Arne Leijon

Supervised and Unsupervised Speech Enhancement Using Nonnegative Matrix Factorization

Reducing the interference noise in a monaural noisy speech signal has been a challenging task for many years. Compared to traditional unsupervised speech enhancement methods, e.g., Wiener filtering, supervised approaches, such as algorithms based on hidden Markov models (HMM), lead to higher-quality enhanced speech signals. However, the main practical difficulty of these approaches is that for each noise type a model is required to be trained a priori. In this paper, we investigate a new class of supervised speech denoising algorithms using nonnegative matrix factorization (NMF). We propose a novel speech enhancement method that is based on a Bayesian formulation of NMF (BNMF). To circumvent the mismatch problem between the training and testing stages, we propose two solutions. First, we use an HMM in combination with BNMF (BNMF-HMM) to derive a minimum mean square error (MMSE) estimator for the speech signal with no information about the underlying noise type. Second, we suggest a scheme to learn the required noise BNMF model online, which is then used to develop an unsupervised speech enhancement system. Extensive experiments are carried out to investigate the performance of the proposed methods under different conditions. Moreover, we compare the performance of the developed algorithms with state-of-the-art speech enhancement schemes using various objective measures. Our simulations show that the proposed BNMF-based methods outperform the competing algorithms substantially.

Bayesian Estimation of Beta Mixture Models with Variational Inference

Bayesian estimation of the parameters in beta mixture models (BMM) is analytically intractable. The numerical solutions to simulate the posterior distribution are available, but incur high computational cost. In this paper, we introduce an approximation to the prior/posterior distribution of the parameters in the beta distribution and propose an analytically tractable (closed form) Bayesian approach to the parameter estimation. The approach is based on the variational inference (VI) framework. Following the principles of the VI framework and utilizing the relative convexity bound, the extended factorized approximation method is applied to approximate the distribution of the parameters in BMM. In a fully Bayesian model where all of the parameters of the BMM are considered as variables and assigned proper distributions, our approach can asymptotically find the optimal estimate of the parameters posterior distribution. Also, the model complexity can be determined based on the data. The closed-form solution is proposed so that no iterative numerical calculation is required. Meanwhile, our approach avoids the drawback of overfitting in the conventional expectation maximization algorithm. The good performance of this approach is verified by experiments with both synthetic and real data.

Bayesian estimation of Dirichlet mixture model with variational inference

In statistical modeling, parameter estimation is an essential and challengeable task. Estimation of the parameters in the Dirichlet mixture model (DMM) is analytically intractable, due to the integral expressions of the gamma function and its corresponding derivatives. We introduce a Bayesian estimation strategy to estimate the posterior distribution of the parameters in DMM. By assuming the gamma distribution as the prior to each parameter, we approximate both the prior and the posterior distribution of the parameters with a product of several mutually independent gamma distributions. The extended factorized approximation method is applied to introduce a single lower-bound to the variational objective function and an analytically tractable estimation solution is derived. Moreover, there is only one function that is maximized during iterations and, therefore, the convergence of the proposed algorithm is theoretically guaranteed. With synthesized data, the proposed method shows the advantages over the EM-based method and the previously proposed Bayesian estimation method. With two important multimedia signal processing applications, the good performance of the proposed Bayesian estimation method is demonstrated.

A new linear MMSE filter for single channel speech enhancement based on Nonnegative Matrix Factorization

In this paper, a linear MMSE filter is derived for single-channel speech enhancement which is based on Nonnegative Matrix Factorization (NMF). Assuming an additive model for the noisy observation, an estimator is obtained by minimizing the mean square error between the clean speech and the estimated speech components in the frequency domain. In addition, the noise power spectral density (PSD) is estimated using NMF and the obtained noise PSD is used in a Wiener filtering framework to enhance the noisy speech. The results of the both algorithms are compared to the result of the same Wiener filtering framework in which the noise PSD is estimated using a recently developed MMSE-based method. NMF based approaches outperform the Wiener filter with the MMSE-based noise PSD tracker for different measures. Compared to the NMF-based Wiener filtering approach, Source to Distortion Ratio (SDR) is improved for the evaluated noise types for different input SNRs using the proposed linear MMSE filter.

