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Proc. of the 6th Int. Conference on Digital Audio Effects (DAFx-03), London, UK, September 8-11, 2003
SOUND SPATIALIZATION BASED ON FAST BEAM TRACING IN THE DUAL SPACE
Marco Foco, Pietro Polotti, Augusto Sarti, Stefano Tubaro
Dipartimento di Elettronica e Informazione
Politecnico di Milano
Piazza L. Da Vinci 32, 20133 Milano, Italy
marco.foco@fastwebnet.it polotti/sarti/tubaro@elet.polimi.it
ABSTRACT
This paper addresses the problem of geometry-based sound rever-
beration for applications of virtual acoustics. In particular, we
propose a novel method that allows us to significantly speed-up
the construction of the beam tree in beam tracing applications, by
avoiding space subdivision. This allows us to dynamically recom-
pute the beam tree as the sound source moves. In order to speed-
up the construction of the beam tree, we determine what portion of
which reflectors the beam “illuminates” by performing visibility
checks in the “dual” of the geometric space.
1. INTRODUCTION
Advanced sound rendering techniques are often based on the phys-
ical modeling of the acoustic reflections and scattering in the en-
vironment. This can be modeled using either numerical methods
(finite elements, boundary elements and finite differences) or geo-
metrical methods (image source, path tracing, beam tracing and
radiosity). With all such techniques, however, computational com-
plexity may easily become an issue. One approach that enables an
efficient auralization of all reflected paths is based on beam trac-
ing [1]. This method is based on a geometric pre-computation of
a tree-like topological representation of the reflections in the envi-
ronment (beam tree) through spatial subdivision techniques. The
beam tree is then used for real-time auralization through a sim-
ple look-up of which beams pass through the auditory points. It
is important to notice, however, that the tree-like structure of the
beam reflections makes the approach suitable for early reverbera-
tions only, as it prevents us from looping beams and implementing
the corresponding IIR structures. In order to overcome this diffi-
culty, we recently proposed an approach [2] that models both early
and late reverberation by cascading a tapped delay line (a FIR)
with a Waveguide Digital Network (WDN) [3] (an IIR). In this ap-
proach, the beam tree is looked-up to generate the coefficients of
a tapped delay line for the auralization of early reverberation. In
order to account for late reverberation as well, we feed the out-
puts of the tapped delay lines into WDN, whose parameters are
determined through path tracing [2].
One problem of the beam tracing approach is that it assumes
that sound sources are fixed and only listeners are allowed to move
around.Infact,everytimethesourcemoves,thebeamtreeneeds
to be re-computed. This may easily become a costly operation,
particularly with environments of complex topology, as it is based
on spatial subdivision algorithms.
In this paper we propose a novel method that allows us to sig-
nificantly speed-up the construction of the beam tree, by avoiding
space subdivision. This allows us to dynamically recompute the
beam tree as the source moves. In order to speed-up the construc-
tion of the beam tree, we determine what portion of which reflec-
tors the beam “illuminates” by performing visibility checks in the
“dual” of the world space. In order to illustrate the approach, we
will assume that the world space be 2-dimensional.
2. TRACING BEAMS IN THE DUAL SPACE
The world space that we consider is made of sources and reflec-
tors. Sources are assumed as point-like, and reflectors are linear
segments. We call “active” that region of a reflector that is directly
“illuminated” by a source. The active portion of a reflector can be
made of one or more active segments (connected sets), due to oc-
clusions. Each one of the active segments generates a beam, which
is defined as that bundle of acoustic rays that connect the source
with points of the active segment. Each acoustic ray wcan be de-
scribed by an equation of the form y=ax +b,whereais the
angular coefficient and bis the offset on the yaxis, all referred to
a world coordinate system (x, y).Thedualˆwof that ray is thus
apoint(a, b)in parameter space. The dual of a point pof coor-
dinates (x, y), on the other hand, is a line ˆpin parameter space,
givenbyallpairs(a, b)that correspond to rays passing through p.
This means that the dual of a source is a ray.
Figure 1: The duals of a set of physical reflector is a collection of
strip-like regions in the dual space.
