Computers have been used for over thirty years to model room acoustics. Nowadays, computational acoustics modeling has become a common practice in many different disciplines: the acoustic design of buildings such as auditoria or concert halls [2] [3] , outdoor acoustics [4] , audio post-production in Digital Cinema [5] or audio for video-games and other interactive applications [6] -[8] .

The most used techniques in computational room acoustics are the so-called geometrical methods which are based on the geometrical theory of acoustics [9] . The general approach of the geometrical methods is to find significant paths along which sound can travel from a source to a receiver through the environment. The most important geometrical methods that have been applied to acoustics are: image source method [10] [11] , ray tracing [12] , beam tracing [13] -[15] and radiosity [16] -[18] . Some of these (sometimes in combination) are the basis of many room acoustics commercial softwares such as EASE [19] , Odeon [20] , Catt-Acoustics [21] or Ramsete [22] .

Recently, Siltanen et al. introduce the Room Acoustic Rendering Equation [1] which is a model for acoustic energy propagation. This approach, borrowed from computer graphics, integrates several geometrical methods within the same theoretical framework. One key concept in the equation is the Acoustic Bidirectional Reflectance Distribution Function [23] (A-BRDF) which is the term that models sound reflections.

In the present paper, we develop a method to compute the analytical solution for the A-BRDF in cases where sound reflections are diffuse, and diffusion is parametrized by one (or more) random variables. The method makes use of various properties of continuous probability functions, and exploits the relation between two and three dimensional probability densities.

The method is applied to the Vector Based Scattering Model [24] (VBS), which is one of the existing ways to efficiently include diffusion in a ray tracing algorithm by means of random vectors. Analytical results are obtained and shown to coincide with the large number of rays limit of the corresponding Monte Carlo simulation.

The paper is organized as follows: Section 2 briefly reviews the Room Acoustic Rendering Equation and the application of BRDFs to acoustics. Section 3 introduces the general methodology to compute analytically the A-BRDF. Section 4 presents the analytical computation of the A-BRDF for the Vector Based Scattering Model, and the comparison with the corresponding Monte Carlo results. Finally, future work and conclusions are discussed in Section 5.

Let us briefly review the Room Acoustic Rendering Equation (RARE) introduced in [1] , which models the propagation of acoustic energy through the environment both in diffusive and general non-diffusive conditions. Following the notation in [1] , the RARE reads:

where is the set of all surface points in the enclosure, is the outgoing time-dependent radiance from point in the direction, is the radiance emitted by the surface from in the direction, and is the reflection kernel where x is the location of the previous reflection. The reflection kernel is a product of three terms:

where is the visibility term, is the geometry term, is the A-BRDF, and

stands for the direction of the rays. Henceforth, directions will also be specified either using unit-norm vectors or using a specific coordinate system on the sphere, , the zenital and azimuthal angles, respectively, as shown in Figure 1.

All the physics of the problem is contained in the A-BRDF, which can be expressed as the following ratio (see Figure 1),

where is the radiance at a point, exiting along the outgoing direction; and is the irradiance at the same point incident along the direction. Again, x represents the location of the previous reflection of the sound ray.

The A-BRDF can be interpreted as the probability that sound energy incident from a direction is reflected in the direction.

As such, the A-BRDF must also be normalized in order to avoid an artificial increase of the acoustic energy in the system. For the sake of simplicity of notation, in the rest of this paper we will not consider explicit dependence of the A-BRDF on the reflection position, and hence write.

In this section we present a method to compute analytically the A-BRDF in cases where reflections suffer diffusion parameterized by a random variable, i.e. in cases where the vector along the outgoing direction, , is determined using a random vector, ,

governed by a probability density. The subindex denotes that is a 3-dimensional probability density, i.e. yields the differential probability that the vector lies in the interval. We have chosen a random vector in rather than a single random variable, to cover more general cases, such as the VBS model discussed in the next section. We have also included dependence on the incident direction of the sound ray, represented by the unitary vector with direction 1 (see Figure 2).

