosity. In order to copute the energy response, we need the knowledge of the position of the virtual source and receiver and the segent on the illuinating reflector fro which rays depart. The energy response H (i) d of the diffusive path fro reflector i is given by the integration of the contributions of each point of the reflector. If the length of the diffusive portion is l, then the auralization filter ay be approxiated as H (i) d = N D(P i )ΔP, () i= where P i is the coordinate on the diffuser, N is the nuber of portions in which the segent has been subdivided and D(P i ) is odeled according to the Labert cosine law: D(P i ) = cos[θ i(p i )] cos[θ o (P i )], (7) π and θ i (P i )andθ o (P i ) are the directions of the incident and outgoing rays referred to the axis of the reflector, respectively. Angles θ i (P i )andθ o (P i ) are related to i and o according to θ i (P) = tan[ i (P)], θ o (P) = tan[ o (P)]... Bea Tracing for Sound Field Rendering. The beatree contains all the inforation that we need to structure a sound field as a superposition of individual beas, each originating fro a different iage-source. A solution was recently proposed for reconstructing an individual bea in a paraetric fashion using a loudspeaker array []. Here, an arbitrary bea of prescribed aperture, orientation origin is reconstructed using an array of loudspeakers. In particular, the least suares difference of the wavefields produced by the array of loudspeakers and by the virtual source is iniized over a set of predefined control points. The iniization returns a spatial filter to be applied at each loudspeaker. It is interesting to notice that the approach described in [] offers the possibility to design a spatial filter that perfors a nonunifor weighting of the rays within the sae bea. This feature enables the rendering of tapered beas, which is particularly useful when dealing with diffractive beas. In fact, the diffraction coefficient (see [] for further details) assigns different levels of energy to rays within the sae bea, according to the reciprocal geoetric configuration of the source, the wedge and the direction of travel of the ray departing fro the wedge. (8) y () x () Figure 3: Exaple of a test environent used for assessing the coputational efficiency of our bea tracing algorith. The 8 roos are all connected together through randoly-located apertures. The rendering of the overall sound field is finally achieved by adding together the spatial filters for the individual beas. More details and a soe preliinary results of this ethod can be found in [35]. 7. Experiental Results In order to test the validity of our solution, we perfored a series of siulation experients as well as a easureent capaign in a real environent. An initial set of siulations was perfored with the goal of assessing the coputational efficiency of our techniues for the auralization of reflective, diffractive and diffusive paths, separately considered. In order to assess the accuracy of our ethod, were constructed the ipulse responses of a given environent (on an assigned grid of points) through both siulation and direct easureents. Fro such ipulse responses, we derived a set of paraeters, typically used for describing reverberation. The coparison between the easured paraeters and the siulated ones was aied at assessing the extent of the iproveent brought by introducing cuulatively diffraction and diffusion into our siulation. 7.. Coputation Tie. In order to assess the coputational efficiency of the proposed ethod, we conducted a series of siulations in environents of controlled coplexity (variable geoetry). We developed a procedure for generating testing environents with an arbitrary nuber of 8 roos, all chained together as shown in Figure 3. An N-roo environent is therefore ade of N reflectors. In order to enable acoustic interaction between roos, each one of the interediate walls exhibits a randolyplaced opening. In our tests, we generated environents with 0, 0, 80, 0, 30, 0 such walls. The siulation platfor was based on an Intel Mobile Pentiu processor euipped with GB of RAM, and the goal was to:

14 EURASIP Journal on Advances in Signal Processing Variable geoetry, 0000 beas 0.0 Bea tracing tie (s) Diffused paths estiation tie (s) Visibility diagra Traditional bea tracing Walls Figure : Bea-tree building tie for variable geoetry for traditional bea tracing (circles) and visibility-based bea tracing (suares) for 0,000 beas. The proposed approach greatly outperfors traditional bea tracing especially when the nuber of traced beas is very large. (i) copare the bea-tree s building tie of visibilitybased bea tracing with that of traditional bea tracing [5]; (ii) easure the diffractive path-tracing tie; (iii) easure the diffusive path-tracing tie. As far as the tie spent by the syste in coputing the visibility diagras is concerned, we invite the reader to []. The bea-tree building tie is intended as the tie spent by the