Utilizing spherical harmonic (SH) domain has been established as the default method of obtaining continuity over space in head-related transfer functions (HRTFs). This paper concerns different variants of extending this solution by replacing SHs with four-dimensional (4D) continuous functional models in which frequency is imagined as another physical dimension. Recently developed hyperspherical harmonic (HSH) representation is compared with models defined in spherindrical coordinate system by merging SHs with one-dimensional basis functions. The efficiency of both approaches is evaluated based on the reproduction errors for individual HRTFs from HUTUBS database, including detailed analysis of its dependency on chosen orders of approximation in frequency and space. Employing continuous functional models defined in 4D coordinate systems allows HRTF magnitude spectra to be expressed as a small set of coefficients which can be decoded back into values at any direction and frequency. The best performance was noted for HSHs and SHs merged with reverse Fourier-Bessel series, with the former featuring better compression abilities, achieving slightly higher accuracy for low number of coefficients. The presented models can serve multiple purposes, such as interpolation, compression or parametrization for machine learning applications, and can be applied not only to HRTFs but also to other types of directivity functions, e.g. sound source directivity.

翻译：利用球谐函数（SH）域已成为实现头相关传递函数（HRTF）空间连续性的默认方法。本文探讨了该方案的多种扩展变体，通过将球谐函数替换为四维（4D）连续函数模型，其中频率被视为另一物理维度。将近期发展的超球谐函数（HSH）表示与通过合并球谐函数及一维基函数在柱球坐标系中定义的模型进行了比较。基于HUTUBS数据库中个体HRTF的重构误差评估了两种方法的效率，并详细分析了其随频率和空间近似阶数选择的依赖关系。采用四维坐标系中定义的连续函数模型可使HRTF幅度谱表示为少量系数，这些系数可解码为任意方向和频率下的数值。最佳性能体现在超球谐函数与球谐函数融合逆傅里叶-贝塞尔级数的组合，前者具有更优的压缩能力，在低系数数量下精度略高。本文提出的模型可服务于插值、压缩或机器学习应用中的参数化等多重目标，且不仅适用于HRTF，还可推广至其他类型的指向性函数（如声源指向性）。