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but it computes early reflections with high efficiency. Therefore, it is often applied in virtual auditory reality to facilitate real-time auralization (Savioja et al., 1999; Vorländer, 2020), whereas at higher order reflections, the
technique quickly becomes intractable.
In a typical room geometry, most of the image sources provided by the image-source method are obstructed such that they can never became audible and thus cause lots of unnecessary computation. Funkhouser et al. (2004) developed an image source-based beam tracing that takes this into account, resulting in high computational efficiency. This is achieved by taking into account the reflecting geometry when new image sources are generated and the only reflections taken into account are those that can produce a visible image source to some receiver location.
Ray-Tracing Methods
Ray tracing (Krokstad et al., 1968), which involves the tracing of sound particles (phonons) traveling at sound speed in enclosures like light rays, represents one major method in room-acoustic computer simulation. Figure 5b illustrates the core of ray tracing. A sound source radiates rays that are traced for each reflection and then registered for valid paths. The source radiates rays using either a predefined distribution or in random directions (Krokstad et al. 1968). A known directivity of the source will weigh the ray distribution.
Reflection paths originating from the source and reaching the receiver are determined using detectors that intersect with the rays. The detectors are typically volumetric objects such as spheres so that they register enough rays to give reliable results (Vorländer, 1989). Use of more rays warrants use of smaller detectors and more accurate results. Another option is to use point-like detectors and volumetric rays, typically of conical or of pyramidal shape.
The ray termination and the energy attenuation of sound propagation rely on each other and can be calculated using different approaches. In a common approach, each ray carries its energy content in given frequency bands on an interior surface reflection. The material properties of the surface determines the energy attenuation. The ray termination is eventually determined for its energy to decay below a preselected threshold or to reach a predefined maximum traveling distance.
In practice, one simulation containing multiple receivers is often computationally advantageous. Each such receiver registers an energy room impulse response, as the one schematically illustrated in Figure 3. Note that the convolution operation applied in auralization requires pressure impulse responses where an energy impulse response needs to be further processed to create a room impulse response. In addition, some approximation techniques need to be involved to achieve as realistic outcome as possible (Kuttruff, 1993).
In comparison with the image-source method, the ability to incorporate diffuse reflections represents one unique feature of ray tracing. Krokstad et al. (1968) and Schroeder (1973) discussed the basic concept of incorporating diffuse reflections into ray tracing, but it was Kuttruff (1971) who first implemented ideally diffuse reflections. A more general method engages a specular reflection component and the other diffuse components; a scattering coefficient determines the ratio of the two components. Comparison studies demonstrate that the ray-tracing technique exhibits a superior performance over the image-source method.
Surface-Based Modeling
The techniques of this kind first engage a sound source to propagate sound energy to the interior surfaces. Subsequently, the energy is further propagated between surfaces until reaching the receiver, and it can be considered as intensity-based boundary-element methods (BEMs). One approach, so-called acoustic radiosity, only accepts ideally diffuse reflections, whereas the path-based image-source technique only incorporates ideally specular reflections. Kuttruff (1971) presented the basic theory for the acoustic radiosity method. The technique is able to simulate the sound propagation in nonrectangular rooms with ideally diffuse surfaces. This is a multipass technique in which much of the simulation can be computed independently from the receiver. Only the last pass considers the sound energy traveling to the receiver, and so it is especially attractive in interactive simulations where the actual interaction needs to be instantaneously determined based on precalculations of all previous passes. The technique is extremely attractive to real-time auralization.
The downside to this technique, however, is the degraded accuracy of an exact reflection path (Savioja and
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