algorith in tracing a preassigned nuber of beas. In order to assess the ipact of the proposed approach on this specific paraeter, in coparison with a traditional beatracing approach, we stopped the algorith when 0,000 beas were traced. The siulation results are shown in Figure. As expected, our approach turns out to outperfor traditional bea tracing. As the bea-tree building tie strongly depends on the source location, we conducted nuerous tests by placing the source at the center of each one of the roos of the odular environents. The beatree building tie was then coputed as the average of the bea-tree building ties over all such siulations. In order to test the tracing tie of the diffusive paths the source has been placed in the center the odular environent. We conducted several experients placing the receiver at the center of each one of the roos. We easured the tie spent by the syste in coputing the diffusive paths. We considered up to three reflections before and after the diffuse interaction. Figure 5 shows the average tracing tie of the diffusive paths as a function of the nuber of walls. If we are interested in an accurate rendering, we should trace at least 000 beas: in this situation we notice that even in a coplex environent (e.g., 0 walls) the auralization of the diffusive paths takes only a fraction of the tie spent by the Walls Figure 5: Tracing tie of the diffusive paths as a function of the nuber of walls of the environent of Figure 3. Diffractive path auralization tie (s) Walls Figure : Auralization tie for diffractive paths (easured in seconds) versus the coplexity (easured by the nuber of walls) of the environent. syste for the auralization of the reflective paths. The last test is aied at assessing the coputational tie for auralizing the diffractive paths. As done above, the source was placed at the center of the environent and we conducted several experients, placing the receiver at the center of each roo. In particular in this situation two tests have been conducted: (i) the nuber of diffractive paths with respect to the nuber of walls in the environent; (ii) the coputational tie for auralizing the diffractive paths. Figure shows the tie that the syste takes to auralize the diffractive paths, as a function of the nuber of walls in the environent. Figure 7 shows the nuber of paths for the sae experient. The nuber of diffractive paths depends upon both the depth of the diffractive bea-trees

15 EURASIP Journal on Advances in Signal Processing 5 Nuber of diffractive paths Walls Figure 7: Nuber of diffractive paths versus the coplexity of the environent (easured by the nuber of walls) in which experients are done Siulated response with reflections: EDT (s) 0 Siulated response with reflection and diffraction: EDT (s) 0 Siulated response with reflection, diffraction and diffusion: EDT (s) 0 (c) y x Figure 8: Floor-ap of the validation environent: the grid represents the 77 acuisition points and the loudspeaker location is arked with a dot. 0.5 Measured response: EDT (s) 0 (d) Figure 9: Early Decay Tie ap: siulated with reflections only, siulated with reflections and diffraction, (c) siulated with reflections, diffraction and diffusion, (d) easured. and the nuber of diffractive edges in the environent in uite an unpredictable fashion. Notice that the auralization for the diffractive paths can exceed the auralization of the reflective paths. However, we should keep in ind that the coputation of over 000 diffractive paths can be uite redundant fro a perceptual standpoint. We propose here to select the diffractive paths arisen fro the closer diffractive wedges, as they are ore perceptually relevant. 7.. Validation. It becoes iportant to assess how well spatial distributions of reverberations are rendered by this approach. In order to do so, we conducted coparative tests between siulated data and real data with corresponding geoetries. We placed a high-uality loudspeaker in a reverberant environent and used it to reproduce a MLS seuence [3]. The environent s floor ap is shown in Figure 8, where the location of the loudspeaker and icrophone are arked with green and red dots, respectively. The acuired signals were then processed to deterine the ipulse responses of the environent in those locations, as described in [3]. The reflectiveand diffusive reflection coefficients of the walls, indeed, could not be directly easured but were coarsely deterined so that the T 0 of the siulated ipulse response would approxiate the estiated one in a given test position. As a saple-to-saple coparison of the acuired and siulated ipulse responses is not very inforative, we copared soe paraeters that describe the ipulse response, which can be readily derived fro the siulated and easured ipulse responses. In particular, we run extensive siulations first with reflections only (up to the th order of the bea-tree), with diffraction (up to the th order) and finally with diffusion (with an order 3).

