Head-related Transfer Function Interpolation Research Articles

Magnitude-Corrected and Time-Aligned Interpolation of Head-Related Transfer Functions

Magnitude-Corrected and Time-Aligned Interpolation of Head-Related Transfer Functions

Estimation of the Optimal Spherical Harmonics Order for the Interpolation of Head-Related Transfer Functions Sampled on Sparse Irregular Grids

Conventional individual head-related transfer function (HRTF) measurements are demanding in terms of measurement time and equipment. For more flexibility, free body movement (FBM) measurement systems provide an easy-to-use way to measure full-spherical HRTF datasets with less effort. However, having no fixed measurement installation implies that the HRTFs are not sampled on a predefined regular grid but rely on the individual movements of the subject. Furthermore, depending on the measurement effort, a rather small number of measurements can be expected, ranging, for example, from 50 to 150 sampling points. Spherical harmonics (SH) interpolation has been extensively studied recently as one method to obtain full-spherical datasets from such sparse measurements, but previous studies primarily focused on regular full-spherical sampling grids. For irregular grids, it remains unclear up to which spatial order meaningful SH coefficients can be calculated and how the resulting interpolation error compares to regular grids. This study investigates SH interpolation of selected irregular grids obtained from HRTF measurements with an FBM system. Intending to derive general constraints for SH interpolation of irregular grids, the study analyzes how the variation of the SH order affects the interpolation results. Moreover, the study demonstrates the importance of Tikhonov regularization for SH interpolation, which is popular for solving ill-posed numerical problems associated with such irregular grids. As a key result, the study shows that the optimal SH order that minimizes the interpolation error depends mainly on the grid and the regularization strength but is almost independent of the selected HRTF set. Based on these results, the study proposes to determine the optimal SH order by minimizing the interpolation error of a reference HRTF set sampled on the sparse and irregular FBM grid. Finally, the study verifies the proposed method for estimating the optimal SH order by comparing interpolation results of irregular and equivalent regular grids, showing that the differences are small when the SH interpolation is optimally parameterized.

Assessing HRTF preprocessing methods for Ambisonics rendering through perceptual models

Binaural rendering of Ambisonics signals is a common way to reproduce spatial audio content. Processing Ambisonics signals at low spatial orders is desirable in order to reduce complexity, although it may degrade the perceived quality, in part due to the mismatch that occurs when a low-order Ambisonics signal is paired with a spatially dense head-related transfer function (HRTF). In order to alleviate this issue, the HRTF may be preprocessed so its spatial order is reduced. Several preprocessing methods have been proposed, but they have not been thoroughly compared yet. In this study, nine HRTF preprocessing methods were used to render anechoic binaural signals from Ambisonics representations of orders 1 to 44, and these were compared through perceptual hearing models in terms of localisation performance, externalisation and speech reception. This assessment was supported by numerical analyses of HRTF interpolation errors, interaural differences, perceptually-relevant spectral differences, and loudness stability. Models predicted that the binaural renderings’ accuracy increased with spatial order, as expected. A notable effect of the preprocessing method was observed: whereas all methods performed similarly at the highest spatial orders, some were considerably better at lower orders. A newly proposed method, BiMagLS, displayed the best performance overall and is recommended for the rendering of bilateral Ambisonics signals. The results, which were in line with previous literature, indirectly validate the perceptual models’ ability to predict listeners’ responses in a consistent and explicable manner.

Interpolation of Head-Related Transfer Functions Using Manifold Learning

We propose a new head-related transfer function (HRTF) interpolation method using Isomap, a nonlinear dimensionality reduction technique. First, we construct a single manifold for all subjects across both azimuth and elevation angles through the construction of an intersubject graph (ISG) that includes important prior knowledge of the HRTFs such as correlations across individuals, directions, and ears. Then, for a new direction, we predict its corresponding low-dimensional HRTF by interpolating over same subject low-dimensional measured HRTFs. Finally, we use a local neighborhood mapping in the manifold to reconstruct the high-dimensional HRTF from measured HRTFs of all subjects. We show that a single manifold representation obtained through the ISG is a powerful way to allow measured HRTFs from different subjects to contribute for reconstructing the HRTFs for new directions. Moreover, our results suggest that a small number of spatial measurements capture most of acoustical properties of HRTFs. Finally, our approach outperforms other linear and nonlinear dimensionality reduction techniques such as principal component analysis, locally linear embedding, and Laplacian eigenmaps.

