Complex NMF under phase constraints based on signal modeling: Application to audio source separation

Abstract

Nonnegative Matrix Factorization (NMF) is a powerful tool for decomposing mixtures of audio signals in the Time-Frequency (TF) domain. In the source separation framework, the phase recovery for each extracted component is necessary for synthesizing time-domain signals. The Complex NMF (CNMF) model aims to jointly estimate the spectrogram and the phase of the sources, but requires to constrain the phase in order to produce satisfactory sounding results. We propose to incorporate phase constraints based on signal models within the CNMF framework: a phase unwrapping constraint that enforces a form of temporal coherence, and a constraint based on the repetition of audio events, which models the phases of the sources within onset frames. We also provide an algorithm for estimating the model parameters. The experimental results highlight the interest of including such constraints in the CNMF framework for separating overlapping components in complex audio mixtures.

Complex ISNMF: A Phase-Aware Model for Monaural Audio Source Separation

“…This model is applied to a musical source separation task in a semiinformed setting. It outperforms both the traditional phaseunaware ISNMF and the phase-constrained CNMF model [25]. This demonstrates the usefulness of such a phase-aware Bayesian AG model to perform the joint estimation of magnitudes and phases for audio source separation.…”

“…We remark that if κ = 0, then λ = ρ = 0: therefore, q j,f t = 0 and p j,f t becomes the posterior power of s j,f t , as mentioned in Section III-C1. Then, we recognize in (25) the IS divergence between P j and W j H j , as in the EM algorithm for ISNMF [46]. Consequently, the updates rules (36) and (37) are similar to those obtained in such a scenario [46], up to an additional power 1/2, which is common when applying the majorizeminimization methodology for estimating ISNMF [44].…”

“…Besides, one can consider an alternative AG model as described in Section II-E. If one considers that the sources are centered (s j ∼ N (0, Γ j )), then Q j = 0: we recognize in (25) the IS divergence between the NMF model and the phase-corrected posterior power. The derivation of the update rules is then easier than in the general case, since it eliminates the need for the majorize-minimization method: one can apply the commonly-used heuristic method described in [2] to obtain alternative multiplicative update rules.…”

Amplitude and phase dereverberation of monocomponent signals

“…The goal is to estimate the quadratic terms in (8) and (9) from the reverberant signal y(t) defined in (3), in order to resynthesize the anechoic signal s(t). To do so, we use the fact that if the RIR h(t) is modeled as in Section II-C, then for any analog signals x 1 (t) and x 2 (t):…”

“…This is the main drawback of these methods, because using this corrupted phase reintroduces reverberation and distortion in the signal, as shown in [8]. In the source separation literature, the idea of modeling the phase has recently been proposed [9] because a similar problem occurs (the phase of the mixture is generally used to synthesize the source signals), but not for dereverberation.…”