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The Room Acoustic Rendering Equation introduced in [1] formalizes a variety of room acoustics modeling algorithms. One key concept in the equation is the Acoustic Bidirectional Reflectance Distribution Function (A-BRDF) which is the term that models sound reflections. In this paper, we present a method to compute analytically the A-BRDF in cases with diffuse reflections parametrized by random variables. As an example, analytical A-BRDFs are obtained for the Vector Based Scattering Model, and are validated against numerical Monte Carlo experiments. The analytical computation of A-BRDFs can be added to a standard acoustic ray tracing engine to obtain valuable data from each ray collision thus reducing significantly the computational cost of generating impulse responses.

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 [

The most used techniques in computational room acoustics are the so-called geometrical methods which are based on the geometrical theory of acoustics [

Recently, Siltanen et al. introduce the Room Acoustic Rendering Equation [

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 [

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 [

where

where

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

where

The A-BRDF can be interpreted as the probability that sound energy incident from a 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

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

governed by a probability density

To compute the A-BRDF in such cases, we will use Equation (4), which relates the outgoing direction and the random vector

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,

One last step is needed to obtain the A-BRDF. Note that

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

where

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) [

where

In this case, the random vector

In this section, we focus on a slight variation of the VBS model whereby the random vector

To obtain the A-BRDF we need to use Equation (7), which in turn depends on the relation between probability densities in Equation (6). The VBS is an example where the random vector has unit norm, and therefore requires Equation (8),

where

To complete the computation in Equation (6), we need the Jacobian of the transformation (10), which turns out to be constant,

To continue, we need to invert Equation (10) and express the modulus

where

The last step to obtain the A-BRDF consists on inserting Equation (15) in Equation (7) and integrating over the radial coordinate. From (15), it is clear that, in this case, the A-BRDF depends on the incident and outgoing directions only through the combination

This integral is straightforward using a general formula for integrals containing Dirac’s delta functions:

where the sum is over all roots

Each of these roots contributes to the integral in Equation (16) only when they are real and positive, the latter restriction due to the fact that the integral range is

The condition that both solutions be real is

which basically states that unless diffusion is high enough, there are outgoing angles that are unattainable for some incident directions. Indeed, it can be seen that Equation (20) is always satisfied if

Similarly, only

whereas both

The results for

The results for

In this section we will validate Equations (21) and (22) by showing that Monte Carlo experiments reproduce the same results in the limit of large number of rays. To avoid subtleties with the divergences explained above, we will compare probability distributions rather than probability densities; the former contains the same information as the latter, but it is necessarily absent of divergences.

The probability of having an outgoing ray contained in a finite solid angle

Given that, as described above,

specular direction less than

Our analytical prediction for

where

whereas for

We will now show that Equations (26) and (27) provide same result as Monte Carlo experiments in the limit of large number of rays. In order to do so, consider a single plane characterized by a diffusion coefficient d. To simplify the experiment, let us set the direction of the incident sound ray to be orthogonal to the plane, which leads to a specular vector,

The usefulness of the analytic results presented here is more apparent when the convergence of the Monte Carlo results is considered.

Acoustic ray tracing enginges find propagation paths between a source and receiver by generating rays emanating from the source position and following them through the environment until a set of rays reach the receiver [

A large number of rays is needed in order to obtain accurated results. For instance, AURA module included in EASE acoustic simulator software generates around 10^{5} rays for medium size rooms [

The resulting ray tracing algorithm incorporates an extra step every time a ray collides with the environment. In addition to compute the outgoing ray direction, it makes use of the analytical solution for the A-BRDF to compute the contribution of the collision to the final Impulse Response.

^{4} rays were needed in our plots to match the analytical solution). Accordingly, the addition of the anaylical solution to the ray tracing algorithm notably reduces the amount of rays needed to obtain the same accuracy in the resulting Impulse Responses.

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 ^{3} - 10^{4} 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