Continuous Head-related Transfer Function Representation Based on Hyperspherical Harmonics
Abstract
Expressing head-related transfer functions (HRTFs) in the spherical harmonic (SH) domain has been thoroughly studied as a method of obtaining continuity over space. However, HRTFs are functions not only of direction but also of frequency. This paper presents an extension of the SH-based method, utilizing hyperspherical harmonics (HSHs) to obtain an HRTF representation that is continuous over both space and frequency. The application of the HSH approximation results in a relatively small set of coefficients which can be decoded into HRTF values at any direction and frequency. The paper discusses results obtained by applying the method to magnitude spectra extracted from exemplary HRTF measurements. The HRTF representations based on SHs and HSHs exhibit similar reproduction accuracy, with the latter one featuring continuity over both space and frequency and requiring much lower number of coefficients. The developed HSH-based continuous functional model can serve multiple purposes, such as interpolation, compression or parametrization for machine-learning applications.Keywords:
hyperspherical harmonics, HRTF, spherical harmonics, continuous functional model, directivityReferences
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24. Kulkarni A., Colburn H.S. (1998), Role of spectral detail in sound-source localization, Nature, 396(6713): 747–749, https://doi.org/10.1038/25526
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26. Kulkarni A., Isabelle S.K., Colburn H.S. (1999), Sensitivity of human subjects to head-related transfer-function phase spectra, The Journal of the Acoustical Society of America, 105(5): 2821–2840, https://doi.org/10.1121/1.426898
27. Langendijk E.H.A., Bronkhorst A.W. (2002), Contribution of spectral cues to human sound localization, The Journal of the Acoustical Society of America, 112(4): 1583–1596, https://doi.org/10.1121/1.1501901
28. Li J., Wu B., Yao D., Yan Y. (2021), A mixed-order modeling approach for head-related transfer function in the spherical harmonic domain, Applied Acoustics, 176: 107828, https://doi.org/10.1016/j.apacoust.2020.107828
29. Liu H., Fang Y., Huang Q. (2019), Efficient representation of head-related transfer functions with combination of spherical harmonics and spherical wavelets, IEEE Access, 7: 78214–78222, https://doi.org/10.1109/ACCESS.2019.2921388
30. Macpherson E.A., Middlebrooks J.C. (2002), Listener weighting of cues for lateral angle: The duplex theory of sound localization revisited, The Journal of the Acoustical Society of America, 111(5): 2219–2236, https://doi.org/10.1121/1.1471898
31. Majdak P. et al. (2013), Spatially oriented format for acoustics: A data exchange format representing head-related transfer functions, [in:] Proceedings of 134th AES Convention.
32. Nishino T., Kajita S., Takeda K., Itakura F. (1999), Interpolating head related transfer functions in the median plane, [in:] Proceedings of the 1999 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, pp. 167–170.
33. Pasha Hosseinbor A. et al. (2015), 4D hyperspherical harmonic (HyperSPHARM) representation of surface anatomy: A holistic treatment of multiple disconnected anatomical structures, Medical Image Analysis, 22(1): 89–101, https://doi.org/10.1016/j.media.2015.02.004
34. Pasqual A.M. (2014), Spherical harmonic analysis of the sound radiation from omnidirectional loudspeaker arrays, Journal of Sound and Vibration, 333(20): 4930–4941, https://doi.org/10.1016/j.jsv.2014.05.006
35. Rasumow E. et al. (2014), Smoothing individual head-related transfer functions in the frequency and spatial domains, The Journal of the Acoustical Society of America, 135(4): 2012–2025, https://doi.org/10.1121/1.4867372
36. Romigh G.D., Brungart D.S., Stern R.M., Simpson B.D. (2015), Efficient real spherical harmonic representation of head-related transfer functions, [in:] IEEE Journal on Selected Topics in Signal Processing, 9(5): 921–930, https://doi.org/10.1109/JSTSP.2015.2421876
37. Shabtai N.R., Behler G., Vorländer M., Weinzierl S. (2017), Generation and analysis of an acoustic radiation pattern database for forty-one musical instruments, The Journal of the Acoustical Society of America, 141(2): 1246–1256, https://doi.org/10.1121/1.4976071
38. Shekarchi S., Hallam J., Christensen-Dalsgaard J. (2013), Compression of head-related transfer function using autoregressive-moving-average models and Legendre polynomials, The Journal of the Acoustical Society of America, 134(5): 3686–3696, https://doi.org/10.1121/1.4822477
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41. Szwajcowski A., Krause D., Snakowska A. (2021), Error analysis of sound source directivity interpolation based on spherical harmonics, Archives of Acoustics, 46(1): 95–104, https://doi.org/10.24425/aoa.2021.136564
