# A Review on Head-Related Transfer Function Generation for Spatial Audio
## 8 Conclusions and Future Directions  
Due to the individual variability in human head and ear shapes, generating and optimising HRTFs for accurate and personalised spatial audio experiences remains a significant research challenge. In this paper, a review about the generation of head-related transfer functions for spatial audio is presented. Starting from the mathematical definitions and settings used for HRTFs, a deep analysis of various methods for their generation is reported. HRTFs can be first obtained by direct measurements conducted using standard HAT simulators or human subjects equipped with in-ear microphones. These measurements can be conducted using several techniques characterised by different excitation signals. However, the measurement procedure is long and time consuming and requires professional equipment. As an alternative to direct measurements, physical modelling can be employed to generate HRTFs by numerically solving the wave equation in the time domain or the Helmholtz equation in the frequency domain. In this context, the boundary element method (BEM) is one of the most widely used techniques. An elementary explanation of the mathematical model based on BEM is provided. Additionally, other approaches, such as the finite element method (FEM), finite-difference time-domain (FDTD) method, and adaptive rectangular decomposition (ARD), are also discussed as alternative solutions. Predictive models have a high computational cost, and they are not able to capture the full anatomical diversity of listeners, reducing precision for individuals with less typical features. Personalised HRTFs can significantly improve spatial binaural performance. They can be obtained by the direct measurement of human subjects using in-ear microphones or by applying anthropometric and morphological approaches. Several anthropometric parameters can be defined and measured to obtain personalised HRTFs through anthropometry matching, adaptation methods, or trained regression models. In some cases, when high-quality HRTF datasets are available and the computational resources are limited, the HRTF interpolation can be applied for enhancing the spatial audio experience. Starting from a limited set of measured HRTFs, interpolation enables spatial upsampling, allowing the generation of HRTFs for positions that were not included in the original measurement points. The interpolated HRTF can be calculated as a linear combination of the nearest neighbors, as a result of functional models that can be derived from several approaches, such as SHD, or using neural networks trained on sparse measurements. Finally, the integration of these HRTF generation techniques is examined, with a focus on their incorporation into consumer technologies and their evaluation in real-world applications. In particular, evaluations of HRTF technologies confirm their benefits in real-world applications, especially in gaming and virtual reality. Personalised HRTFs improve spatial awareness and immersion, while efficient interpolation methods help maintain audio quality with low latency. Techniques for matching headphone responses make HRTFs more accessible for consumer devices. The proposed review can be a valuable reference for future research in this field since it investigates the innovations and the limitations of the current techniques. Despite the significant advancements introduced in the field of HRTF generation, several challenges remain. Achieving real-time performance without compromising accuracy requires further innovation in computational methods. Additionally, the integration of machine learning techniques, while promising, poses challenges related to generalisation across different listener anatomies and environments. Future research should focus on developing scalable methods incorporating low-latency computational models that can operate in real-time. Adaptive machine learning algorithms can be investigated for developing an immersive audio system capable of dynamically adjusting themselves to anatomical and environmental changes.