Let us now consider a reflector, which is a segment in world
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Proc. of the 6th Int. Conference on Digital Audio Effects (DAFx-03), London, UK, September 8-11, 2003
space. In order to determine its dual, we can consider the duals ˆp
and ˆqof its two extremes p and q, respectively. Such points will
correspond to two lines ˆpand ˆqin the dual space. As the reflector
r is made of all points in between p and q, its dual ˆrwill be made
of all the lines in between ˆpand ˆq. In other words, the dual of a
reflector is a beam-like region. We recall, however, that an active
segment of a reflector, together with a virtual source s, completely
specifies a beam. The dual of this beam can be determined as the
intersection between the dual ˆrof the active segment (a beam-like
region) and the dual ˆsof the virtual source (a line). In conclusion,
the dual of a beam is a segment, just like the dual of a segment
(reflector) is a beam.
Pushing this formalism even further, we can define active re-
flecting segments at infinity (just like in projective geometry) and
beams at infinity, which are those beams that are not reflected by
any surface. If we have a collection of reflectors in world space,
their duals will form a collection of beam-like regions (see Fig. 1).
We would now be tempted to say that the intersection between
the dual of a source and the dual of the reflectors is the dual of the
branching beams, therefore it can be used to determine the beam
tree. This would be true if there were no occlusions between re-
flectors (i.e. if all points of the reflectors were active). In fact, mu-
tual occlusions cause the duals of reflectors to overlap each other,
therefore in order to determine the branching of the beams we need
first to figure out the layering order of such regions, according to
mutual occlusions (i.e. “which region eats which”).
The layering order is not the only problem, as we need to con-
struct the tree of all possible reflections in the environment while
keeping track of the occlusions. In order to do so, we propose an
iterative approach that starts from the source (root) and tracks the
branching of all beams in dual space. At each iteration we con-
sider a beam at a time and we determine how this beam is split
into various sub-beams as it encounters reflectors on its way. This
splitting characterizes the branching of the beam tree.
Figure 2: Beam tracing in world-space. a. The source silluminates
the reflector r3and partially the reflector r4. b. The beam that
describes the reflection from r4(s0is the virtual source for this
reflection). In this figure sub-beams illuminating r3,r1and r2are
shown.
At the first iteration we consider the source and the physical
reflectors (Fig. 2a). From this source a number of beams will
depart, each corresponding to an active segment of the various re-
flectors. Such active segments can be determined quite straightfor-
wardly by tracing all rays that depart from the source. At this point
we consider all the beams one by one and figure out how each of
them branches out as it encounters the reflectors (Fig. 2b). Each
beam will keep branching out until its cumulative attenuation or
its aperture angle falls beneath some pre-assigned threshold. Let
us consider a beam originating from a virtual source s.Ifwewant
to characterize how this beam branches out, start from the active
reflector segment r0(Fig. 3a) that defines that beam (aperture) and
we perform a change of reference frame in order for its extremes
p0and q0to fall in the coordinates (0,1) and (0,−1), respectively
(Fig. 3b).
Figure 3: A reflective segment: a. in the world space, b. after
normalization relative to ri, c. in the dual space
This way, the dual of the considered aperture will be the ref-
erence strip −1≤b≤1in parameter space (see Fig. 3c). The
intersection in the dual space between the refence strip and the rep-
resentation oof the othe considered reflection is the area named ˆr1
that represents the visibility of r0
1from r0
0.
In Fig. 4 and 5 we can see some other examples of a reflector
visibility from a reference segment.
Figure 4: The visibility of r0
1with respect of r0
0. The visibility is
limited to the right half-plane of the reference reflector.
Let us now consider again Fig. 1, which has four physical re-
flectors. Let us assume that, as the iterations go alone, r4is found
to be an active reflector, and we want to show how to determine the
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Proc. of the 6th Int. Conference on Digital Audio Effects (DAFx-03), London, UK, September 8-11, 2003
Figure 5: Another particular case of visibility. Varying the obser-
vation point on the reference segment, the two different sides of
the reflector r0
1are seen. This is clear from the observation of the
dual space.
branching of the beams. Therefore we will consider r4as an active
aperture, and a coordinate change is used to normalize its repre-
sentation in the space domain. We will name the new coordinate
system (x4,y
4)(superscript denotes which reflector is currently
the reference for the normalization), and the new representation of
the other reflectors will become r4
1,r4
2and r4
3. In the dual space
we obtain the set of regions of Fig. 6. Notice that r4
3partially oc-
cludes r4
1. In the dual space, when the areas that represents two
reflectors overlap, ther is an occlusion. By analyzing the slopes of
the dual representation of the reflector endpoints, it is possible to
define in the correct way the relative visibility.