16 EURASIP Journal on Advances in Signal Processing Siulated response with reflections: Noralized energy Siulated response with reflections: Centre tie (s) Siulated response with reflection and diffraction: Noralized energy 0 Siulated response with reflection, diffraction and diffusion: Noralized energy 0 (c) Measured response: Noralized energy 0 (d) Figure 0: Noralized energy ap: siulated with reflections only, siulated with reflections and diffraction, (c) siulated with reflections, diffraction and diffusion, (d) easured Siulated response with reflection and diffraction: Centre tie (s) 0 Siulated response with reflection, diffraction and diffusion: Centre tie (s) 0 (c) Measured response: Centre tie (s) 0 (d) Figure : Center Tie ap: easured siulated with reflections only, siulated with reflections and diffraction, (c) siulated with reflections, diffraction and diffusion, (d) easured. Early Decay Tie (EDT). EDT is the tie that the Schroeder envelope of the ipulse response takes to drop of 0 db. A coparison of the EDT aps (geographical distribution of the EDT in the easured and siulated cases) is shown in Figure 9. Notice that the ipact of diffusion in this experient is uite significant: in fact, as expected, including diffusion increases the energy in the tail of the ipulse response, with the result of obtaining a ore realistic reverberation. Noralized Energy of the Ipulse Response. where the noralization is perfored with respect to the ipulse response of axiu energy. A coparison between noralized energy aps is shown in Figure 0. Once again we notice that a siulation of all phenoena (reflections, diffraction and diffusion) produces ore realistic results. Center Tie. It is first-order oentu of the suared ipulse response. The Center Tie (CT) ap is shown in Figure, where we see that a siulation based on reflections, diffraction and diffusion allows us to achieve, once again, a good atch. We notice, in particular, that odeling diffusion increases the level of soothness of the siulated ap, which akes it ore siilar to the easured one. 8. Conclusions In this paper we proposed an extension of the visibilitybased bea tracing ethod proposed in [], which now allows us to odel and render propagation phenoena such as diffraction and diffusion, without significantly affecting the coputational efficiency. We also iproved the ethod

17 EURASIP Journal on Advances in Signal Processing 7 in [] by showing that not just the construction of the bea-tree but also the whole path-tracing process can be entirely perfored on the visibility aps. We finally showed that this approach produces uite accurate results when coparing siulated data with real acuisitions. Thanks to that, this odeling tool proves particularly useful every tie there is a need for an accurate and fast siulation of acoustic propagation in environents of variable geoetry and variable physical characteristics. Acknowledgent The research leading to these results has received funding fro the European Counity s Seventh Fraework Prograe (FP7/007-03) under grant agreeent no References [] S. Kopuz and N. Lalor, Analysis of interior acoustic fields using the finite eleent ethod and the boundary eleent ethod, Applied Acoustics, vol. 5, no. 3, pp. 93 0, 995. [] A. Kludszuweit, Tie iterative boundary eleent ethod (TIBEM) a new nuerical ethod of four-diensional syste analysis for thecalculation of the spatial ipulse response, Acustica, vol. 75, pp. 7 7, 99. [3] R. Ciskowski and C. Brebbia, Boundary Eleent Methods in Acoustics, Elsevier, Asterda, The Netherlands, 99. [] U. Krockstadt, Calculating the acoustical roo response by the use ofa ray tracing techniue, Journal of Sound and Vibrations, vol. 8, no., pp. 8 5, 98. [5] Paul S. Heckbert and Hanrahan, Bea tracing polygonal objects, in Proceedings of the th Annual Conference on Coputer Graphics and Interactive Techniues, pp. 9 7, July 98. [] N. Dadoun, D. Kirkpatrick, and J. Walsh, The geoetry of bea tracing, in Proceedings of the ACM Syposiu on Coputational Geoetry, pp. 