The room acoustic rendering equation is an integral equation that can be used to formalize the behavior of different geometrical acoustics modeling techniques. The basic formulation of this equation was presented by Kuttruff in 1971 for diffuse reflections [1], and it was later extended by Joyce to cover arbitrary reflection properties [2]. Nosal was the first to bring in time to the equation [3]. The current presentation is based on the formulation by Siltanen et al [4].
Most of the GA-based modeling techniques can be covered by this formulation as long as reflections can be assumed to be of local reaction type. The radiosity and acoustic radiance transfer methods fall best in this scope as both of those techniques assume a tessellated geometry that is similar to the surface elements used in this equation, but even the plain image-source method can be written using the same terms.
In the following the equation is built step-by-step starting from the form factors, and individual reflections. The final result is the equation that recursively covers the whole sound propagation from the source to the listener.
The form factor tells what is fraction of the solid angle covered by one surface when seen from the other one in absence of any occluder. The form factor between two small surface patches represented by the points $x$ and $x'$ can be written as
\begin{equation} F(x,x') = \frac{cos(\Theta) \cos(\Gamma)}{|x-x'|^2}. \label{eq:F} \end{equation}
where $\Theta$ and $\Gamma$ are the angles formed between a line connecting the points and the surfaces. A more complete formula considering the finite area of the surfaces would involve integrals over both surfaces. That version is practical if the surfaces are large, i.e., the evaluation is performed only on coarse points.
The actual reflection is fully covered by the BRDF that tells the fraction of energy received in one angle that is reflected into a given angle, and can be written as
\begin{equation} BRDF(\Theta, \Omega) = \frac{E_{in}(\Theta)}{E_{out}(\Omega)} \label{eq:BRDF} \end{equation}
where $E_{in}$ and $E_{out}$ are the incoming and outgoing energies at given angles $\Theta$ and $\Omega$. For an ideal diffuse reflection the $BRDF$ has a constant value determined solely by the absorption coefficient of the surface. An ideal specular reflection has a value of 1 for the pair of angles where $\Omega$ corresponds to the specular reflection direction of a ray with incident angle $\Theta$, and has a value of 0 for all other values of $\Omega$ for the given $\Theta$
To make a complete reflection operator we combine here the propagation from one patch to another and the actual reflection. This all can be represented with the reflection kernel, $R$, as follows:
\begin{equation} R(x, x', \Omega) = V(x, x') F(x, x') p(x, x') BRDF(\Theta, \Omega) \label{eq:R} \end{equation}
where $\Theta$ is the incident angle of sound coming from $x$ to $x'$, $V(x, x')$ is the visibility term that tells if the two points have a line-of-sight, $F(x, x')$ and BRDF(\Theta, \Omega) are the form factor and BRDF as introduced previously. The sound propagation term $p(x, x')$ models the delay and air absorption taking place while the sound propagates from $x$ to $x'$.
Here we present an energy-based formulation of the room acoustic rendering equation. It could be done for pressures as well, but as the BRDF formulation presented before is valid only for energies, the presentation here is for energies as well.
The given formulation computes recursively the outgoing time-dependent sound energy $l$ at a given surface point $x'$ in a direction $\Omega$:
\begin{equation} l(x', \Omega) = l_0(x', \Omega) + \int_G R(x, x', \Omega) l(x, \Gamma) dx \label{eq:RARE} \end{equation}
where $l_0$ is the energy emitted by the surface itself; the integral describes the reflected energy whereby $\Gamma$ is the outgoing angle from the surface point $x$ toward $x'$. Note that time is not explicitly written out in the equation above.
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