42. Varshalovich D.A., Moskalev A.N., Khersonskii V.K. (1988), Quantum Theory of Angular Momentum, World Scientific.
43. Wenzel E.M., Arruda M., Kistler D.J., Wightman F.L. (1993), Localization using nonindividualized head-related transfer functions, The Journal of the Acoustical Society of America, 94(1): 111–123, https://doi.org/10.1121/1.407089
44. Xie B., Zhang T. (2010), The audibility of spectral detail of head-related transfer functions at high frequency, Acta Acustica United with Acustica, 96(2): 328–339.
45. Zagala F., Zotter F. (2019), Idea for sign-change retrieval in magnitude directivity patterns, [in:] Conference: Fortschritte der Akustik – DAGA.
46. Zhang M., Kennedy R.A., Abhayapala T.D. (2015), Empirical determination of frequency representation in spherical harmonics-based HRTF functional modeling, [in:] IEEE/ACM Transactions on Audio Speech and Language Processing, 23(2): 351–360, https://doi.org/10.1109/TASLP.2014.2381881
47. Zhang W., Abhayapala T.D., Kennedy R.A., Duraiswami R. (2010), Insights into head-related transfer function: Spatial dimensionality and continuous representation, The Journal of the Acoustical Society of America, 127(4): 2347–2357, https://doi.org/10.1121/1.3336399
48. Zhang W., Kennedy R.A., Abhayapala T.D. (2009), Efficient continuous HRTF model using data independent basis functions: Experimentally guided approach, IEEE Transactions on Audio, Speech and Language Processing, 17(4): 819–829, https://doi.org/10.1109/TASL.2009.2014265
49. Zhang W., Zhang M., Kennedy R.A., Abhayapala T.D. (2012), On high-resolution head-related transfer function measurements: An efficient sampling scheme, IEEE Transactions on Audio, Speech and Language Processing, 20(2), 575–584, https://doi.org/10.1109/TASL.2011.2162404
50. Ziegler J.D., Rau M., Schilling A., Koch A. (2017), Interpolation and display of microphone directivity measurements using higher order spherical harmonics, [in:] Proceedings of 143rd AES Convention.
51. Zotkin D.N., Duraiswami R., Grassi E., Gumerov N.A. (2006), Fast head-related transfer function measurement via reciprocity, The Journal of the Acoustical Society of America, 120(4): 2202–2215, https://doi.org/10.1121/1.2207578
52. Zotkin D.N., Duraiswami R., Gumerov N.A. (2009), Regularized HRTF fitting using spherical harmonics, [in:] IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, pp. 257–260.
2. Agterberg M.J.H, Snik A.F.M, Hol M.K.S, Van Wanrooij, M.M., Van Opstal A.J. (2012), Contribution of monaural and binaural cues to sound localization in listeners with acquired unilateral conductive hearing loss: Improved directional hearing with a bone-conduction device, Hearing Research, 286(1–2): 9–18, https://doi.org/10.1016/j.heares.2012.02.012
3. Ahrens J., Thomas M.R.P., Tashev I. (2012), HRTF magnitude modeling using a non-regularized least-squares fit of spherical harmonics coefficients on incomplete data, [in:] Proceedings of the 2012 Asia Pacific Signal and Information Processing Association Annual Summit and Conference (IEEE).