A spatial compression technique for head-related transfer function interpolation and complexity estimation.

A head-related transfer function (HRTF) model employing Legendre polynomials (LPs) is evaluated as an HRTF spatial complexity indicator and interpolation technique in the azimuth plane. LPs are a set of orthogonal functions derived on the sphere which can be used to compress an HRTF dataset by transforming it into a lower dimensional space. The LP compression technique was applied to various HRTF datasets, both real and synthetic, to determine how much different HRTFs can be compressed with respect to their structural complexity and their spatial resolution. The spatial complexity of different datasets was evaluated quantitatively by defining an HRTF spatial complexity index, which considers the rate of change in HRTF power spectrum with respect to spatial position. The results indicate that the compression realized by the LP technique is largely independent of the number of spatial samples in the HRTF dataset, while compressibility tracks the HRTF spatial complexity index so that more LP coefficients are needed to represent an HRTF dataset with a larger complexity index. The slope of the complexity index with respect to sub-sampling density can be used as a predictor for high interpolation error.

Head-related transfer function interpolation in azimuth, elevation, and distance.

Although distance-dependent head-related transfer function (HRTF) databases provide interesting possibilities, e.g., for rendering virtual sounds in the near-field, there is a lack of algorithms and tools to make use of them. Here, a framework is proposed for interpolating HRTF measurements in 3-D (i.e., azimuth, elevation, and distance) using tetrahedral interpolation with barycentric weights. For interpolation, a tetrahedral mesh is generated via Delaunay triangulation and searched via an adjacency walk, making the framework robust with respect to irregularly positioned HRTF measurements and computationally efficient. An objective evaluation of the proposed framework indicates good accordance between measured and interpolated near-field HRTFs.

Statistical analysis of head related transfer function data

The head related transfer function (HRTF) is a function that characterizes the response of a given individual to sound from a particular location in an egocentric coordinate system. The range dependence is often neglected, and the HRTF is approximated as a function of frequency and direction, H(θ,φ,ω). The HRTF displays considerable inter-personal variability, and a major open problem is the development of a generative model for the HRTF from anthropometry. Further, the sampling used in measuring HRTF data varies widely from database to database, and moreover often there are no measurements for elevations below the subject. This raises associated questions of optimal sampling, interpolation, hole-filling, and others. In this work, we model the HRTF via a non-parametric, data-driven, Gaussian Process Regression model. We develop efficient regression techniques to perform inference using this model on measured HRTF data. We then suggest methods for HRTF interpolation, HRTF extrapolation, feature extraction, sampling, and personalization. The methods are tested on the CIPIC database and results presented. [Partial NSF support is gratefully acknowledged.]

An improved method for head-related transfer function interpolation and grid matching

Today, there exists a growing number of HRTF datasets available with each set often proposing a unique variation on the spatial discretization measurement grid. These differing grids, typically determined by the mechanical system employed, result in datasets, which are not directly comparable or exploitable. To alleviate the limitation of incompatible grids and assist in the adaptation of measurements performed on one grid to another, facilitating the inter-exchange of HRTF sets, a fixed radius HRTF interpolation method is proposed. The approach is based on decomposition of the sound field using spherical harmonics, allowing for a global spatial recomposition. The frequency domain HRTF is separated into its complex components and interpolations are performed independently before reconstitution. Spherical harmonic truncation order is chosen to provide a system which is roughly square, improving matrix inversion with Tikhonov regularization. A high spatial density HRTF was used as a test case for evaluating the interpolation method. A series of measures are employed to quantitatively compare the quality of the interpolation as compared to traditional interpolation methods in both the time and frequency domains.