Figure 6: Beam tracing in the dual space. The intersection between
ˆsand the colored regions generates a collection of segments, which
are duals of the branching beams.
A particular ordering situation between reflectors occurs when
the sorting between two (or more) reflectors is ambiguous with re-
spect to a third reflector (see Fig. ??). This happens when the
reflector’s order changes with respect to a moving point on the ref-
erence reflector. This could be a problem when working in metric
space, but it becomes easy to solve in the dual space. In fact, in
the dual space the visibility information is obtained by partitioning
the reference strip (−1≤b≤1), by looking at the intersections
of the segments that define the reflector’s endpoints.
As a last foremark it is important to notice that, without loss
of generality, the source can always be placed on the left of the
normalized aperture, as we can always replace a (virtual) source
with its specular with respect to the aperture. As we can see in
Fig. 6, the scan of the rays that define a source beam corresponds
to the scan of the points of its dual along its extension.
Figure 7: The sorting problem: a.world-space sorting is ambigu-
ous; b. solution in dual-space.
3. IMPLEMENTATION
In order to test the method that we propose, we implemented a
software application that includes a 2D CAD for creating test en-
vironments, and an auralization system based on both a traditional
beam tracer and a dual-space beam tracer. Auralization is based on
tapped delay lines, and includes HRTF and cross-fading between
beam configurations.
The geometries considered by this application are only two-
dimensional, but the method can be extended to the 3D space. The
basic structure of our auralization system is shown in Fig. 8.
Figure 8: A schematic view of the auralization algorithm embed-
ded in the beam tracing application.
In a first step, the algorithm analyses the geometric descrip-
tion of the reflectors of the environment, and it extracts all the
necessary information on visibility by working in the dual of the
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Proc. of the 6th Int. Conference on Digital Audio Effects (DAFx-03), London, UK, September 8-11, 2003
geometric space. As this step depends only on the geometry of the
reflectors,as long as the environment is fixed, this operation can
be performed beforehand, as a preprocessing step.
The location of the sound source and the visibility information
can now be used to construct the beam tree, or to update it every
time the source moves.
Figure 9: Building a tree level
Fig. 9 refers to the construction of one level of the beam
tree (a reflectiononther4reflector, as already shown in Fig. 6).
In this portion of the beam tree, we describe the splitting of the
generating beam (the one that is incident to reflector r4)intofive
sub-beams (numbered from 1 to 5). Sub-beams 1 and 3 are “un-
bounded” beams (they keep propagating undisturbed), while sub-
beams 2, 3 and 5 are “bounded”, as they end up onto reflectors r2,
r1and r3, respectively. The process is repeated for each one of
the “bounded” sub-beams, until the number of reflections reaches
apre-fixed maximum or until the energy associated to the beam
becomes negligible.
The beam-tree lookup block of Fig. 8 generates the filter taps
according to what beams reach the listener’s location. This update
operation is fast and can be performed frequently [1]. Through
this lookup process we obtain the direction of arrival (DOA), the
path length and the list of the reflectors encountered so far. This
phase requires all of the tree nodes and leaves to be visited, and the
relative beam to be tested for listener’s inclusion. If the listener’s
location falls within the beam area, the DOA and the path length
are computed. From there we can go down the beam tree towards
the root, in order to obtain the list of reflectors encountered along
the propagation of the wavefront.
Since it is impossible to produce one stream for every DOA,
we grouped together angular intervals in order to generate a limited
number of audio streams, one per interval. Each one of these an-
gular intervals will be attributed a separate delay line (a FIR filter).
The taps are computed considering all path lengths and relative
attenuations.
