Time-of-arrival Estimation and Phase Unwrapping of Head-related Transfer Functions With Integer Linear Programming

In binaural audio synthesis, aligning head-related impulse responses (HRIRs) in time has been an important pre-processing step, enabling accurate spatial interpolation and efficient data compression. The maximum correlation time delay between spatially nearby HRIRs has previously been used to get accurate and smooth alignment by solving a matrix equation in which the solution has the minimum Euclidean distance to the time delay. However, the Euclidean criterion could lead to an over-smoothing solution in practice. In this paper, we solve the smoothing issue by formulating the task as solving an integer linear programming problem equivalent to minimising an $L^1$-norm. Moreover, we incorporate 1) the cross-correlation of inter-aural HRIRs, and 2) HRIRs with their minimum-phase responses to have more reference measurements for optimisation. We show the proposed method can get more accurate alignments than the Euclidean-based method by comparing the spectral reconstruction loss of time-aligned HRIRs using spherical harmonics representation on seven HRIRs consisting of human and dummy heads. The extra correlation features and the $L^1$-norm are also beneficial in extremely noisy conditions. In addition, this method can be applied to phase unwrapping of head-related transfer functions, where the unwrapped phase could be a compact feature for downstream tasks.