Multi-target pitch tracking of vibrato sources in noise using the GM-PHD filter

Probabilistic approaches to tracking often use single-source Bayesian models; applying these to multi-source tasks is problematic. We apply a principled multi-object tracking implementation, the Gaussian mixture probability hypothesis density filter, to track multiple sources having fixed pitch plus vibrato. We demonstrate high-quality filtering in a synthetic experiment, and find improved tracking using a richer feature set which captures underlying dynamics. Our implementation is available as open-source Python code. Probabilistic modelling of audio objects is useful because Bayesian methods can be used to make principled inferences about the content of audio signals. For reasons of simplicity and tractability, inferences based on single-source models are widely used, such as the standard Hidden Markov Model (HMM) approach to speech recognition and music modelling. However, music is very often polyphonic, so there is a need to analyse acoustic scenes in which multiple sources may be simultaneously active. Multi-source tracking can be achieved by repeated application of single-source models, but this does not reflect the true scene and may yield sub-optimal results (Mahler, 2007). Existing multi-source approaches in music informatics often use non-probabilistic techniques. Probabilistic approaches exist, such as Probabilistic Latent Component Analysis (PLCA) which characterises sources as time-varying activations of spectral bases. However, such models are not always well-matched to audio objects with structured variability over time, and are poorly suited to causal (e.g. real-time) tracking. In this paper we investigate an alternative multiple Appearing in Proceedings of the 29 th International Conference on Machine Learning, Edinburgh, Scotland, UK, 2012. Copyright 2012 by the author(s)/owner(s). tracking paradigm, which models a set-valued random variable having multiple objects (Mahler, 2007). The probability hypothesis density filter (PHD filter) is one practical realisation of this approach. Given a system with linear Markov state updates and a linear observation model, it propagates a density through time which is an estimate of the underlying system state. The PHD filter was originally formulated as a particletype filter. Later work introduced the Gaussian mixture PHD filter (GM-PHD filter), using a Gaussian mixture (GM) to represent state and having improved performance (Vo & Ma, 2006; Mahler, 2007). The GM-PHD filter has similarities to a HMMor Kalman-type filter with hidden state represented as a GM, propagated from one time-frame to the next. However, the GM does not represent a probability density but the “intensity”: the first moment of the setvalued system state. The intensity does not integrate to 1, but to a total reflecting the expected number of objects present; its value at a location can be thought of as the expected number of objects at that location.