55, Sedona, Ariz, USA, June 985. [7] M. Monks, B. Oh, and J. Dorsey, Acoustic siulation and visualisationusing a new unified bea tracing and iage source approach, in Proceedings of the 00th Audio Engineering Society Convention (AES 9), Los Angeles, Ariz, USA, 99. [8] U. Stephenson and U. Kristiansen, Pyraidal bea tracing and tie dependent radiosity, in Proceedings of the 5th InternationalCongressonAcoustics, pp. 57 0, Trondhei, Norway, June 995. [9] J. P. Walsh and N. Dadoun, What are we waiting for? The developentof Godot, II, Journal of the Acoustical Society of Aerica, vol. 7, no. S, p. S5, 98. [0] T. Funkhouser, I. Carlbo, G. Elko, G. Pingali, M. Sondhi, and J. West, Bea tracing approach to acoustic odeling for interactive virtual environents, in Proceedings of the Annual Conference on Coputer Graphics (SIGGRAPH 98), pp. 3, July 998. [] S. Laine, S. Siltanen, T. Lokki, and L. Savioja, Accelerated bea tracing algorith, Applied Acoustics, vol. 70, no., pp. 7 8, 009. [] F. Antonacci, M. Foco, A. Sarti, and S. Tubaro, Fast tracing of acoustic beas and paths through visibility lookup, IEEE Transactions on Audio, Speech and Language Processing, vol., no., pp. 8 8, 008. [3] T. Funkhouser, P. Min, and I. Carlbo, Real-tie acoustic odeling for distributed virtual environents, in Proceedings of ACM Coputer Graphics (SIGGRAPH 99), A. Rockwood, Ed., pp , Los Angeles, Ariz, USA, 999. [] N. Tsingos, T. Funkhouser, A. Ngan, and I. Carlbo, Modeling acoustics in virtual environents using the unifor theory of diffraction, in Proceedings of the 8th International Conference on Coputer Graphics and Interactive Techniues (SIGGRAPH 00), pp , August 00. [5] R. G. Kouyoujian and P. H. Pathak, A unifor geoetrical theory of diffraction for an edge in a perfectly conducting surface, Proceedings of the IEEE, vol., no., pp. 8, 97. [] A. Canclini, A. Galbiati, A. Calatroni, F. Antonacci, A. Sarti, and S. Tubaro, Rendering of an acoustic bea through an array of loud-speakers, in Proceedings of the th International Conference on Digital Audio Effects (DAFx 09), 009. [7] A. J. Berkhout, Holographic approach to acoustic control, Journal of the Audio Engineering Society, vol. 3, no., pp , 988. [8] M. Foco, P. Polotti, A. Sarti, and S. Tubaro, Sound spatialization basedon fast bea tracing in the dual space, in Proceedings of the th International Conference on Digital Audio Effects (DAFX 03), pp. 98 0, London, UK, Septeber 003. [9] B. Brouard, D. Lafarge, J. -F. Allard, and M. Taura, Measureent and prediction of the reflection coefficient of porous layers at obliue incidence and for inhoogeneous waves, Journal of the Acoustical Society of Aerica, vol. 99, no., pp , 99. [0] M. Kleiner, H. Gustafsson, and J. Backan, Measureent of directional scattering coefficients using near-field acoustic holography and spatial transforation of sound fields, Journal of the Audio Engineering Society, vol. 5, no. 5, pp. 33 3, 997. [] C. Nocke, In-situ acoustic ipedance easureent using a free-field transfer function ethod, Applied Acoustics, vol. 59, no. 3, pp. 53, 000. [] S. I. Thoasson, Reflection of waves fro a point source by an ipedance boundary, Journal of the Acoustical Society of Aerica, vol. 59, no., pp , 97. [3] T. Funkhouser, N. Tsingos, and J. M. Jot, Sounds good to e! coputational sound for graphics, virtual reality, and interactive systes, in Proceedings of ACM Coputer Graphics (SIGGRAPH 0), San Antonio, Tex, USA, July 00. [] H. Kuttruff, Roo Acoustics, Elsevier, Asterda, The Netherlands, 3rd edition, 99. [5] L. L. Beranek, Concert and Opera Halls: How they Sound, Acoustical Society of Aerica through the Aerican Institute of Physics, 99. [] N. Tsingos, Siulating high uality virtual sound fields for interactive graphics applications, Ph.D. dissertation, Universite J. Fourier, Grenoble, France, 998. [7] M. A. Biot and I. Tolstoy, Forulation of wave propagation in infinite edia by noral coordinates with an application to diffraction, Journal of the Acoustical Society of Aerica, vol. 9, pp , 957.

