Examples of Noise Reduction Using Gabor Regression

In all of the experiments described here, a tight, normalised Hanning window was employed as the Gabor window function:

A regular time-frequency lattice was constructed to yield a redundancy of two (corresponding to the common practice of a 50% window overlap in the overlap-add method.) The arithmetic mean of the signal reconstructions from 1000 iterations (following 1000 iterations of ``burn-in'', by which time the sampler appeared to have reached a stationary regime in each case) was taken to be the final result. Complete details can be found in the associated paper accepted for NIPS 2002, available here.

A zipfile containing the full data from Gabor experiments in Matlab 6.1 format (.mat) is available here, and from the Wilson experiments here. Comparisons with short-time spectral attenuation (STSA) techniques were made over a wide range of SNR, as detailed in the paper; a file containing these data is available here. The STSA experiments were performed using the Matlab STSA toolbox (Installation instructions and bug fixes may be found here.) Matlab code for the Gabor and Wilson experiments detailed below may be found here.

Examples using a redundancy-2 tight Gabor system

Clean Signal

Noisy Signal (SNR = 20 dB) Restoration (SNR gain = 2.10 dB)

Noisy Signal (SNR = 10 dB) Restoration (SNR gain = 4.07 dB)

Noisy Signal (SNR = 00 dB) Restoration (SNR gain = 5.91 dB)

In all cases, the noise variance has been treated as an unknown parameter, and hence estimated by the algorithm from the data. Plots detailing the estimated and true noise variances can be found in the NIPS Submission above.

Examples using the associated Wilson basis

Noisy Signal (SNR = 10 dB)

Restoration using diffuse priors (as in the above examples)

Restoration using frequency-dependent priors (scale factor inversely proportional to frequency)

Here, too, the noise variance is treated as unknown. Note that the latter restoration exhibits a significantly lower amount of residual "musical noise" while yielding almose the same SNR gain (3.07 vs. 2.85 dB). Spectrograms of these signals, along with the residuals, are shown below:

The below diagram demonstrates how the estimation depends on the overall prior scale factor, in the case of it being set inversely proportional to frequency. Here, the scale factor increases from left to right and top to bottom:

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