To compute the A-BRDF in such cases, we will use Equation (4), which relates the outgoing direction and the random vector, to obtain a relation between the probability density of the outgoing direction, and. In other words, we will use Equation (4) to obtain the probability that the outgoing ray is in a small neighborhood, which is precisely the information provided by.

The relation between probability densities can be obtained recalling the way differential forms transform under a change of coordinates,

which yields the following relation

Here, is the Jacobian of the transformation defined by Equation (4).

One last step is needed to obtain the A-BRDF. Note that contains information about the probability that the outgoing vector has modulus between, besides the probability that it has direction between. Thus, to obtain, all that is left to do is an integration over all possible moduli:

Note that the need for this last integration arises from the fact that relations of the form (4) are often written in a manner where the outgoing vector is not guaranteed to have unit norm, as will be shown in the next section.

The general method presented here can also apply to cases where the random vector is unitary, which is the case of VBS. In such cases, the probability density of the random unit vector is actually two-dimen- sional, and it can be seen as the restriction to the unit sphere of the three-dimensional probability density,

where. This relation guarantees that probabilities can be indistinctly computed integrating or over two or three-dimensional intervals, respectively,

Although several ray tracing simulations propagate rays using only specular reflections, one of the existing choices in room acoustic simulations is to include a diffusion model to determine the direction of the outgoing rays. The Vector Based Scattering (VBS) [24] is a method for room acoustics ray-tracing computations that makes use of a specific model to parameterize diffusion of sound waves by obstacles. Within VBS, the vector along the outgoing ray, , is calculated via a linear combination of two vectors

where is a unit vector along the specular direction, is a random unit vector, and is the diffusion coefficient of the material (see Figure 3). Clearly, is entirely determined by the incident vector, and thus, Equation (10) is an explicit example of the general Equation (4).

In this case, the random vector is uniformly distributed along the hemisphere whose equator is the plane of collision, and which contains the incoming and outgoing rays. The problem of determining the outgoing direction can be thought of that of adding a random vector of length d to the outgoing unit vector, after rescaling by (see Figure 3).

A method to derive analytical solutions for the Acoustic Bidirectional Reflectance Distribution Function (A-BRDF) in cases of diffuse reflections parametrized by random variables has been presented and discussed. The method works for generic relations between outgoing, incoming and random vectors of the form (4), and makes use of various properties of continuous probability functions, exploiting the relation between two and three dimensional probability densities.

The method has been applied to the well-known Vector Based Scattering Model, for which exact analytical A-BRDF has been obtained, Equations (21) and (22). The results provide, by means of only one analytical formula evaluation, the same results as the corresponding Monte Carlo simulation in the limit of large number of rays.

The computation of analytical solutions for the A-BRDF can be added to a standard acoustic ray tracing engine by introducing an extra step in the algorithm to compute the contribution of every ray collision with the environment to the final Impulse Response. That is, instead of only using the information coming from the rays that reach the listener position to compute the Impulse Response, valuable data can be obtained from each collision that contributes to the computation of the Impulse Response thus reducing significantly the computational cost, as shown in Figure 7. The use of this methodology can imply a reduction of about 10³ - 10⁴ rays in any of the existing comercial acoustic ray tracing engines. Further work will focus on the application of the method discussed here to other ray tracing diffusion models for acoustics and on the computation of real time Impulse Responses for simple environments through the method introduced in this paper.

The authors wish to thank Jordi Arqués, Daniel Arteaga, Pau Arumí, Giulio Cengarle, David Garcia, Natanael Olaiz, Ferran Orriols and Carlos Spa for help and discussions.

JaumeDurany,ToniMateos,AdanGarriga, (2015) Analytical Computation of Acoustic Bidirectional Reflectance Distribution Functions. Open Journal of Acoustics,05,207-217. doi: 10.4236/oja.2015.54016