## 1 Introduction  
Auditory sensing is essential for human beings to perceive the three-dimensional (3D) environment. The human hearing system can localize sound sources by using of the spatialization cues of the sources, for which head-related transfer functions (HRTFs) and the corresponding impulse responses (head-related impulse responses, HRIRs) play a vital role. HRTF is the transfer function of a linear system describing the filtering effect of the pinna, head and torso of a listener as a sound propagates from the source to the ear drum in free space [1]. Consequently, HRTFs are highly individualized, as the aforementioned anthropometric features are listener-dependent. HRTFs are defined as the ratio of the Fourier transform of the signal at the listener's eardrum to that at the center of the listener's head with the listener absent [1]. Obtaining accurate binaural HRTFs is of great significance in generating 3D sound. However, experimental measurement for HRTFs is difficult and time-consuming, which can not sufficiently support the practical application. Therefore, fast and efficiently individualizing HRTFs have drawn more and more attention from researchers. Theoretically, the estimation of HRTFs is to decompose the parts that reflect anthropometric features from sound waves. It is the solution to a wave equation under a certain boundary condition. The most straightforward way to estimate the HRTF is to solve Rayleigh scattering of rigid sphere on plane wave, in which head and ears are simplified as a rigid sphere and two symmetrical points on the surface of the horizontal diameter respectively. However, the simplified head model fails to catch the diversity of individuals. [PERSON] et al. [2] used multipole reexpansion and boundary element methods to compute the HRTF of the models in both the frequency domain and the time domain. [PERSON] [3] used boundary element method to calculate a portion the HRTF of an individual based on precise geometrical data. The calculation is time-consuming; special devices are needed to obtain the grid sampling data of human body surface. [PERSON] et al. [4] implemented the numerical simulation of acoustic propagation with Adaptive Rectangular Decomposition, which saved about 20 min when calculating HRTFs on a computer with eight cores and 3.4 GHz. Unfortunately, this is still time-consuming and difficult to implement in auralization [5]. Given the limitations of the above methods, many researchers tried to select a pair of HRTFs from an HRTF database as the individualized HRTFs of a new subject. According to this idea, the selection criterion is the similarity of anthropometric features. The more similar the anthropometric features, the closer the filtering effect on sound waves, and HRTFs can be shared among subjects with similar anthropometric features [6]. Based on the similarity and relativity of anthropometric structures, [PERSON] et al. [7] proposed a hybrid HRTF individualization algorithm, which has combined the method of principal component analysis (PCA), multiple linear regression (MLR) and database matching. [PERSON] and [PERSON] [8] proposed a method of database matching by using sparsely measured HRTF datasets. It minimized data collection durations by applying a selective measurement procedure without weakening spatialization accuracy, which provides users the best-fitting densely measured HRTFs from an existing repository. Even though such methods can obtain individualized HRTFs fast, there are always differences in the same anthropometric features among different subjects. Thus, HRTFs are also different. Such methods still need improvement. Considering the filtering effect of anthropometric features on sound waves, many researchers tried to construct a functional relationship between anthropometric features and HRTFs with the help of signal processing. By reducing the dimensions of HRTFs to five basic vectors, [PERSON] et al. [9] established a mapping system model from anthropometric features to basic vectors to obtain individualized HRTFs.