18 8 EURASIP Journal on Advances in Signal Processing [8] T. Lokki, L. Savioja, and P. Svensson, An efficient auralization of edge diffraction, in Proceedings of st International Conference of Audio Engineering Society (AES 0), May 00. [9] U. P. Svensson, R. I. Fred, and J. Vanderkooy, An analytic secondary source odel of edge diffraction ipulse responses, Journal of the Acoustical Society of Aerica, vol. 0, no. 5, pp. 33 3, 999. [30] R. R. Torres, U. P. Svensson, and M. Kleiner, Coputation of edge diffraction for ore accurate roo acoustics auralization, Journal of the Acoustical Society of Aerica, vol. 09, no., pp. 00 0, 00. [3] J. B. Keller, Geoetrical theory of diffraction, Journal of the Optical Society of Aerica, vol. 5, pp. 30, 9. [3] D. McNaara, C. Pistorius, and J. Malherbe, Introduction to the Unifor Geoetrical Theory of Diffraction, ArtechHouse, 990. [33] J. H. Rindel, The use of coputer odeling in roo acoustics, Journal of Vibroengineering, vol. 3, pp. 7, 000. [3] T. Lewers, A cobined bea tracing and radiatn exchange coputer odel of roo acoustics, Applied Acoustics, vol. 38, no., pp. 78, 993. [35] F. Antonacci, A. Calatroni, A. Canclini, A. Galbiati, A. Sarti, and S. Tubaro, Soundfield rendering with loudspeaker arrays through ultiple bea shaping, in Proceedings of IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA 09), pp. 33 3, 009. [3] D. D. Rife and J. Vanderkooy, Transfer-function easureent with axiu-length seuences, Journal of the Audio Engineering Society, vol. 37, no., pp. 9, 989.

19 Photograph Turise de Barcelona / J. Trullàs Preliinary call for papers The 0 European Signal Processing Conference (EUSIPCO 0) is the nineteenth in a series of conferences prooted by the European Association for Signal Processing (EURASIP, This year edition will take place in Barcelona, capital city of Catalonia (Spain), and will be jointly organized by the Centre Tecnològic de Telecounicacions de Catalunya (CTTC) and the Universitat Politècnica de Catalunya (UPC). EUSIPCO 0 will focus on key aspects of signal processing theory and applications as listed below. Acceptance of subissions i will be based on uality, relevance and originality. Accepted papers will be published in the EUSIPCO proceedings and presented during the conference. Paper subissions, proposals for tutorials and proposals for special sessions are invited in, but not liited to, the following areas of interest. Areas of Interest Audio and electro acoustics. Design, ipleentation, and applications of signal processing systes. Multiedial d signal processing and coding. Iage and ultidiensional signal processing. Signal detection and estiation. Sensor array and ulti channel signal processing. Sensor fusion in networked systes. Signal processing for counications. Medical iaging and iage analysis. Non stationary, non linear and non Gaussian signal processing. Subissions Procedures to subit a paper and proposals for special sessions and tutorials will be detailed at Subitted papers ust be caera ready, no ore than 5 pages long, and conforing to the standard specified on the EUSIPCO 0 web site. First authors who are registered students can participate in the best student paper copetition. Iportant Deadlines: Organizing Coittee Honorary Chair Miguel A. Lagunas (CTTC) General Chair Ana I. Pérez Neira (UPC) General Vice Chair Carles Antón Haro (CTTC) Technical Progra Chair Xavier Mestre (CTTC) Technical Progra Co Co ChairsChairs Javier Hernando (UPC) Montserrat Pardàs (UPC) Plenary Talks Ferran Marués (UPC) Yonina Eldar (Technion) Special Sessions Ignacio Santaaría (Unversidad de Cantabria) Mats Bengtsson (KTH) Finances Montserrat Nájar (UPC) Tutorials Daniel P. Paloar (Hong Kong UST) Beatrice Pesuet Popescu (ENST) Publicity Stephan Pfletschinger (CTTC) Mònica Navarro (CTTC) Publications Antonio Pascual (UPC) Carles Fernández (CTTC) Industrial Liaison & Exhibits i Angeliki Alexiou (University of Piraeus) Albert Sitjà (CTTC) International Liaison Ju Liu (Shandong University China) Jinhong Yuan (UNSW Australia) Taas Sziranyi (SZTAKI Hungary) Rich Stern (CMU USA) Ricardo L. de Queiroz (UNB Brazil) Proposals for special sessions l i 5 Dec 00D Proposals for tutorials 8 Feb 0 Electronic subission of full papers Feb 0 Notification of acceptance 3 May 0 Subission of caera ready papers Jun 0 Webpage: