Speech Enhancement Using Gaussian Scale Mixture Models

Abstract

This paper presents a novel probabilistic approach to speech enhancement. Instead of a deterministic logarithmic relationship, we assume a probabilistic relationship between the frequency coefficients and the log-spectra. The speech model in the log-spectral domain is a Gaussian mixture model (GMM). The frequency coefficients obey a zero-mean Gaussian whose covariance equals to the exponential of the log-spectra. This results in a Gaussian scale mixture model (GSMM) for the speech signal in the frequency domain, since the log-spectra can be regarded as scaling factors. The probabilistic relation between frequency coefficients and logspectra allows these to be treated as two random variables, both to be estimated from the noisy signals. Expectation-maximization (EM) was used to train the GSMM and Bayesian inference was used to compute the posterior signal distribution. Because exact inference of this full probabilistic model is computationally intractable, we developed two approaches to enhance the efficiency: the Laplace method and a variational approximation. The proposed methods were applied to enhance speech corrupted by Gaussian noise and speech-shaped noise (SSN). For both approximations, signals reconstructed from the estimated frequency coefficients provided higher signal-to-noise ratio (SNR) and those reconstructed from the estimated log-spectra produced lower word recognition error rate because the log-spectra fit the inputs to the recognizer better. Our algorithms effectively reduced the SSN, which algorithms based on spectral analysis were not able to suppress. Index TermsGaussian scale mixture model (GSMM); Laplace method; speech enhancement; variational approximation

A Bayesian Modeling Approach to Situated Design of Personalized Soundscaping Algorithms

“…The two example models consist of two distinct source mixing models and similar submodels for the constituent signals. First, we introduce the source mixing models, based on the Algonquin algorithm [9] and Gaussian scale models [30], respectively. In other words, we provide explicit specifications for the source mixing model p(x n | s n ) in (1).…”

“…x . Here we present an alternative novel source mixing model that does not require any tuning parameters, inspired by the Gaussian scale models from [30].…”

“…In order to keep inference tractable, independence is assumed between the real and imaginary parts of the coefficients, requiring C = 0. Following [30], the covariance matrix Γ is modelled as a diagonal matrix with exponentiated auxiliary variables s k, f n , leading to the model…”

“…All message passing update rules are readily available within ReactiveMP, based on Tables 1 and 2 and published update rule tables elsewhere [32,37,62]. However, in order to improve convergence the message passing-based inference is preceded by other algorithms for training the Gaussian mixtures, similarly to [30]. In the remainder of this subsection we will describe in detail how both signal models from Section 2.2 have been initialized.…”

“…Furthermore, we provide an intuitive explanation for the derived messages. Additionally, in Section 2.2.2 we introduce an alternative source mixing model based on Gaussian scale models [30] for acoustic signals, represented by their pseudo log-power spectrum. We also frame this model as a re-usable factor graph node and describe how to perform message passing-based inference in this model in Section 2.3.6.…”