4. Algazi V.R., Avendano C., Duda R.O. (2001), Elevation localization and head-related transfer function analysis at low frequencies, The Journal of the Acoustical Society of America, 109(3): 1110–1122, https://doi.org/10.1121/1.1349185
5. Alon D.L., Ben-Hur Z., Rafaely B., Mehra R. (2018), Sparse head-related transfer function representation with spatial aliasing cancellation, [in:] 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 6792–6796, https://doi.org/10.1109/ICASSP.2018.8462101
6. Andreopoulou A., Begault D.R., Katz B.F. (2015), Inter-Laboratory round robin HRTF measurement comparison, IEEE Journal on Selected Topics in Signal Processing, 9(5): 895–906, https://doi.org/10.1109/JSTSP.2015.2400417
7. Andreopoulou A., Roginska A., Mohanraj H. (2013), Analysis of the spectral variations in repeated head-related transfer function measurements, [in:] International Conference on Auditory Display, pp. 213–218.
8. Bates A.P., Khalid Z., Kennedy R.A. (2015), On the use of Slepian functions for the reconstruction of the head-related transfer function on the sphere, [in:] 2015 9th International Conference on Signal Processing and Communication Systems (ICSPCS).
9. Begault D.R., Wenzel E.M., Anderson M.R. (2001), Direct comparison of the impact of head tracking, reverberation, and individualized head-related transfer functions on the spatial perception of a virtual speech source, Journal of the Audio Engineering Society, 49(10): 904–916.
10. Ben-Hur Z., Alon D.L., Rafaely B., Mehra R. (2019), Loudness stability of binaural sound with spherical harmonic representation of sparse head-related transfer functions, EURASIP Journal on Audio and Music Processing, 2019: 5, https://doi.org/10.1186/s13636-019-0148-x
11. Blommer M.A., Wakefield G.H. (1997), Pole-zero approximations for head-related transfer functions using a logarithmic error criterion, IEEE Transactions on Speech and Audio Processing, 5(3): 278–287, https://doi.org/10.1109/89.568734
12. Bonvallet B., Griffin N., Li J. (2007), A 3D shape descriptor: 4D hyperspherical harmonics an exploration into the fourth dimension, [in:] Proceedings of IASTED International Conference on Graphics and Visualization in Engineering, pp. 113–116.
13. Breebaart J., Kohlrausch A. (2001), Perceptual (ir)relevance of HRTF magnitude and phase spectra, [in:] Proceedings of 110th AES Convention.
14. Brinkmann F., Weinzierl S. (2018), Comparison of head-related transfer functions pre-processing techniques for spherical harmonics decomposition, [in:] 2018 AES International Conference on Audio for Virtual and Augmented Reality.
15. Chen J., Van Veen B.D., Hecox K.E. (1995), A spatial feature extraction and regularization model for the head-related transfer function, The Journal of the Acoustical Society of America, 97(1): 439–452, https://doi.org/10.1121/1.413110
16. Domokos G. (1967), Four-dimensional symmetry, Physical Review, 159(5): 1387–1403, https://doi.org/10.1103/PhysRev.159.1387
17. Evans M.J., Angus J.A.S., Tew A.I. (1998), Analyzing head-related transfer function measurements using surface spherical harmonics, The Journal of the Acoustical Society of America, 104(4): 2400–2411, https://doi.org/10.1121/1.423749
18. Gardner W.G., Martin K.D. (1995), HRTF measurements of a KEMAR, The Journal of the Acoustical Society of America, 97(6): 3907–3908, https://doi.org/10.1121/1.412407
19. Hartung K., Braasch J., Sterbing S.J. (1999), Comparison of different methods for the interpolation of head-related transfer functions, [in:] 16th International Conference: Spatial Sound Reproduction, pp. 319–3129.
