# A Review on Head-Related Transfer Function Generation for Spatial Audio
## 6 Interpolation
### Functional Models for HRTF Interpolation
The first work based on SHD was presented by [PERSON] et al. in 1998 [197]. In the following years, several approaches using SHD were applied. The performance of this method varies with the order of the SHD and the employed measurement grid. Noting \(N\) as the SHD order, at least \(N^{2}\) measured HRTFs are necessary to guarantee an accurate representation, especially at higher frequencies [200]. Moreover, measurement point distribution, i.e., the grid, is also important for reconstruction accuracy. Thus, this method requires the entire dataset, not only the nearest neighbour positions. In this way, interpolation performance can be improved at the expense of the computational complexity, especially for dense datasets. In 2008, [PERSON] et al. explored the challenge of missing data at low elevations by proposing an iterative algorithm and using a fourth-order decomposition [199]. However, results showed that an order of four does not produce sufficiently accurate responses. In 2009, [PERSON] et al. tested SHD interpolation performance with different types of grids, proving that the best results are obtained with closed grids, i.e., the measurement points cover the entire sphere surface [200]. In 2012, [PERSON] et al. applied spherical harmonics decomposition to interpolate ITD values of HRTFs measured with a Kemar mannequin; then, they used PCA to derive a full-sphere model [30; 201]. In 2018, [PERSON] and [PERSON] investigated different pre-processing algorithms for time alignment applied before SHD [202]. The effectiveness of the pre-processing allows for a reduction in the order of the SHD and the number of HRTF measurements. Results show that the error is below the just noticeable difference (see Section 2.2) for orders \(\geq 4\)[202]. In 2020, [PERSON] et al. compared different HRTF interpolation methods, including nearest-neighbour techniques and SHD. The results demonstrate that the performance of the SHD increases with the spatial resolution of the measurements [203]. Thus, the nearest-neighbour interpolation performs better with a low spatial resolution, while the accuracy is comparable to SHD when a higher spatial resolution is considered [203]. In 2021, [PERSON] obtained similar conclusions by comparing the SHD with the binaural interpolation of the four closest and evaluating the reconstruction error and the computational complexity [204]. The study shows that the bilinear interpolation outperforms the SHD when sparse HRTF datasets are involved, while SHD exhibits better performance with dense measurement grids. This emphasises the importance of selecting appropriate interpolation methods depending on the specific application, such as real-time audio rendering or offline audio production. [PERSON] also discusses the computational trade-offs associated with these methods, highlighting the balance between interpolation accuracy and processing speed [204]. In the same year, [PERSON] et al. proved that SH representation is affected by spatial aliasing and truncation errors [205]. These problems are reduced by introducing a time-alignment procedure prior to interpolation [205]. In 2023, the same authors presented a new approach in which a magnitude correction procedure is applied after a time-aligned interpolation method, thus improving performance [29].