Head-related transfer functions (HRTFs) are essential for virtual acoustic realities because they contain all cues for localizing sound sources in three-dimensional space. Acoustic measurements are one way to obtain high-quality HRTFs. To reduce measurement time, cost, and complexity of measurement systems, a promising approach is to capture only a few HRTFs on a sparse sampling grid and then upsample them to a dense HRTF set by interpolation. However, HRTF interpolation is challenging because small changes in source position can result in significant changes in the HRTF phase and magnitude response. Previous studies have greatly improved the interpolation by time-aligning the HRTFs in pre-processing, but magnitude interpolation errors remain a problem, especially in contralateral regions. Building on time-aligned interpolation, we propose a post-interpolation magnitude correction derived from a frequency-smoothed HRTF representation. Our technical evaluation based on 96 individual simulated HRTF sets shows that the magnitude correction reduces subject-averaged magnitude errors in the higher frequency range by up to 1.5 dB when averaged over all directions and by up to 4 dB in the contralateral region. As a result, interaural level differences in the upsampled HRTFs are also improved. The accompanying perceptual evaluation shows that the magnitude correction significantly reduces perceived coloration and results in a more stable and accurate perceived source position. Additional technical evaluations show that the proposed method outperforms current machine learning based algorithms, can be used with measured HRTFs, and is superior to using a dummy head HRTF set even when only six source positions are used for upsampling.

We present a method for spatial upsampling of signals captured with spherical microphone arrays with a limited number of microphones. The upsampling is performed by adding virtual microphone signals using interpolation between the measured array signals. For interpolation, the differences in amplitude and time of neighbored signals are weighted separately. For this, the time differences are determined with a broadband cross-correlation. After the upsampling, the array signals can be represented as spherical harmonics (SH) coefficients at significantly higher orders. This work introduces the proposed upsampling procedure and presents a technical evaluation based on simulated and measured sound fields. Thereby, we focus on the binaural decoding of the upsampled SH signals. For perceptual evaluation, we present the results of a listening experiment evaluating the binaural reproduction of the upsampled signals compared to a baseline Ambisonics decoding and the state-of-the-art decoding with Magnitude Least-Squares optimization. Both the technical and perceptual evaluations consider only array impulse responses, but the algorithm could generally be applied to array streams. The evaluation shows that the proposed method significantly improves the binaural reproduction of low-order spherical microphone array captures compared to established methods.

In the paper “Directional Equalization of Sparse Head-Related Transfer Function Sets for Spatial Upsampling”, published in IEEE/ACM Transactions on Audio, Speech, andLanguage Processing (Volume: 27, Issue: 6, June 2019) [1],Eq. (2) and Eq. (6) are incorrect. In this Comment Correspondence we provide a correction of these Errors.

Acquiring decent full-spherical sets of head-related transfer functions (HRTFs) based on a small number of measurements is highly desirable. For spatial upsampling, HRTF interpolation in the spatially continuous spherical harmonics (SH) domain is a common approach. However, the number of measured HRTFs limits the assessable SH order, resulting in order-limited HRTFs when transformed to the SH domain. Thus, the SH representation of sparse HRTF sets shows restricted spatial resolution and suffers from order-limitation errors. We present a method that reduces these errors by a directional equalization prior to the SH transform. This is done by a spectral division of each HRTF with a corresponding directional rigid sphere transfer function. The processing removes direction-dependent temporal and spectral components and, therefore, significantly reduces the spatial complexity of the HRTF set, allowing for an enhanced interpolation of HRTFs at reduced SH orders. Spatial upsampling is achieved by an inverse SH transform on an arbitrary dense sampling grid. A subsequent de-equalization by a spectral multiplication with the rigid sphere transfer function recovers the energy in higher spatial orders that was not inherent in the sparse HRTF set. For evaluation, HRTFs were calculated for various limited orders from sparse datasets and compared to a reference. The results show that the proposed method clearly outperforms common SH interpolation of HRTF spectra regarding the overall spectral and temporal structure as well as modeled localization performance.