# Ear Centering for Accurate Synthesis of Near-Field Head-Related Transfer Functions +
## 1 Introduction  
The head-related transfer functions (HRTF) are a major tool in spatial sound technology for personal use [1; 2; 3; 4]. They are linear filters describing the transmission of sound from a point in space to the eardrums of a listener [5]. The HRTF for a point source is defined as the ratio between the sound pressure at the ear position and the free-field sound pressure at a reference position. The reference is typically placed at the center of the listener's head. The HRTFs are commonly obtained for a sparse set of points at a single distance measured from the center of the head. Such distance is greater than 1 m and lies in a region of space referred to as the far field. Besides far-field HRTF datasets, there is a growing interest in accurately synthesizing HRTFs for arbitrary points in the near field, that is, in the region of space within 1 m from the center of the head [6; 7; 8]. Interests include the development of near-field auditory displays [9] and the consideration of distance in auditory attention experiments [10]. A promising synthesis approach extrapolates near-field HRTFs starting from far-field ones using spherical Fourier transforms (SFT) and distance-varying filters (DVF) [11; 12; 13]. When using the SFT to represent spherical HRTF datasets, the default spherical symmetry of the SFT is specified with respect to the head center. The default spherical symmetry, however, does not allow for distinguishing between the reference (the head center) andthe measurement positions (the ears). Such mismatch produces a demand of a high number of basis functions in the SFT representation and, therefore, affects the synthesis accuracy. Ear centering is the name adopted in this paper to address the mismatch between the default SFT center and the ear position in the framework of a more general technique called acoustic centering [14, 15, 16, 17, 18]. Ear centering is performed by means of acoustic operators that translate the center of symmetry of the SFT from the head center to the ears [19, 20, 21, 22, 23]. Table 1 summarizes the most relevant research interpreted in terms of translation operators for HRTF synthesis. The first column from the left shows the references. The second column indicates whether the proposed translation operation incorporates distance information or not. The third column highlights the translation point used for the translation process. It is worth noting that only [24] used the ear position provided in the HRTF dataset as the translation point. The fourth column indicates the mathematical domain where the translation is applied, either the unit sphere (the natural domain) or the SFT domain (the transform domain). Finally, the fifth column classifies the translation models. [PERSON] et al. [19] used a ratio of Hankel functions to compensate for the difference in distance between each source and the ears, and then extrapolated the HRTF to a different distance from the head center. In our context, this operation is equivalent to a translation operator applied in the SFT domain. [PERSON] et al. [20] calculated the time difference between the arrivals from each source to the ears and the head center, and then applied it as a phase difference to the HRTF dataset before the SFT. This can be interpreted as a plane-wave translation operator applied on the unit sphere domain. [PERSON] et al. [21] applied a ratio of pressures using plane waves arriving to the head center and the ears. This, again, corresponds to a plane-wave translation operator because it is changing the reference point of the HRTF. [PERSON] et al. [22] applied a directional equalization to the HRTF using a rigid spherical scatterer, removing direction-dependent spectral components, and reducing the spatial complexity of the HRTF dataset. This whole process can be interpreted as a plane-wave translation operator, but rather than being applied in the free-field, it was applied in the presence of a rigid sphere, which emulated the most simple model of the human head. [PERSON] et al.