The filter bank whose parameters are generated at the previous
step, constitutes the auralization algorithm, which generates the
directional streams from an anechoic (dry) source. These streams
are then mixed for stereo headphone auralization using a HRTF. In
our implementation we used OpenAL [7] to mix 16 virtual “equiv-
alent sources” placed in circle around the listener (to simulate 16
discretized DOAs). The listener’s head orientation is accounted
foronlyinthislaststep.
In the following table we summarise what kind of computation
is required for source-listener geometry changes.
Type of motion Recomputation required
Source beam tree re-computation
Listener’s location beam tree lookup, filter bank update
Listener’s orientation no re-computation required
Every time a re-computation is required, the taps in the de-
lay lines are changed, and this can cause some clicking sounds to
be produced in the output. This can be avoided by smoothly in-
terpolating parameters in place of just switching them. A simple
method to avoid clicks is a linear interpolation in the audio streams
generated by using the “old” and “new” filters (see Fig. 10).
Figure 10: Mixing audio streams to avoid clicks.
4. PERFORMANCE EVALUATION
In order to test the effectiveness of the proposed solution, we com-
pared our implementation (dual space approach) with an optimized
one that we developed, which works in geometry space but does
not require space subdivision. In spite of the computational ef-
ficiency of the approach based working in metric space, the per-
formance improvement of our approach in the dual space is quite
apparent.
In the geometry space, the complexity of tracing a beam tree
level without space subdivisions depends on n, the number of re-
flectors. In the dual space, the complexity depends on m(<n), the
number of “visible” reflectors from the reference. In most cases,
such as a typical office configuration, mdepends only on the local
geometry, and it is independent from n. For example, in a system
of square rooms the average visible faces is 6-7, but in a structure
characterized by long corridors the number of visible faces can be
higher.
In the test we performed with the geometry space approach,
the time used to rebuild the beam tree depends quadratically on
thenumberofreflector in the world, while our method (dual space
analysis) grows almost in a linear way (Fig. 11).
These tests have been executed on a modular set of rooms
connected by not aligned doorways: an example of this model (5
rooms, 20 reflectors) is shown in the program’s screenshot in Fig.
12.
5. CONCLUSIONS AND FUTURE WORK
In this paper we showed a novel, very effective algorithm to trace
beams in the 2-dimensional space to be used for virtual acoustics.
The computational efficiency has been achieved by exploiting par-
ticular properties of the representation in the dual space.
We also showed that the above procedure, implemented on a
standard PC platform, turns out to be very effective, and the com-
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Proc. of the 6th Int. Conference on Digital Audio Effects (DAFx-03), London, UK, September 8-11, 2003
Figure 11: Execution time vs. number of reflectors in the scene for
the standard beam tracing (diamonds), proposed method (squares).
Figure 12: Test application screenshot. The room’s model is the
5-rooms test model (20 reflectors).
putational saving compared with standard beam tree methods in-
creases as the complexity of the environment grows.
We are currently working on a 3-dimensional version of this
algorithm, to be used with more complex environments.
6. REFERENCES
[1] T. Funkhouser, I. Carlbom, G. Elko, G. Pingali, M. Sondhi,
J.West,“BeamTracingApproachtoAcousticModelingfor
Interactive Virtual Environments”. Computer Graphics (SIG-
GRAPH ’98), Orlando, FL, July 1998, pp. 21–32.
[2] Augusto Sarti and Stefano Tubaro: “Efficient geometry-based
sound reverberation”, XEUSIPCO, Toulouse, France, 2002.
[3] J.O. Smith, “Principles of digital waveguide models of musi-
cal instruments”, in Applications of digital signal processing
to audio and acoustics,editedbyM.KahrsandK.Branden-
burg, Kluwer, 1998, pp. 417–466.
[4] S. Teller and P. Hanrahan, “Global visibility algorithms for
illumination computations”, in Computer Graphics (SIG-
GRAPH ’93 Proceedings), pp. 239–246.
[5] P.S. Heckbert and P. Hanrahan, “Beam Tracing Polygonal Ob-
jects”, in Computer Graphics (SIGGRAPH ’84 Proceedings),
pp. 119–127.
[6] L.Savioja, “Modeling Techniques for Virtual Acoustics”, PhD
dissertation, .Espoo 1999.
[7] OpenAL Documentation, available via
http://www.openal.org/.
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