20. Hu S., Trevino J., Salvador C., Sakamoto S., Suzuki Y. (2019), Modeling head-related transfer functions with spherical wavelets, Applied Acoustics, 146: 81–88, https://doi.org/10.1016/j.apacoust.2018.10.026
21. Huopaniemi J., Zacharov N., Karjalainen M. (1999), Objective and subjective evaluation of head-related transfer function filter design, Journal of the Audio Engineering Society, 47(4): 218–239.
22. Kistler D.J., Wightman F.L. (1992), A model of head-related transfer functions based on principal components analysis and minimum-phase reconstruction, The Journal of the Acoustical Society of America, 91(3): 1637–1647, https://doi.org/10.1121/1.402444
23. Kulkarni A., Colburn H.S. (1995), Efficient finite-impulse-response filter models of the head-related transfer function, The Journal of the Acoustical Society of America, 97(5): 3278–3278, https://doi.org/10.1121/1.411579
24. Kulkarni A., Colburn H.S. (1998), Role of spectral detail in sound-source localization, Nature, 396(6713): 747–749, https://doi.org/10.1038/25526
25. Kulkarni A., Colburn H.S. (2004), Infinite-impulse-response models of the head-related transfer function, The Journal of the Acoustical Society of America, 115(4): 1714–1728, https://doi.org/10.1121/1.1650332
26. Kulkarni A., Isabelle S.K., Colburn H.S. (1999), Sensitivity of human subjects to head-related transfer-function phase spectra, The Journal of the Acoustical Society of America, 105(5): 2821–2840, https://doi.org/10.1121/1.426898
27. Langendijk E.H.A., Bronkhorst A.W. (2002), Contribution of spectral cues to human sound localization, The Journal of the Acoustical Society of America, 112(4): 1583–1596, https://doi.org/10.1121/1.1501901
28. Li J., Wu B., Yao D., Yan Y. (2021), A mixed-order modeling approach for head-related transfer function in the spherical harmonic domain, Applied Acoustics, 176: 107828, https://doi.org/10.1016/j.apacoust.2020.107828
29. Liu H., Fang Y., Huang Q. (2019), Efficient representation of head-related transfer functions with combination of spherical harmonics and spherical wavelets, IEEE Access, 7: 78214–78222, https://doi.org/10.1109/ACCESS.2019.2921388
30. Macpherson E.A., Middlebrooks J.C. (2002), Listener weighting of cues for lateral angle: The duplex theory of sound localization revisited, The Journal of the Acoustical Society of America, 111(5): 2219–2236, https://doi.org/10.1121/1.1471898
31. Majdak P. et al. (2013), Spatially oriented format for acoustics: A data exchange format representing head-related transfer functions, [in:] Proceedings of 134th AES Convention.
32. Nishino T., Kajita S., Takeda K., Itakura F. (1999), Interpolating head related transfer functions in the median plane, [in:] Proceedings of the 1999 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, pp. 167–170.
