Patrick J. Wolfe

Efficient alternatives to the Ephraim and Malah suppression rule for audio signal enhancement

Audio signal enhancement often involves the application of a time-varying filter, or suppression rule, to the frequency-domain transform of a corrupted signal. Here we address suppression rules derived under a Gaussian model and interpret them as spectral estimators in a Bayesian statistical framework. With regard to the optimal spectral amplitude estimator of Ephraim and Malah, we show that under the same modelling assumptions, alternative methods of Bayesian estimation lead to much simpler suppression rules exhibiting similarly effective behaviour. We derive three of such rules and demonstrate that, in addition to permitting a more straightforward implementation, they yield a more intuitive interpretation of the Ephraim and Malah solution.

Simple alternatives to the Ephraim and Malah suppression rule for speech enhancement

Short-time spectral attenuation is a common form of audio signal enhancement in which a time-varying filter, or suppression rule, is applied to the frequency-domain transform of a corrupted signal. The suppression rule (see Ephraim, Y. and Malah, D., IEEE Trans. on Acoustics, Speech and Signal Proc., vol.ASSP-32, no.6, p.1109-21, 1984) for speech enhancement is both optimal in the minimum mean-square error sense and well-known for its associated colourless residual noise; however, it requires the computation of exponential and Bessel functions. We show that, under the same modelling assumptions, alternative Bayesian approaches lead to suppression rules exhibiting almost identical behaviour. We derive three such rules and show that they are efficient to implement and yield a more intuitive interpretation.

Towards a perceptually optimal spectral amplitude estimator for audio signal enhancement

We present a statistical model-based approach to signal enhancement in the case of additive broadband noise. Because broadband noise is localised in neither time nor frequency, its removal is one of the most pervasive and difficult signal enhancement tasks. In order to improve perceived signal quality, we take advantage of human perception and define a best estimate of the original signal in terms of a cost function incorporating perceptual optimality criteria. We derive the resultant signal estimator and implement it in a short-time spectral attenuation framework.

Analysis of reassigned spectrograms for musical transcription

The reassignment method for the short-time Fourier transform is proposed as a technique for improving the time and frequency estimates of musical audio data. Based on this representation, four classes of expected objects (sinusoid, unresolved sinusoid, transient and noise) are proposed and explained. Pattern classification methods are then used to extract objects conforming to these classes from individual frames of the reassigned spectrogram, with each frame being examined independently. Results for several simple real-world examples are presented, showing the capability of this method even without the aid of tracking from frame to frame. The main benefits of the proposed reassignment stage are that it yields an improved time-frequency localisation estimate relative to standard methods, and that it produces a measure of the variance of these estimates to be used as an aid in later processing.

Multi-Gabor dictionaries for audio time-frequency analysis

We consider the construction of multiresolution Gabor dictionaries appropriate for audio signal analysis. Motivated by a desire for parsimony and efficiency, we propose and formalise the idea of reduced multi-Gabor systems, showing that they constitute a frame for L/sup 2/(R) and other Hilbert spaces of interest. In order to demonstrate the practicality of such a scheme, we apply it to the atomic decomposition of music and speech signals observed in noise. Qualitative results indicate the potential of this method to yield a salient representation of typical audio signals while at the same time reducing computational costs as compared to a full multiresolution decomposition.

Bayesian Approach for Blind Separation of Underdetermined Mixtures of Sparse Sources

We address in this paper the problem of blind separation of underdetermined mixtures of sparse sources. The sources are given a Student t distribution, in a transformed domain, and we propose a bayesian approach using Gibbs sampling. Results are given on synthetic and audio signals.

Perceptually Motivated Approaches to Music Restoration

Spurred by the success of perceptual models in audio coding applications, researchers have recently begun to address audio signal enhancement in a similar manner. Here we consider the case of musical recordings degraded by additive broadband noise such as tape hiss, in which the prevention of signal distortion is tantamount to noise removal. We review perceptually motivated approaches to music restoration and describe a statistical model based framework we have recently proposed. By integrating psychoacoustics into the restoration process through the use of perceptual optimality criteria, our method aims to take advantage of human auditory perception to yield improvements in both noise reduction and perceived signal fidelity. Audio examples and related software may be found at http://www.sigproc.eng. cam.ac.uk/~pjw47.

A Gabor regression scheme for audio signal analysis

We describe novel Bayesian models for time-frequency analysis of non-stationary audio waveforms. These models are based on the idea of a Gabor regression, in which a time series is represented as a superposition of time-frequency shifted versions of a simple window function. Prior distributions over the corresponding time-frequency coefficients are constructed in a manner which favours both smoothness of the estimated function and sparseness of the coefficient representation (either indirectly through scale mixtures of normals, or directly through prior probability mass at zero). In this way, prior regularisation may induce a parsimonious, meaningful representation of the underlying audio time series.

A perceptually balanced loss function for short-time spectral amplitude estimation

We present a novel approach to audio signal enhancement based on psychoacoustic principles. Specifically, we describe a short-time spectral amplitude estimator whose form comprises a weighted sum of the minimum mean-square error solution and the observed spectral value, where the weighting factor is given by the ratio of the masked threshold and this observed value. We then explore the connection between our approach and the idea of so-called balanced loss functions in statistics, showing the former to be an instance of the latter with a very special choice of weighting factor. Lastly, we present results indicating the relative merits of our approach in both objective and subjective terms, as compared to standard minimum mean-square error estimation under the assumed model.

Nonlinear perceptual audio filtering using support vector machines

The perceptually based loss functions for audio filtering used by P.J. Wolfe and S.J. Godsill (see Proc. IEEE ICASSP, vol.2, p.821-4, 2000) are shown to fit well within a complex-valued support vector machine (SVM) framework. SVM regression is extended to the estimation of complex-valued functions, including the derivation of a variant of the sequential minimal optimisation (SMO) algorithm. Audio filters are derived using this, based on an autoregressive (AR) model used for audio and two different Hermitian kernel functions. Results are found to be promising, and further improvements are discussed.