Extended Nonnegative Tensor Factorisation Models for Musical Sound Source Separation

Abstract

Recently, shift-invariant tensor factorisation algorithms have been proposed for the purposes of sound source separation of pitched musical instruments. However, in practice, existing algorithms require the use of log-frequency spectrograms to allow shift invariance in frequency which causes problems when attempting to resynthesise the separated sources. Further, it is difficult to impose harmonicity constraints on the recovered basis functions. This paper proposes a new additive synthesis-based approach which allows the use of linear-frequency spectrograms as well as imposing strict harmonic constraints, resulting in an improved model. Further, these additional constraints allow the addition of a source filter model to the factorisation framework, and an extended model which is capable of separating mixtures of pitched and percussive instruments simultaneously.

On the use of the beta divergence for musical source separation

“…The musical source separation algorithm used in this paper is a harmonicity enforcing additive synthesis based model, where each instrument or source is modelled by a set of harmonic weights [2]. These weights are invariant to pitch and so each note played by an instrument will use the same weights regardless of pitch.…”

“…the update equations, which have not been previously presented in published work, for the model parameters are as follows: [2].…”

“…As no results comparing the performance of the generalised model described above with the model described in [2] have previously been published, these are provided in Table 1 to provide a baseline against which to compare the performance of the Beta Divergence, where Standard refers to the model used in [2] and Generalised the model presented in this paper. In both cases KLD was used as a cost function.…”

“…Much research has been carried out on the use of non-negative matrix factorisation (NMF) and nonnegative tensor factorisation (NTF) for the purposes of musical sound source separation [1,2]. The majority of this work has focused on the use of the generalised Kullback-Liebler divergence (KLD) as a cost function as it has been found to work reliably for sound source separation.…”

“…However, the Beta Divergence has yet to be evaluated for musical signals, particularly in the context of the extended non-negative tensor factorisation models proposed for musical signals proposed in [2,7].…”