33. Pasha Hosseinbor A. et al. (2015), 4D hyperspherical harmonic (HyperSPHARM) representation of surface anatomy: A holistic treatment of multiple disconnected anatomical structures, Medical Image Analysis, 22(1): 89–101, https://doi.org/10.1016/j.media.2015.02.004
34. Pasqual A.M. (2014), Spherical harmonic analysis of the sound radiation from omnidirectional loudspeaker arrays, Journal of Sound and Vibration, 333(20): 4930–4941, https://doi.org/10.1016/j.jsv.2014.05.006
35. Rasumow E. et al. (2014), Smoothing individual head-related transfer functions in the frequency and spatial domains, The Journal of the Acoustical Society of America, 135(4): 2012–2025, https://doi.org/10.1121/1.4867372
36. Romigh G.D., Brungart D.S., Stern R.M., Simpson B.D. (2015), Efficient real spherical harmonic representation of head-related transfer functions, [in:] IEEE Journal on Selected Topics in Signal Processing, 9(5): 921–930, https://doi.org/10.1109/JSTSP.2015.2421876
37. Shabtai N.R., Behler G., Vorländer M., Weinzierl S. (2017), Generation and analysis of an acoustic radiation pattern database for forty-one musical instruments, The Journal of the Acoustical Society of America, 141(2): 1246–1256, https://doi.org/10.1121/1.4976071
38. Shekarchi S., Hallam J., Christensen-Dalsgaard J. (2013), Compression of head-related transfer function using autoregressive-moving-average models and Legendre polynomials, The Journal of the Acoustical Society of America, 134(5): 3686–3696, https://doi.org/10.1121/1.4822477
39. Ultraspherical polynomials (n.d.), Encyclopedia of Mathematics, http://encyclopediaofmath.org/index.php?title=Ultraspherical_polynomials&oldid=52128
40. Szwajcowski A. (2021), Objective-Oriented Directivity. MATLAB toolbox for processing directivity models (access: 3.06.2022).
41. Szwajcowski A., Krause D., Snakowska A. (2021), Error analysis of sound source directivity interpolation based on spherical harmonics, Archives of Acoustics, 46(1): 95–104, https://doi.org/10.24425/aoa.2021.136564
42. Varshalovich D.A., Moskalev A.N., Khersonskii V.K. (1988), Quantum Theory of Angular Momentum, World Scientific.
43. Wenzel E.M., Arruda M., Kistler D.J., Wightman F.L. (1993), Localization using nonindividualized head-related transfer functions, The Journal of the Acoustical Society of America, 94(1): 111–123, https://doi.org/10.1121/1.407089
44. Xie B., Zhang T. (2010), The audibility of spectral detail of head-related transfer functions at high frequency, Acta Acustica United with Acustica, 96(2): 328–339.
45. Zagala F., Zotter F. (2019), Idea for sign-change retrieval in magnitude directivity patterns, [in:] Conference: Fortschritte der Akustik – DAGA.
46. Zhang M., Kennedy R.A., Abhayapala T.D. (2015), Empirical determination of frequency representation in spherical harmonics-based HRTF functional modeling, [in:] IEEE/ACM Transactions on Audio Speech and Language Processing, 23(2): 351–360, https://doi.org/10.1109/TASLP.2014.2381881
47. Zhang W., Abhayapala T.D., Kennedy R.A., Duraiswami R. (2010), Insights into head-related transfer function: Spatial dimensionality and continuous representation, The Journal of the Acoustical Society of America, 127(4): 2347–2357, https://doi.org/10.1121/1.3336399
48. Zhang W., Kennedy R.A., Abhayapala T.D. (2009), Efficient continuous HRTF model using data independent basis functions: Experimentally guided approach, IEEE Transactions on Audio, Speech and Language Processing, 17(4): 819–829, https://doi.org/10.1109/TASL.2009.2014265
49. Zhang W., Zhang M., Kennedy R.A., Abhayapala T.D. (2012), On high-resolution head-related transfer function measurements: An efficient sampling scheme, IEEE Transactions on Audio, Speech and Language Processing, 20(2), 575–584, https://doi.org/10.1109/TASL.2011.2162404
50. Ziegler J.D., Rau M., Schilling A., Koch A. (2017), Interpolation and display of microphone directivity measurements using higher order spherical harmonics, [in:] Proceedings of 143rd AES Convention.
51. Zotkin D.N., Duraiswami R., Grassi E., Gumerov N.A. (2006), Fast head-related transfer function measurement via reciprocity, The Journal of the Acoustical Society of America, 120(4): 2202–2215, https://doi.org/10.1121/1.2207578
52. Zotkin D.N., Duraiswami R., Gumerov N.A. (2009), Regularized HRTF fitting using spherical harmonics, [in:] IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, pp. 257–260.