Efficient Binaural Rendering of Moving Sound Sources Using HRTF Interpolation

This paper presents efficient techniques for the simulation of continuously moving sound sources, to be presented over headphones, based on the interpolation of Head-Related Transfer Functions (HRTFs). Two novel approaches are presented, one based on interpolating impulse responses in time-domain, and another based on interpolating poles and zeros of low-order Infinite Impulse Response (IIR) approximations of measured HRTF filters. These techniques are apt both for real-time implementation and for multi-user applications where several independent binaural renderings are required. We discuss some of the difficulties of assessing the quality of the results from both objective and subjective points-of-view, and provide objective measurements of the errors caused by the interpolation schemes, as well as of the errors introduced by the use of low-order IIR approximations. A subjective evaluation of these techniques using non-individualized HRTFs suggests that they are able to preserve spatial impression of directions in binaural rendering of static and moving sound sources.

A Relevant Distance Criterion for Interpolation of Head-Related Transfer Functions

In binaural synthesis, in order to realize more precise and accurate spatial sound, it would be desirable to measure a large number of the head-related transfer functions (HRTFs) in various directions. To reduce the size of the HRTFs, interpolation is often employed, where the HRTF for any direction can be obtained by a limited number of the representative HRTFs. In this paper, it is determined which distortion measure for interpolation of the HRTFs in the horizontal plane is most suitable for predicting audible differences in sound location. Four kinds of HRTF sets, measured using three human heads and one mannequin (KEMAR), were prepared for this study. Using various objective distortion criteria, the differences between interpolated and measured HRTFs were computed. These were then related to the results from the listening tests through receiver operator characteristic (ROC) curves. The results of the present study indicated that for the HRTF sets measured from three human heads, the best predictor of performance was obtained using the distortion measurement computed from the mel-cepstral coefficients, whereas the distortion measurement associated with interaural time delay predicted audible differences in sound location reasonably well for the KEMAR HRTF set. A feasibility test was conducted to verify the usefulness of the selected distortion measurement.

Interpolation of head-related transfer functions using spherical fourier expansion

A new interpolation algorithm for Head-Related Transfer Function (HRTF) is proposed to realize 3D sound reproduction via headphones in arbitrary spatial direction. HRTFs are modeled as a weighted sum of spherical harmonics on a spherical surface. Truncated Singular Value Decomposition (SVD) is adopted to calculate the weights of the model. The truncation number is chosen according to Frobenius norm ratio and the partial condition number. Compared with other interpolated methods, our proposed approach not only is continuous but exploits global information of available directions. The HRTF from any desired direction can be obtained more accurately and robustly. Reconstructed and interpolated results demonstrate that our proposed algorithm acquired better performance.

Head-related transfer function interpolation through multivariate polynomial fitting of principal component weights

Head-related transfer function (HRTF) interpolation plays an important role for implementation of 3D sound system because it can not only reduce the number of measurements for HRTFs, but also reduce the data of HRTFs for seamless binaural synthesis. This paper addresses the problem of accurately realizing the interpolation of HRTF for synthesis of virtual auditory space, and proposes a HRTF interpolation method based on principal component analysis. Firstly, the HRTF is decomposed into principal components and corresponding principal component weights, where principal components are direction-independent and principal component weights are direction- dependent; then the directional variation of the principal component weight is multivariate polynomial fitted with a bivariate function of two spatial angulus (azimuth and elevation). Moreover, a sphere- partitioning optimization scheme is employed to improve the approximation precision. Experiment results demonstrate that HRTFs in the entire sphere surface can be interpolated by the proposed method with small distortion, and the proposed method performs better than conventional methods. Therefore the proposed method gives a promising way for HRTF interpolation.

Interpolation of Head-Related Transfer Functions based on the Common-Acoustical-Pole and Residue model

Interpolation of Head-Related Transfer Functions based on the Common-Acoustical-Pole and Residue model