On the use of Slepian Functions for the Reconstruction of the Head-Related Transfer Function on the Sphere

In this work we investigate using the Slepian basis for the reconstruction of the head-related transfer function (HRTF) on the sphere. Measurements of the HRTF are unavailable over the south polar cap which tends to result in a large reconstruction error when reconstruction is performed in the traditionally used spherical harmonic basis. While the spherical harmonic basis is well-suited to applications where data is taken over the whole sphere, it is not a natural basis when considering a region on the sphere. The Slepian basis is a set of functions which are optimally concentrated and orthogonal within a region, unlike the spherical harmonic basis. We demonstrate through numerical experiments on randomly generated data and synthetic HRTF measurements that reconstruction of the HRTF in the Slepian basis is significantly more accurate at sample locations; the reconstruction error is up to 11 orders of magnitude smaller. The reconstruction error obtained using the Slepian basis is also smaller at other locations on the sphere, both within and outside of the region where measurements are taken, than in the spherical harmonic basis. We also briefly investigate truncation of the Slepian basis as a means of denoising the HRTF measurements and find that this reduces the reconstruction error. Our analysis suggests that the Slepian basis allows more accurate reconstruction than the spherical harmonic basis.