Accurate, fast and stable denoising source separation algorithms

The above adaptation of γ has turned out to be very useful and it can both
stabilise and accelerate convergence. According to (11), γ keeps increasing as
long as the steps are taken to the same direction and decreases if they are taken
backwards.

4 Denoising based on estimated signal variance

Several denoising procedures based on masking the source estimate were pro-
posed in [1]. The basic idea is to multiply the source estimate by a positive
envelope, a mask which has low values when SNR is low and vice versa. Depend-
ing on how the mask is computed, several types of prior information about the
sources can be used for separation.

A simple and well-founded mask can be obtained from the maximum-a-
posteriori (MAP) estimate. Assuming that the signals are Gaussian with chang-
ing variance σ_s²(t) (for related methods, see, e.g., [3]) and additive Gaussian noise
σ_n² , the MAP estimate of the signal is
where σ_t²_ot(t) = σ_s²(t) + σ_n² (t) is the total variance of the observation. Masking
then boils down to estimating σ_s² (t) and σ_t²_ot (t) from the observations.

A naive estimate of the signal variance is σ2 (t) ≈ s² (t). It can be improved
by low-pass filtering in time, e.g., by convolving with a Gaussian kernel. Simple
estimation of the baseline noise-level σ_n² was suggested in [1] resulting in a sim-
ple DSS algorithm. However, from the viewpoint of the estimated signal, other
signals should be treated as noise. DSS algorithm using the above approximation
separates easily the signal subspace from noise but the separation in the signal
subspace is slow and may even fail. In [1], this was solved by using σ2^μ(t) with
μ > 1 in (12). This way the mask does not saturate so quickly for large signal
variances, giving competitive edge to the source which is strongest. A close con-
nection to the familiar tanh-nonlinearity was shown: f(s) = s - tanh s has the
same fixed points as f(s) = tanh s but the former can be interpreted as s masked
by a slowly saturating envelope.

In this paper, we propose a new and better founded solution to the sepa-
ration problem. One can simply whiten the estimated total variance σtot (t) by
a symmetric whitening matrix. This bares resemblance to proposals of the role
of divisive normalisation on cortex [4] and to the classical ICA-method called
JADE [5]. Whitening naturally requires that all sources are estimated simul-
taneously and deflation approach is thus not applicable. The total variance is
obtained by smoothing s² (t) as described above. We obtain σ_s² (t) by taking
the positive part of the whitened σ_t²_ot (t). Whitening here includes removing the
mean. Separation by (12) is accelerated significantly because the differences be-
tween the envelopes of source estimates are actively emphasised.

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