# A Review on Head-Related Transfer Function Generation for Spatial Audio
## 4 Physical Modelling
### Boundary Element Method  
In this section, the boundary element method for the solution of scattering problems in acoustics is introduced. The aim of this part is to give a general overview of the method as a useful starting point for a more in-depth investigation. Furthermore, some key terms and concepts are introduced. For more details, we refer to [92; 93; 94; 95], and the references therein. The appendix of the book [96] contains also a good starting point for understanding the theory of BEM. The first step in BEM computation for an HRTF is to reformulate the problem described in Equations (8)-(10) as a boundary problem on \(\Sigma\) using [PERSON]'s identity. The result is the Kirchhoff-Helmholtz integral equation for the scattered wave \(u_{\text{sc}}(\mathbf{r})\):  
\[c(\mathbf{r})u_{\text{sc}}(\mathbf{r})=L[q]-M[u_{\text{sc}}]\qquad\quad q= \frac{\partial u_{\text{sc}}}{\partial\mathbf{n}}, \tag{14}\]  
where the functional operators \(L\) and \(M\) are defined as follows:  
\[L[q]=\int_{\Sigma}q(\mathbf{r}^{\prime})G(|\mathbf{r}^{\prime}-\mathbf{r}|)d \Sigma(\mathbf{r}^{\prime}),\quad\quad M[u]=\int_{\Sigma}u(\mathbf{r}^{\prime })\frac{\partial G(|\mathbf{r}^{\prime}-\mathbf{r}|)}{\partial\mathbf{n}}d \Sigma(\mathbf{r}^{\prime}), \tag{15}\]  
where \(\mathbf{r}\) is an arbitrary observation point in space, \(\frac{\partial}{\partial\mathbf{n}}\) is the normal derivative to the surface \(\Sigma\), and \(G(r)\) is the Green function (12) for the Helmholtz equation. The normal to \(\Sigma\) is directed toward the external of \(\Omega\) (the internal of the head) as established in Section 2.3. The factor \(c(\mathbf{r})\) on the left-hand side of Equation (14) is one for \(\mathbf{r}\) internal to \(\Omega\), \(\frac{1}{2}\) for \(\mathbf{r}\) on the boundary \(\Sigma\) and zero for \(\mathbf{r}\) internal to the closed surface bounded by \(\Sigma\). The integral Equation (14) gives a representation of the scattered pressure field in terms of its values on the boundary \(\Sigma\). The numerical computation of an HRTF for different source positions requires repeated solutions of Equation (14). Hence, determining HRTFs in a sufficiently dense set of directions may be an expensive computational task. Furthermore, only pressure field values at the receiver are retained. This computation can be made much more efficient using the reciprocal principle of acoustic: the pressure field at the receiver is not affected if the position \(S\) of the source and the position \(R\) of the receiver are swapped [97]. When the reciprocity principle is applied, the source is kept fixed at the entrance of the auditory channel and a single simulation enables obtaining the pressure fields at different positions of the receiver. The effectiveness of this approach has been experimentally validated using a rigid sphere and a Kemar dummy-head with a source located at the ear entrance and arrays of microphones in several positions in the laboratory [98]. However, this reversed approach requires that the source is placed on the boundary \(\Sigma\), and this generates conceptual and computational difficulties that must be addressed. In [99], the authors show how the singular part of the scattered field can be separated using an image source internal to the region bounded by \(\Sigma\). Though computationally more efficient, reciprocity-based approaches produce a pressure field on the surface \(\Sigma\) that does not correspond to the propagation from a source placed in the true position \(S\). Furthermore, the method based on the reciprocity principle is well suited for applications with point sources but cannot straightforwardly be applied with spatially complex sources. Hence, reciprocity-based approaches are not useful if the distribution of the pressure field on the head and torso is the objective of the simulation and in cases where the acoustic solicitation is complex.