Variational Bayesian Matrix Factorization for Bounded Support Data

A novel Bayesian matrix factorization method for bounded support data is presented. Each entry in the observation matrix is assumed to be beta distributed. As the beta distribution has two parameters, two parameter matrices can be obtained, which matrices contain only nonnegative values. In order to provide low-rank matrix factorization, the nonnegative matrix factorization (NMF) technique is applied. Furthermore, each entry in the factorized matrices, i.e., the basis and excitation matrices, is assigned with gamma prior. Therefore, we name this method as beta-gamma NMF (BG-NMF). Due to the integral expression of the gamma function, estimation of the posterior distribution in the BG-NMF model can not be presented by an analytically tractable solution. With the variational inference framework and the relative convexity property of the log-inverse-beta function, we propose a new lower-bound to approximate the objective function. With this new lower-bound, we derive an analytically tractable solution to approximately calculate the posterior distributions. Each of the approximated posterior distributions is also gamma distributed, which retains the conjugacy of the Bayesian estimation. In addition, a sparse BG-NMF can be obtained by including a sparseness constraint to the gamma prior. Evaluations with synthetic data and real life data demonstrate the good performance of the proposed method.

Preferred overall loudness. II: Listening through hearing aids in field and laboratory tests

In a laboratory study, we found that normal-hearing and hearing-impaired listeners preferred less than normal overall calculated loudness (according to a loudness model of Moore & Glasberg, ). The current study verified those results using a research hearing aid. Fifteen hearing-impaired and eight normal-hearing participants used the hearing aid in the field and adjusted a volume control to give preferred in median loudness. The hearing aid logged the preferred volume control setting and the calculated loudness at that setting. The hearing-impaired participants preferred loudness levels of −14 phon re normal for input levels from 50 to 89 dB SPL. The normal-hearing participants preferred close to normal overall loudness. In subsequent laboratory tests, using the same hearing aid, both hearing-impaired and normal-hearing listeners preferred less than normal overall calculated loudness, and larger reductions for higher input levels. In summary, the hearing-impaired listeners preferred less than normal overall calculated loudness, whereas the results for the normal-hearing listeners were inconclusive. Sumario En un estudio de laboratorio encontramos que normoacúsicos o hipoacúsicos prefieren una intensidad subjetiva global menor que la calculada (de acuerdo con el modelo de intensidad subjetiva de Moore & Glasberg, ). Este estudio verificó esos resultados usando un auxiliar auditivo de investigación. Quince hipoacúsicos y ocho normoacúsicos usaron el auxiliar auditivo en campo libre y ajustaron el control de volumen hasta el nivel de intensidad subjetiva preferido. El auxiliar auditivo conservó el nivel de control de volumen preferido y el nivel de intensidad subjetiva calculado para esa posición. Los hipoacúsicos prefirieron niveles de intensidad subjetiva de −14 phon re normal, para niveles de presentación de 50 a 89 dB SPL. Los normoacúsicos prefirieron niveles de intensidad subjetiva globales cerca de lo normal. En unas pruebas de laboratorio subsecuentes, usando el mismo auxiliar auditivo tanto los hipoacúsicos como los normoacúsicos prefirieron una intensidad subjetiva global menor que la calculada como normal así como amplias reducciones para los niveles de presentación mayores. En resumen, los hipoacúsicos prefieren intensidades subjetivas globales menores que las calculadas como normales, mientras que los resultados para los normoacúsicos no fueron concluyentes.

