# A Review on Head-Related Transfer Function Generation for Spatial Audio
## 4 Physical Modelling
### Other Approaches  
BEM-based approaches to solving the Helmholtz equation are especially attractive because only the values of the pressure field on the boundary \(\Sigma\) of the domain \(\Omega\) are considered. This avoids the discretisation of large volumes, which results in cubic growth in the number of elements. Another appealing feature of BEM is that the Sommerfeld radiation condition (7) is already considered in the integral formulation (8) of the problem. Nevertheless, approaches with FEM, finite differences in the time domain (FDTD), and adaptive rectangular decomposition (ARD) are also considered. In this section, we mention some of these alternative approaches. We also mention some other approaches, for instance, ray tracing, that are only indirectly connected to the wave equation. As stated above, the integral formulation in the frequency domain of the scattering problem in Equation (14) incorporates the Sommerfeld radiation condition (7), making an exact formulation of the problem for an unbounded domain possible. On the contrary, approaches in the time domain or based on FEM require a truncation of the unbounded domain with an external surface equipped with boundary conditions suited to absorb outgoing waves. Perfectly matched layers (PMLs) [124], where a fictitious absorbing material is placed at the external boundary of the computational domain, provide such conditions. PMLs are applied in [125] to absorb outgoing waves at the truncated computational domain of the unbounded medium. The proposed system is based on FDTD to simulate the scattering of a sound wave on a simplified spherical model of a human head and a more realistic model up to a frequency of 7 kHz. The model is validated with the analytical solution for a spherical surface. It is also applied to compute an HRTF in the presence of a wall. In [126], an FDTD approach is used to compute HRTFs from head shapes to investigate the mechanism generating peaks and notches. This study shows that the pinna is responsible for generating such spectral structures. The method used in computer simulations is also explained in [127]. In [128], the effect of varying the complexity of the head and torso model is investigated by means of FDTD simulations. The objective is to assess the accuracy of the computed HRTFs with respect to the complexity of the geometry. In this paper, the simulations with a single sphere, a double sphere (the so-called snowman model), and a simplified model of a human head and torso are compared. The study shows that, for the single and double spheres, the simulations are accurate up to relatively high frequencies, whereas for the more complex model of a human head and torso, the same accuracy can be reached only at lower frequencies. In [129], the quality and reliability of FDTD-based approaches in HRTF generation are investigated. Limitations in boundary modelling are also faced in this work. The study is conducted both in the near and far fields. An efficient algorithm for the solution of the wave equation in the time domain is adaptive rectangular decomposition (ARD) [130]. This algorithm is based on the well-known analytical solution of the wave equation in homogeneous media and in a rectangular decomposition of the simulation domain by means of large rectangular regions of empty space. The methods can be efficiently implemented on GPUs, achieving considerable performance. Furthermore, these approaches reduce the errors accumulated during computation. In [131], the authors propose an efficient computation framework for HRTF generation making use of ARD. Accurate 3D meshes of the head and torso are generated with the use of consumer-grade digital cameras. The meshes are subsequently used in an efficient computational pipeline for the numerical generation of HRTFs. In order to overcome the high computational cost associated with large volumes, they limit the simulation domain to a small cuboid enclosing the head and torso. In this way, they can gain an estimate of the pressure field on the scattering surface.