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Spherical correlation as a similarity measure for 3D radiation patterns of musical instruments

Thibaut Carpentier
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Aaron Einbond
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This work is part of an artistic-research residency where composer Aaron Einbond seeks to apply audio descriptor analysis and corpus-based synthesis techniques to the spatial manipulation of instrumental radiation patterns for projection with a compact spherical loudspeaker array. Starting from a database of 3D directivity patterns of orchestral instruments, measured with spherical microphone arrays in anechoic conditions, we wish to derive spatial descriptors in order to classify the corpus. This paper investigates the use of spherical cross-correlation as a similarity measure between radiation patterns. Considering two directivity patterns f and g as bandlimited, square integrable functions on the 2-sphere, their correlation can be computed from their spherical harmonic spectra via a spatial inverse discrete Fourier transform. The magnitudes of these Fourier coefficients provide a rotation-invariant representation of the functions on the sphere. One can therefore search for the transformation matrix m, in the 3D rotation group SO(3), which maximizes the cross-correlation, i.e. which offers the optimal spherical shape matching between f and g. The mathematical foundations of these tools are well established in the literature ; however, their practical use in the field of acoustics remains limited and challenging. In this study, we apply these techniques to both simulated and measured radiation data, attempting to answer a number of practical questions : How does the similarity measure behave when f and g are not rotated cousins ? How can we adapt the cross-correlation formalism established for complex-valued harmonics to real-valued harmonics, as the latter are predominantly used in the field of Ambisonics ? Can we compute the correlation of spherical spectra of different bandwidths ? What is the impact of the finite sampling distribution used for integration on the SO(3) space? And most importantly, is the cross-correlation an efficient measure for the classification of 3D radiation patterns ?
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hal-03649913 , version 1 (23-04-2022)


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Thibaut Carpentier, Aaron Einbond. Spherical correlation as a similarity measure for 3D radiation patterns of musical instruments. 16ème Congrès Français d’Acoustique, Apr 2022, Marseille, France. ⟨hal-03649913⟩
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