Single channel speech enhancement using Bayesian NMF with recursive temporal updates of prior distributions

We present a speech enhancement algorithm which is based on a Bayesian Nonnegative Matrix Factorization (NMF). Both Minimum Mean Square Error (MMSE) and Maximum a-Posteriori (MAP) estimates of the magnitude of the clean speech DFT coefficients are derived. To exploit the temporal continuity of the speech and noise signals, a proper prior distribution is introduced by widening the posterior distribution of the NMF coefficients at the previous time frames. To do so, a recursive temporal update scheme is proposed to obtain the mean value of the prior distribution; also, the uncertainty of the prior information is governed by the shape parameter of the distribution which is learnt automatically based on the nonstationarity of the signals. Simulations show a considerable improvement compared to the maximum likelihood NMF based speech enhancement algorithm for different input SNRs.

Vector quantization of LSF parameters with a mixture of dirichlet distributions

Quantization of the linear predictive coding parameters is an important part in speech coding. Probability density function (PDF)-optimized vector quantization (VQ) has been previously shown to be more efficient than VQ based only on training data. For data with bounded support, some well-defined bounded-support distributions (e.g., the Dirichlet distribution) have been proven to outperform the conventional Gaussian mixture model (GMM), with the same number of free parameters required to describe the model. When exploiting both the boundary and the order properties of the line spectral frequency (LSF) parameters, the distribution of LSF differences LSF can be modelled with a Dirichlet mixture model (DMM). We propose a corresponding DMM based VQ. The elements in a Dirichlet vector variable are highly mutually correlated. Motivated by the Dirichlet vector variables neutrality property, a practical non-linear transformation scheme for the Dirichlet vector variable can be obtained. Similar to the Karhunen-Loève transform for Gaussian variables, this non-linear transformation decomposes the Dirichlet vector variable into a set of independent beta-distributed variables. Using high rate quantization theory and by the entropy constraint, the optimal inter- and intra-component bit allocation strategies are proposed. In the implementation of scalar quantizers, we use the constrained-resolution coding to approximate the derived constrained-entropy coding. A practical coding scheme for DVQ is designed for the purpose of reducing the quantization error accumulation. The theoretical and practical quantization performance of DVQ is evaluated. Compared to the state-of-the-art GMM-based VQ and recently proposed beta mixture model (BMM) based VQ, DVQ performs better, with even fewer free parameters and lower computational cost

Beta mixture models and the application to image classification

Statistical pattern recognition is one of the most studied and applied approaches in the area of pattern recognition. Mixture modelling of densities is an efficient statistical pattern recognition method for continuous data. We propose a classifier based on the beta mixture models for strictly bounded and asymmetrically distributed data. Due to the property of the mixture modelling, the statistical dependence in a multi-dimensional variable is captured, even with the conditional independence assumption in each mixture component. A synthetic example and the USPS handwriting digit data was used to verify the effectiveness of this approach. Compared to the conventional Gaussian mixture models (GMM), the beta mixture models has a better performance on data which has strictly bounded value and asymmetric distribution. The performance of beta mixture models is about equivalent to that of GMM applied to data transformed via a strictly increasing link function.

Decorrelation of Neutral Vector Variables: Theory and Applications

In this paper, we propose novel strategies for neutral vector variable decorrelation. Two fundamental invertible transformations, namely, serial nonlinear transformation and parallel nonlinear transformation, are proposed to carry out the decorrelation. For a neutral vector variable, which is not multivariate-Gaussian distributed, the conventional principal component analysis cannot yield mutually independent scalar variables. With the two proposed transformations, a highly negatively correlated neutral vector can be transformed to a set of mutually independent scalar variables with the same degrees of freedom. We also evaluate the decorrelation performances for the vectors generated from a single Dirichlet distribution and a mixture of Dirichlet distributions. The mutual independence is verified with the distance correlation measurement. The advantages of the proposed decorrelation strategies are intensively studied and demonstrated with synthesized data and practical application evaluations.